24411:
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5727:
365:
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1617:
11574:
718:
72:
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1414:
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251:
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360:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
15862:{\displaystyle {\begin{aligned}\left\{\log w^{z}\right\}&=\left\{z\cdot \operatorname {Log} w+z\cdot 2\pi in+2\pi im\mid m,n\in \mathbb {Z} \right\}\\\left\{z\log w\right\}&=\left\{z\operatorname {Log} w+z\cdot 2\pi in\mid n\in \mathbb {Z} \right\}\end{aligned}}}
1612:{\displaystyle {\begin{aligned}b^{n+m}&=\underbrace {b\times \dots \times b} _{n+m{\text{ times}}}\\&=\underbrace {b\times \dots \times b} _{n{\text{ times}}}\times \underbrace {b\times \dots \times b} _{m{\text{ times}}}\\&=b^{n}\times b^{m}\end{aligned}}}
523:
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Primum ergo considerandæ sunt quantitates exponentiales, seu
Potestates, quarum Exponens ipse est quantitas variabilis. Perspicuum enim est hujusmodi quantitates ad Functiones algebraicas referri non posse, cum in his Exponentes non nisi constantes locum
15517:
8469:
713:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
11280:
408:
1265:
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Archimedes. (2009). THE SAND-RECKONER. In T. Heath (Ed.), The Works of
Archimedes: Edited in Modern Notation with Introductory Chapters (Cambridge Library Collection - Mathematics, pp. 229-232). Cambridge: Cambridge University Press.
19544:. Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the
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802:
725:
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10875:
19539:
These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of
9144:
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17071:
4202:
17832:{\displaystyle {\begin{aligned}x^{0}&=1\\x^{m+n}&=x^{m}x^{n}\\(x^{m})^{n}&=x^{mn}\\(xy)^{n}&=x^{n}y^{n}\quad {\text{if }}xy=yx,{\text{and, in particular, if the multiplication is commutative.}}\end{aligned}}}
18618:
are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely
14231:
1035:
7650:
8223:
On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values. One may choose one of these values, called the
2005:
949:
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11030:
8997:
and of Euler's number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has
14716:
9480:{\displaystyle \exp(x)\exp(y)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}\left(1+{\frac {y}{n}}\right)^{n}=\lim _{n\rightarrow \infty }\left(1+{\frac {x+y}{n}}+{\frac {xy}{n^{2}}}\right)^{n},}
1419:
15215:{\displaystyle {\begin{aligned}(-2)^{3+4i}&=2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2+3(\pi +2k\pi ))+i\sin(4\ln 2+3(\pi +2k\pi )))\\&=-2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2)+i\sin(4\ln 2)).\end{aligned}}}
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Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or
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Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for
26013:(1987) . "2.4.1.1. Definition arithmetischer Ausdrücke" [Definition of arithmetic expressions]. Written at Leipzig, Germany. In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.).
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In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of
590:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
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8287:
621:
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1621:
In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that
23068:), but many reasonably efficient heuristic algorithms are available. However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.
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974:
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When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if
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steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using
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7645:
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2795:
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1852:
25970:(1994). "The infinite in Leibniz's mathematics – The historiographical method of comprehension in context". In Kostas Gavroglu; Jean Christianidis; Efthymios Nicolaidis (eds.).
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shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to
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20452:
19841:
18302:
13230:
475:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
22105:
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8926:
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11148:
9047:
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9684:
2988:—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"—and كَعْبَة (
2683:
23671:
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3189:
Some mathematicians (such as
Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write
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16450:
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13140:
7907:
On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real
6507:
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6004:
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19556:, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the
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15283:
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2182:
1791:
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Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined
14113:
13816:
13457:
13330:
12477:
11173:
The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an
10918:
7499:
6214:
4101:
863:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}
19405:
15521:
Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:
14441:
13166:
11923:
11877:
6455:
5411:
3895:
778:{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}
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26846:
22181:. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at
18978:
11897:
11168:
8192:
For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (
7507:
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is an integer, the identities are valid for all nonzero complex numbers. If exponentiation is considered as a multivalued function then the possible values of
10139:. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.
9820:. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.
11674:
10707:
13557:
9062:
27318:
27185:
11553:
4371:{\displaystyle {\begin{aligned}b^{m+n}&=b^{m}\cdot b^{n}\\\left(b^{m}\right)^{n}&=b^{m\cdot n}\\(b\cdot c)^{n}&=b^{n}\cdot c^{n}\end{aligned}}}
14627:
24518:
16180:{\displaystyle \left({\frac {1}{-1}}\right)^{\frac {1}{2}}=(-1)^{\frac {1}{2}}=i\neq {\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i}
19586:
is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is
2043:
The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what
14560:
12147:
12116:
7828:{\displaystyle (x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.}
992:
1087:
223:
26059:
1893:
914:
15524:
17129:
9053:
10957:
26023:] (in German). Vol. 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun, Switzerland / Frankfurt am Main, Germany:
23597:
For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the
14748:
26842:
26781:
26332:
18987:, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of
14410:{\displaystyle z^{w}=\rho ^{c}e^{-d(\theta +2k\pi )}\left(\cos(d\ln \rho +c\theta +2ck\pi )+i\sin(d\ln \rho +c\theta +2ck\pi )\right),}
23498:
When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example
23050:. For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal
26988:
26655:
16335:
2999:
27311:
27178:
18082:
26393:
22702:
multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute
17960:
16592:
26053:
25646:
25544:
10581:
10010:
8549:
8514:
of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain
21494:. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example,
24511:
22716:
19861:
18583:
16242:
17385:
consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by
12949:
12406:
11441:
th root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal
5444:. This is because a negative number multiplied by another negative number cancels the sign, and thus gives a positive number.
26705:
26449:
26346:
26300:
26271:
26234:
26147:
26075:
25951:
25924:
25481:
23961:
or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the
3254:
Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not
23770:
which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.
22957:
22284:
21932:
21369:
16457:
23803:
5669:
as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
28118:
27304:
27171:
26307:
24475:
18574:. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the
13685:
11786:
2076:
1080:
216:
10395:
2804:
exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including
28113:
27096:
27060:
26954:
26338:
4761:{\displaystyle (a+b)^{n}=\sum _{i=0}^{n}{\binom {n}{i}}a^{i}b^{n-i}=\sum _{i=0}^{n}{\frac {n!}{i!(n-i)!}}a^{i}b^{n-i}.}
4423:
3019:
20335:{\displaystyle {\begin{aligned}F\colon {}&\mathbb {F} _{q}\to \mathbb {F} _{q}\\&x\mapsto x^{p}\end{aligned}}}
486:
28128:
26833:
26772:
26209:
26173:
26036:
26028:
25987:
25516:
25341:
is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing.
24504:
21306:
8162:{\displaystyle \left((-1)^{2}\right)^{\frac {1}{2}}=1^{\frac {1}{2}}=1\neq (-1)^{2\cdot {\frac {1}{2}}}=(-1)^{1}=-1.}
2450:
1697:
1041:
17:
28108:
26516:
23886:
21870:
19057:
4494:
28821:
28401:
26129:
26010:
24548:
21532:
19811:
16532:
11628:
8234:
602:
23501:
23358:, the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration
23350:
When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the
21140:
16736:
26643:
26591:
25539:
25469:
24910:
23958:
21497:
20515:
18578:
of the group). The possible orders of group elements are important in the study of the structure of a group (see
13635:
874:
23216:
equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the
13751:
11402:
955:
28123:
26736:
24306:
24269:
24163:
24119:
24037:
17552:
16230:
13494:. Although this can be described by a single formula, it is clearer to split the computation in several steps.
4809:
involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose
3941:
3615:
1073:
371:
209:
23432:
23362:
the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication
21039:" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example
9532:
9196:
5019:
followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example,
4018:
3812:
28907:
27137:
26981:
26697:
26139:
25692:
25473:
24316:
24167:
24135:
23051:
22032:
17364:
The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any
17211:
15512:{\displaystyle \log((-i)^{2})=\log(-1)=i\pi \neq 2\log(-i)=2\log(e^{-i\pi /2})=2\,{\frac {-i\pi }{2}}=-i\pi }
12868:
denotes one of the values of the multivalued logarithm (typically its principal value), the other values are
9875:
8464:{\displaystyle b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} ),}
4015:
This definition of exponentiation with negative exponents is the only one that allows extending the identity
3992:
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (
23552:
19773:
19594:. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of
11984:
10476:
28573:
28223:
27892:
27685:
27152:
27147:
24960:
23136:
22178:
21152:
20948:
20122:
19244:
18988:
10924:
is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to
3912:
but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.
3546:
23676:
20522:. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the
18678:
13384:
consists of several sheets that define each a holomorphic function in the neighborhood of every point. If
28749:
28608:
28439:
28253:
28243:
27897:
27877:
26090:
26063:
26006:
24425:
24352:
24326:
24143:
22220:
0. The limits in these examples exist, but have different values, showing that the two-variable function
21296:. This exponential notation is justified by the following canonical isomorphisms (for the first one, see
20074:
12393:
12029:
11084:
10276:
10175:
9948:
9493:
7609:
3927:
Exponentiation with negative exponents is defined by the following identity, which holds for any integer
3683:
2755:
2510:
1796:
1409:
all multiplied by each other, several other properties of exponentiation directly follow. In particular:
28578:
18622:
17304:
16678:
15593:
This identity does not hold even when considering log as a multivalued function. The possible values of
14123:
12568:
12166:
10338:
7965:
7851:
2366:
The properties of fractional exponents also follow from the same rule. For example, suppose we consider
29245:
28698:
28321:
28163:
28078:
27887:
27869:
27763:
27753:
27743:
27579:
27219:
27194:
26608:, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall: 8–26 .
26067:
25508:
24460:
24209:
24181:
24071:
24041:
23369:
23012:
20590:
20533:
20352:
20227:
20083:
20024:
19957:
19700:
14179:
11604:
7930:
3116:
2840:
2720:
2413:
28603:
21212:
21042:
20477:
20428:
19817:
19321:
18262:
13202:
11275:{\displaystyle \left(\rho e^{i\theta }\right)^{\frac {1}{n}}={\sqrt{\rho }}\,e^{\frac {i\theta }{n}}.}
29250:
28826:
28371:
27992:
27778:
27773:
27768:
27758:
27735:
26014:
25861:
25641:
24635:
24491:
22391:
22213:
22075:
21762:
19119:
12871:
8895:
8298:. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a
5536:
5417:
3915:
31:
28583:
23726:
22114:
21566:
the vector space of those sequences that have a finite number of nonzero elements. The latter has a
18501:
17440:
16893:
14842:
14487:
12649:
11121:
9004:
8931:
7430:
2315:
2187:
28248:
28158:
27811:
26974:
26536:
25651:
25549:
24553:
24528:
24288:
24199:
24147:
24013:
23351:
21097:
19989:
17871:
17076:
12498:
12350:
11457:
th root for positive real radicands. For negative real radicands, and odd exponents, the principal
10568:
9697:
9690:
is an integer (this results from the repeated-multiplication definition of the exponentiation). If
9644:
5500:
of powers of a number greater than one diverges; in other words, the sequence grows without bound:
3235:
2642:
26674:
26201:
25696:
23637:
22236:
21071:
18459:
12254:
28937:
28902:
28688:
28598:
28472:
28447:
28356:
28346:
28068:
27958:
27940:
27860:
26907:
26882:
26605:
25974:. Boston Studies in the Philosophy of Science. Vol. 151. Springer Netherlands. p. 276.
25866:
25577:
24885:
24045:
24033:
23598:
23301:
21718:
18837:
17923:
17375:
16414:
14792:
13105:
8870:
8500:
6462:
6289:
5964:
5757:
4870:
4810:
4178:
4119:
3803:
3135:. Early in the 17th century, the first form of our modern exponential notation was introduced by
2848:
2369:
1109:
23604:
22042:
21829:
20816:
20770:
19549:
18181:
10096:
9152:
7344:
6114:
2238:
29197:
28467:
28341:
27972:
27748:
27528:
27455:
27106:
27070:
26531:
26441:
25312:
25265:
24685:
24560:
24543:
24465:
24245:
24217:
23997:
23880:
23104:
23090:
21567:
21206:
20386:
19591:
18307:
18223:
17879:
17859:
17530:
11474:
9739:
results from the definitions given in preceding sections, by using the exponential identity if
9606:
8792:
8786:
8517:
6332:
6007:
4854:
4185:
3487:
2556:
1260:{\displaystyle b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.}
628:
26251:
25941:
25498:
25318:
23265:
23018:
21666:
21592:
20863:
20392:
20056:
19636:
17403:
17115:, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity
16831:
13862:
13354:
11331:
8965:
7388:
6378:
6053:
5342:
3761:
29240:
29161:
28801:
28452:
28306:
28233:
27388:
27285:
27142:
26872:
25914:
25874:
24585:
24533:
24480:
24189:
23043:
21454:
21434:
20723:
20601:
is large, while there is no known computationally practical algorithm that allows retrieving
20194:
19737:
19557:
18811:
18751:
18355:
17365:
17279:
15225:
13938:
13825:
13246:
13171:
12495:
is real and positive, the principal value of the complex logarithm is the natural logarithm:
12055:
11484:
If the complex number is moved around zero by increasing its argument, after an increment of
11466:
11311:
11101:
11066:
9757:
8962:, this definition cannot be used here. So, a definition of the exponential function, denoted
7309:
5852:
5802:
4406:
4140:
3995:
3503:
3003:
2282:
2010:
1860:
1651:
1179:
1118:
599:
202:
26647:
26193:
26052:
Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010).
25775:
25728:
22185:, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and
20697:
19529:{\displaystyle \left({\frac {d}{dx}}\right)^{n}f(x)={\frac {d^{n}}{dx^{n}}}f(x)=f^{(n)}(x).}
19177:
19147:
18399:
14534:
13895:
13510:
12845:
12781:
12732:
12620:
12320:
12221:
11487:
9789:
6521:
6263:
5918:
3470:
The exponentiation operation with integer exponents may be defined directly from elementary
3389:, that exponent indicates how many copies of the base are multiplied together. For example,
2046:
29094:
28988:
28952:
28693:
28416:
28396:
28213:
27882:
27670:
26841:(Preliminary report). New York, USA: Programming Research Group, Applied Science Division,
26085:
26024:
25573:
25089:
24985:
24835:
24538:
24021:
23954:
23230:
23098:
23082:
21802:
21627:
21491:
21272:
21011:
20910:
20665:
20608:
20565:
20519:
20463:
19213:
19024:
18884:
18798:
18483:
18372:
18351:
17951:
17896:
17867:
17501:
17249:
16866:
16804:
16226:
16025:{\displaystyle (-1\cdot -1)^{\frac {1}{2}}=1\neq (-1)^{\frac {1}{2}}(-1)^{\frac {1}{2}}=-1}
15261:
14721:
14455:
13462:
13395:
13275:
13058:
12919:
12836:
12807:
12705:
12541:
12298:
12245:
12150:
11515:
11288:
11038:
10543:
10383:
is not a real number. If a meaning is given to the exponentiation of a complex number (see
10236:
8883:
8824:
8299:
8203:
5222:
4786:
3471:
2907:
2805:
2688:
2615:
2160:
1769:
1624:
1345:
28173:
27642:
26367:
19292:
19261:
14092:
13801:
13433:
13306:
12453:
10892:
7596:{\displaystyle x^{\frac {p}{q}}=\left(x^{p}\right)^{\frac {1}{q}}=(x^{\frac {1}{q}})^{p}.}
6190:
4077:
8:
28816:
28680:
28675:
28643:
28406:
28381:
28376:
28351:
28281:
28277:
28208:
28098:
27930:
27726:
27695:
26398:
26194:
26125:
26101:
25893:
25638:
25567:
25536:
25351:
25169:
24935:
24470:
24277:
24009:
23355:
20113:
19583:
19565:
19561:
18981:
18907:
18794:
17851:
17843:
17382:
17235:
14420:
13381:
13145:
12480:
11902:
11856:
11446:
8857:
8352:
8340:
7469:
6434:
5497:
5388:
5049:
4996:
4782:
4402:
4115:
3874:
3799:
3537:
3050:
26667:
25749:
25686:
25666:
25616:
25291:
23776:
11764:{\displaystyle 1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},\omega ^{n-1}.}
10870:{\displaystyle b^{x+iy}=b^{x}b^{iy}=b^{x}e^{iy\ln b}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)).}
3061:
would use Roman numerals for exponents in a way similar to that of
Chuquet, for example
29219:
28973:
28968:
28882:
28856:
28754:
28733:
28505:
28386:
28336:
28258:
28228:
28168:
27935:
27915:
27846:
27559:
27132:
27011:
26829:
26825:
26774:
The FORTRAN Automatic Coding System for the IBM 704 EDPM: Programmer's
Reference Manual
26752:
26625:
26617:
26434:
26121:
26105:
25144:
24989:
24660:
24440:
24416:
23065:
22422:
22217:
22174:
21128:
20523:
20382:
18928:
18876:
18745:
17847:
12778:
In some contexts, there is a problem with the discontinuity of the principal values of
11882:
11608:
11153:
10467:
8959:
6414:
6358:
6338:
6243:
6223:
6170:
6089:
6033:
6013:
5944:
5898:
5878:
5828:
5703:
4955:(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)
4131:
4107:
3255:
2871:
2595:
2490:
2393:
1677:
1392:
1372:
1325:
28103:
26654:. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 .
26117:
25855:
10163:
is defined by means of the exponential function with complex argument (see the end of
9139:{\displaystyle \exp(x)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}.}
29215:
29113:
29058:
28912:
28887:
28861:
28316:
28311:
28238:
28218:
28203:
27925:
27907:
27826:
27816:
27801:
27564:
26950:
26921:
26732:
26701:
26629:
26445:
26389:
26364:
26342:
26296:
26287:
Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001).
26267:
26230:
26205:
26169:
26143:
26135:
26071:
26032:
25983:
25967:
25947:
25920:
25770:
25734:
25512:
25477:
25410:
25385:
25064:
24435:
24430:
24410:
24083:
24058:
character set did not include lowercase characters or punctuation symbols other than
23077:
23058:
sequence of multiplications (the minimal-length addition chain for the exponent) for
22826:
Then compute the following terms in order, reading Horner's rule from right to left.
21526:
21124:
20649:
20645:
17863:
17257:
14082:{\displaystyle w\log z=(c\ln \rho -d\theta -2dk\pi )+i(d\ln \rho +c\theta +2ck\pi ),}
13819:
13736:
13099:
12314:
12157:
10929:
10260:
9849:
8955:
8511:
8336:
8310:
5068:
4858:
3141:
2936:
2929:
2899:
2836:
2832:
28638:
25765:
23981:
in 1967, so the caret became usual in programming languages. The notations include:
5261:), and the negative exponents are determined by the rank on the right of the point.
4560:
The powers of a sum can normally be computed from the powers of the summands by the
3136:
2073:
should mean. In order to respect the "exponents add" rule, it must be the case that
29149:
28942:
28528:
28500:
28490:
28482:
28366:
28331:
28326:
28293:
27987:
27950:
27841:
27836:
27831:
27821:
27793:
27680:
27627:
27584:
27523:
27051:
26756:
26609:
26541:
26259:
26109:
25994:
A positive power of zero is infinitely small, a negative power of zero is infinite.
25975:
25592:
25563:
25457:
25119:
24005:
23797:
23086:
23064:
is a difficult problem, for which no efficient algorithms are currently known (see
22707:
22403:
21768:
18790:
18359:
17221:
15334:
13240:
11846:
10928:
th roots, this case deserves to be considered first, since it does not need to use
9860:
5192:
4802:
4561:
3386:
3282:
3100:
2985:
2828:
1113:
27632:
26652:
A Collection of
Examples of the Applications of the Calculus of Finite Differences
25503:. Synthese Historical Library. Vol. 17. Translated by A.M. Ungar. Dordrecht:
17066:{\displaystyle \exp \left((1+2\pi in)\log \exp(1+2\pi in)\right)=\exp(1+2\pi in).}
11521:
3904:
is controversial. In contexts where only integer powers are considered, the value
3424:
can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation
29125:
29014:
28947:
28873:
28796:
28770:
28588:
28301:
28093:
28063:
28053:
28048:
27714:
27622:
27569:
27413:
27353:
27116:
26691:
26314:
26164:
Hass, Joel R.; Heil, Christopher E.; Weir, Maurice D.; Thomas, George B. (2018).
26081:
25979:
25612:
25596:
25039:
24185:
24017:
21579:
19607:
19251:
18923:
18880:
18590:
13745:
13087:
12310:
12119:
12049:
11971:
8225:
7455:
3250:
introduced variable exponents, and, implicitly, non-integer exponents by writing:
3035:
2801:
26263:
12244:
is the variant of the complex logarithm that is used, which is, a function or a
3082:
2956:
to estimate the number of grains of sand that can be contained in the universe.
29130:
28998:
28983:
28847:
28811:
28786:
28662:
28633:
28618:
28495:
28391:
28361:
28088:
28043:
27920:
27518:
27513:
27508:
27480:
27465:
27378:
27363:
27328:
27021:
26997:
26780:. New York, USA: Applied Science Division and Programming Research Department,
26687:
26573:
25889:
25724:
25682:
25259:
25014:
24445:
22170:
22160:
22036:
21578:
of the former cannot be explicitly described (because their existence involves
21472:
21036:
19599:
18579:
18454:
11392:
11056:
10694:
10156:
9747:
8217:
6161:
6050:. All graphs from the family of even power functions have the general shape of
5551:
5064:
3355:
3247:
3092:
3054:
2995:
2883:
1169:
511:
27163:
26396:[Problems and propositions, the former to solve, the later to prove].
24236:
and thus not as useful. In some languages, it is left-associative, notably in
11603:
is a positive integer. They arise in various areas of mathematics, such as in
6375:. All graphs from the family of odd power functions have the general shape of
1030:{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
29234:
29053:
29037:
28978:
28932:
28628:
28613:
28523:
27806:
27675:
27637:
27594:
27475:
27460:
27450:
27408:
27398:
27373:
27296:
26925:
26917:
25898:(in Latin). Vol. I. Lausanne: Marc-Michel Bousquet. pp. 69, 98–99.
24107:
22169:. Iterating tetration leads to another operation, and so on, a concept named
22018:
21707:
21483:
20761:
20518:
is an application of exponentiation in finite fields that is widely used for
19553:
19545:
18919:
17890:
17855:
15612:
11967:
11568:
5210:
5107:
are also used to describe small or large quantities. For example, the prefix
5086:
4798:
4130:
of a given dimension). In particular, in such a structure, the inverse of an
4127:
3795:
3491:
2891:
26835:
Specifications for: The IBM Mathematical FORmula TRANSlating System, FORTRAN
26729:
Unicode
Demystified: A Practical Programmer's Guide to the Encoding Standard
17842:
These definitions are widely used in many areas of mathematics, notably for
3058:
29089:
29078:
28993:
28831:
28806:
28723:
28623:
28593:
28568:
28552:
28457:
28424:
28147:
28058:
27997:
27574:
27470:
27403:
27383:
27358:
26613:
26545:
26328:
25440:
25413:
25288:
is most commonly used for explicit numbers and at a very elementary level;
25093:
24216:
In most programming languages with an infix exponentiation operator, it is
24091:
21487:
21120:
20660:
20504:
20471:
19626:
19611:
19577:
19541:
19141:
19018:
18872:
18571:
18479:
18438:
17875:
12129:
12098:
10698:
6512:
5238:
5134:
3601:
The associativity of multiplication implies that for any positive integers
3483:
3219:
3096:
2977:
2964:
21478:
One can use sets as exponents for other operations on sets, typically for
18433:. A group (or subgroup) that consists of all powers of a specific element
12770:
is an integer, this principal value is the same as the one defined above.
3482:
The definition of the exponentiation as an iterated multiplication can be
29048:
28923:
28728:
28192:
28083:
28038:
28033:
27783:
27690:
27589:
27418:
27393:
27368:
27249:
27244:
27229:
27101:
27065:
26768:
26764:
26759:; Herrick, Harlan L.; Hughes, R. A.; Mitchell, L. B.; Nelson, Robert A.;
26549:
24348:
24344:
24259:
21575:
21570:
consisting of the sequences with exactly one nonzero element that equals
20764:
20754:
20750:
19693:, which allows, in general, working as if there were only one field with
19630:
19603:
19587:
17073:
Therefore, when expanding the exponent, one has implicitly supposed that
12488:
11963:
7299:
5249:
that appears in the sum; the exponent is determined by the place of this
5044:
5002:
4390:
4382:
2000:{\displaystyle (b^{1})^{3}=b^{1}\times b^{1}\times b^{1}=b^{1+1+1}=b^{3}}
1322:
Starting from the basic fact stated above that, for any positive integer
1270:
1101:
944:{\displaystyle \scriptstyle {\sqrt{\scriptstyle {\text{radicand}}}}\,=\,}
396:
22948:
In general, the number of multiplication operations required to compute
18482:(it has a finite number of elements), and its number of elements is the
15285:
have the same argument. More generally, this is true if and only if the
5656:
alternates between even and odd, and thus does not tend to any limit as
3420:
The word "raised" is usually omitted, and sometimes "power" as well, so
29185:
29166:
28462:
28073:
27280:
27275:
27270:
27239:
27234:
27224:
27080:
26113:
25462:
25194:
24336:
24111:
21479:
21035:
is naturally endowed with a similar structure. In this case, the term "
20346:
19690:
19622:
19255:
19237:
is the transition matrix between the state now and the state at a time
18915:
18805:
18450:
13499:
12943:
is one value of the exponentiation, then the other values are given by
10948:
10537:
9743:
is rational, and the continuity of the exponential function otherwise.
8228:, but there is no choice of the principal value for which the identity
7916:
6217:
5163:
5151:
4963:(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
3190:
3099:), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and
2941:
26828:; Herrick, Harlan L.; Nelson, Robert A.; Ziller, Irving (1954-11-10).
26812:
Introduction to
Digital Computing and FORTRAN IV with MTS Applications
26621:
16934:
which is a multi-valued function. Thus the notation is ambiguous when
13625:{\displaystyle z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),}
11025:{\displaystyle z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),}
5650:
alternates between larger and larger positive and negative numbers as
5628:
alternates between even and odd, and thus do not tend to any limit as
4771:
However, this formula is true only if the summands commute (i.e. that
4196:, hold for all integer exponents, provided that the base is non-zero:
2800:
The definition of exponentiation can be extended to allow any real or
28791:
28718:
28710:
28515:
28429:
27547:
27111:
27036:
27031:
26877:
26372:
26292:
26258:. Texts and Readings in Mathematics. Vol. 37. pp. 126–154.
26134:(in German). Vol. I (1 ed.). Berlin / Heidelberg, Germany:
25504:
25418:
24860:
24785:
24610:
24455:
24450:
24079:
23978:
22166:
22156:
21652:
18786:
15286:
9845:
8199:
5721:
5173:
5100:
4806:
3038:
used a form of exponential notation in the 15th century, for example
2820:
2812:
980:
25710:
22945:
This series of steps only requires 8 multiplications instead of 99.
17602:
Exponentiation with integer exponents obeys the following laws, for
17272:
are algebraic, Gelfond–Schneider theorem asserts that all values of
16225:, but assuming its truth for complex numbers leads to the following
14711:{\displaystyle i^{i}=e^{i\log i}=e^{-{\frac {\pi }{2}}}e^{-2k\pi }.}
9746:
The limit that defines the exponential function converges for every
8785:
So, the upper bounds and the lower bounds of the intervals form two
4821:) for exponentiation with non-commuting bases, which is then called
28892:
27075:
27041:
27016:
26912:
26760:
26648:"Part III. Section I. Examples of the Direct Method of Differences"
25852:
The most recent usage in this sense cited by the OED is from 1806 (
25219:
24735:
24099:
24029:
23094:
22230:. One may consider at what points this function does have a limit.
21772:
21297:
18430:
11481:
th root to the complex plane without the nonpositive real numbers.
11450:
11449:
in the whole complex plane, except for negative real values of the
7371:
6107:
5473:
5145:
899:
722:
239:
26966:
26767:; Sheridan, Peter B.; Stern, Harold; Ziller, Irving (1956-10-15).
26394:"Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen"
26187:
26185:
23970:
4801:
of the same size, this formula cannot be used. It follows that in
28897:
28556:
28550:
24103:
24095:
24075:
24055:
16944:. Here, before expanding the exponent, the second line should be
11150:
is also an argument of the same complex number for every integer
7244:
6431:
increases and losing all flatness there in the straight line for
5846:
5740:
5726:
5702:
Other limits, in particular those of expressions that take on an
5008:
4389:. Also unlike addition and multiplication, exponentiation is not
2824:
2816:
1165:
26362:
23220:
th power of the function under composition, commonly called the
13344:
is an integer, there is only one value that agrees with that of
12450:
The principal value of the complex logarithm is not defined for
11775:
th roots of unity that have this generating property are called
11559:
th root function that is continuous in the whole complex plane.
27265:
26182:
24241:
24001:
22684:
21699:
19174:
is the state of the system after two time steps, and so forth:
17920:
denotes the exponentiation with respect of multiplication, and
17390:
16404:{\displaystyle \left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad }
12691:
is holomorphic except in the neighbourhood of the points where
11510:
the complex number comes back to its initial position, and its
8212:, below). The result is always a positive real number, and the
5177:
4111:
3018:(k), respectively, by the 15th century, as seen in the work of
2903:
649:
27612:
26100:
25913:
Hodge, Jonathan K.; Schlicker, Steven; Sundstorm, Ted (2014).
14617:{\displaystyle \log i=i\left({\frac {\pi }{2}}+2k\pi \right).}
10142:
8343:
of rational numbers, exponentiation of a positive real number
8294:
7848:
All these definitions are required for extending the identity
3014:
3008:
2390:
and ask if there is some suitable exponent, which we may call
26809:
25711:"Earliest Known Uses of Some of the Words of Mathematics (E)"
24292:
24276:
Still others only provide exponentiation as part of standard
24237:
24151:
24115:
24087:
23993:
23962:
20943:
20758:
18168:{\displaystyle (f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots )).}
13728:
11088:
5108:
4878:
3327:
squared" are so ingrained by tradition and convenience that "
2990:
2864:
2811:
Exponentiation is used extensively in many fields, including
18066:{\displaystyle (f^{n})(x)=(f(x))^{n}=f(x)\,f(x)\cdots f(x),}
16668:{\displaystyle e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad }
15296:
12835:. In this case, it is useful to consider these functions as
3241:
3234:, but had declined in usage and should not be confused with
2553:. Using the fact that multiplying makes exponents add gives
26594:(1813) . "On a Remarkable Application of Cotes's Theorem".
24171:
24159:
24139:
24131:
24123:
21090:
denotes the real numbers) denotes the
Cartesian product of
13822:. The other values of the logarithm are obtained by adding
11573:
11181:
th root of the absolute value and dividing its argument by
10683:{\displaystyle b^{x+iy}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)),}
10083:{\displaystyle b^{x}=\left(e^{\ln b}\right)^{x}=e^{x\ln b}}
9754:, and therefore it can be used to extend the definition of
8775:{\displaystyle \left,\left,\left,\left,\left,\left,\ldots }
5476:. For a similar discussion of powers of the complex number
5214:
3091:
was coined in 1544 by
Michael Stifel. In the 16th century,
2844:
808:
771:
646:
529:
414:
257:
26286:
24355:, exponentiation is performed via a multitude of methods:
11177:
th root of a complex number can be obtained by taking the
10385:
8180:
4893:-letter alphabet). Some examples for particular values of
4381:
Unlike addition and multiplication, exponentiation is not
1890:
can similarly be derived from the same rule. For example,
190:
equals the base because any number raised to the power of
26824:
26597:
Philosophical Transactions of the Royal Society of London
24155:
24127:
24025:
23989:
22815:{\displaystyle 100=2^{2}+2^{5}+2^{6}=2^{2}(1+2^{3}(1+2))}
19947:{\displaystyle \{g^{1}=g,g^{2},\ldots ,g^{p-1}=g^{0}=1\}}
17822:
and, in particular, if the multiplication is commutative.
16324:{\displaystyle e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e}
15916:
is a real number. But, for the principal values, one has
6544:, the opposite asymptotic behavior is true in each case.
5206:
71:
26751:
26051:
23097:
of the function written on the right is included in the
22648:, including negative ones. This may make the definition
19140:
of the system. This is the standard interpretation of a
17234:
is an algebraic number. This results from the theory of
15584:{\displaystyle \log w^{z}\equiv z\log w{\pmod {2\pi i}}}
13034:{\displaystyle e^{w(2ik\pi +\log z)}=z^{w}e^{2ik\pi w},}
12440:{\displaystyle -\pi <\operatorname {Arg} z\leq \pi .}
10880:
3333:
to the second power" tends to sound unusual or clumsy.)
3297:
squared", because the area of a square with side-length
26464:
Chapter 1, Elementary Linear Algebra, 8E, Howard Anton.
25912:
22406:), which will contain the points at which the function
17200:
which is a true identity between multivalued functions.
17193:{\displaystyle \left(e^{x}\right)^{y}=e^{y\log e^{x}},}
16801:
is nonzero. The error is the following: by definition,
15639:
as any integers the possible values of both sides are:
9054:
many equivalent ways to define the exponential function
3747:
As mentioned earlier, a (nonzero) number raised to the
2447:. From the definition of the square root, we have that
23004:{\displaystyle \sharp n+\lfloor \log _{2}n\rfloor -1,}
22345:{\displaystyle D=\{(x,y)\in \mathbf {R} ^{2}:x>0\}}
18429:
The set of all powers of an element of a group form a
17242:
is any algebraic number, in which case, all values of
12122:
is defined, which is not continuous for the values of
10341:
8184:
for details on the way these problems may be handled.
7472:
3370:
cubed", because the volume of a cube with side-length
2731:
1045:
996:
959:
921:
918:
878:
832:
814:
811:
806:
760:
749:
738:
735:
729:
686:
679:
676:
662:
655:
652:
644:
606:
556:
535:
532:
527:
490:
441:
420:
417:
412:
375:
326:
305:
284:
263:
260:
255:
28276:
25441:"Exponent | Etymology of exponent by etymonline"
25321:
25294:
25268:
24251:
Other programming languages use functional notation:
23889:
23806:
23779:
23729:
23679:
23640:
23607:
23555:
23504:
23435:
23372:
23304:
23268:
23233:
23139:
23107:
23021:
22960:
22719:
22287:
22239:
22216:
gives a number of examples of limits that are of the
22117:
22078:
22045:
22000:{\displaystyle \hom(U,S^{T})\cong \hom(T\times U,S).}
21935:
21873:
21832:
21805:
21669:
21630:
21595:
21535:
21500:
21457:
21437:
21421:{\displaystyle S^{T\sqcup U}\cong S^{T}\times S^{U},}
21372:
21309:
21275:
21215:
21155:
21100:
21074:
21045:
21014:
20951:
20913:
20866:
20819:
20773:
20726:
20700:
20668:
20611:
20568:
20536:
20480:
20431:
20395:
20355:
20265:
20230:
20197:
20125:
20086:
20059:
20027:
19992:
19960:
19864:
19820:
19776:
19740:
19703:
19639:
19408:
19324:
19295:
19264:
19216:
19180:
19150:
19060:
19027:
18931:
18840:
18814:
18754:
18681:
18625:
18504:
18462:
18402:
18375:
18310:
18265:
18226:
18184:
18085:
17963:
17926:
17899:
17627:
17555:
17504:
17443:
17406:
17307:
17132:
17079:
16950:
16896:
16869:
16834:
16807:
16739:
16681:
16595:
16535:
16522:{\displaystyle e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad }
16460:
16417:
16338:
16245:
16038:
15922:
15646:
15527:
15344:
15264:
15228:
14890:
14845:
14795:
14781:{\displaystyle e^{-{\frac {\pi }{2}}}\approx 0.2079.}
14751:
14724:
14630:
14563:
14537:
14490:
14458:
14423:
14234:
14182:
14126:
14095:
13970:
13941:
13898:
13865:
13828:
13804:
13754:
13688:
13638:
13560:
13513:
13465:
13436:
13398:
13357:
13309:
13278:
13249:
13205:
13174:
13148:
13108:
13102:
of the exponential function, more specifically, that
13061:
12952:
12922:
12874:
12848:
12810:
12784:
12735:
12708:
12652:
12623:
12571:
12544:
12501:
12456:
12409:
12353:
12323:
12257:
12224:
12169:
12058:
12048:, although this terminology can be confused with the
11987:
11905:
11885:
11859:
11789:
11677:
11631:
11490:
11405:
11334:
11314:
11291:
11194:
11156:
11124:
11104:
11069:
11041:
10960:
10895:
10710:
10584:
10546:
10479:
10398:
10279:
10239:
10178:
10099:
10013:
9951:
9878:
9792:
9760:
9700:
9647:
9609:
9535:
9496:
9273:
9199:
9155:
9065:
9007:
8968:
8934:
8898:
8827:
8795:
8552:
8520:
8364:
8237:
8027:
7968:
7933:
7854:
7653:
7612:
7510:
7433:
7391:
7347:
7312:
6524:
6465:
6437:
6417:
6381:
6361:
6341:
6292:
6266:
6246:
6226:
6193:
6173:
6117:
6092:
6056:
6036:
6030:, and also towards positive infinity with decreasing
6016:
5967:
5947:
5921:
5901:
5881:
5855:
5831:
5805:
5760:
5391:
5345:
4572:
4497:
4426:
4205:
4177:"Laws of Indices" redirects here. For the horse, see
4143:
4080:
4021:
3998:
3944:
3877:
3815:
3764:
3686:
3618:
3549:
3506:
3490:, and this definition can be used as soon one has an
3401:
times in the multiplication, because the exponent is
2758:
2723:
2691:
2645:
2618:
2598:
2559:
2513:
2493:
2453:
2416:
2396:
2372:
2318:
2285:
2241:
2190:
2163:
2079:
2049:
2013:
1896:
1863:
1799:
1772:
1700:
1680:
1654:
1627:
1417:
1395:
1375:
1348:
1328:
1192:
1044:
995:
958:
917:
877:
805:
728:
643:
605:
526:
489:
411:
374:
254:
28661:
26112:; Blath, Jochen; Schied, Alexander; Dempe, Stephan;
26005:
25853:
24406:
24198:: Haskell (for fractional base, integer exponents),
23957:
generally express exponentiation either as an infix
23872:{\displaystyle \sin ^{-1}x=\sin ^{-1}(x)=\arcsin x.}
22447:. Accordingly, this allows one to define the powers
22010:
This means the functor "exponentiation to the power
21008:
is endowed with some structure, it is frequent that
19610:, considered earlier in this article, which are all
15222:
In this case, all the values have the same argument
12317:
is the unique continuous function, commonly denoted
10151:
is a positive real number, exponentiation with base
7606:
The equality on the right may be derived by setting
6547:
5665:
If the exponentiated number varies while tending to
3107:
has been used to refer to the fourth power as well.
26192:Anton, Howard; Bivens, Irl; Davis, Stephen (2012).
25754:. Vol. 1. The Open Court Company. p. 204.
25671:. Vol. 1. The Open Court Company. p. 102.
23354:, which induces another exponentiation. When using
23101:of the function written on the left. It is denoted
20526:, is computationally expensive. More precisely, if
13719:{\displaystyle \theta =\operatorname {atan2} (a,b)}
11835:{\displaystyle \omega ^{k}=e^{\frac {2k\pi i}{n}},}
11328:, the complex number is not changed, but this adds
2150:{\displaystyle b^{-1}\times b^{1}=b^{-1+1}=b^{0}=1}
26433:
26387:
26163:
25854:
25461:
25333:
25303:
25280:
23939:
23871:
23788:
23762:
23715:
23665:
23626:
23589:
23541:
23490:
23421:
23340:
23290:
23246:
23190:
23122:
23030:
23003:
22814:
22344:
22273:
22173:. This sequence of operations is expressed by the
22135:
22099:
22064:
21999:
21918:
21859:
21818:
21690:
21643:
21616:
21558:
21517:
21463:
21443:
21420:
21357:
21288:
21242:
21193:
21111:
21082:
21060:
21027:
20989:
20926:
20893:
20852:
20806:
20741:
20712:
20686:
20624:
20581:
20554:
20495:
20446:
20417:
20373:
20334:
20248:
20216:
20176:
20104:
20065:
20045:
20013:
19978:
19946:
19835:
19800:
19759:
19721:
19661:
19528:
19379:
19310:
19281:
19229:
19196:
19166:
19107:
19040:
18972:
18859:
18826:
18773:
18748:, it may occur that some nonzero elements satisfy
18729:
18667:
18539:
18470:
18414:
18388:
18335:
18296:
18251:
18212:
18167:
18065:
17942:
17912:
17831:
17592:
17517:
17478:
17428:
17334:
17205:
17192:
17103:
17065:
16926:
16882:
16855:
16820:
16782:
16723:
16667:
16579:
16521:
16444:
16403:
16323:
16179:
16024:
15861:
15583:
15511:
15277:
15246:
15214:
14876:
14826:
14780:
14737:
14710:
14616:
14549:
14523:
14471:
14435:
14409:
14220:
14168:
14107:
14081:
13956:
13919:
13883:
13843:
13810:
13790:
13718:
13674:
13624:
13534:
13478:
13451:
13411:
13372:
13351:The multivalued exponentiation is holomorphic for
13333:
13324:
13291:
13264:
13224:
13189:
13160:
13134:
13074:
13033:
12935:
12904:
12860:
12823:
12796:
12758:
12721:
12683:
12635:
12609:
12557:
12528:
12471:
12439:
12381:
12332:
12282:
12236:
12207:
12141:
12110:
12079:
12016:
11917:
11891:
11871:
11834:
11763:
11663:
11555:). This shows that it is not possible to define a
11547:
11502:
11429:
11354:
11320:
11300:
11274:
11162:
11142:
11110:
11075:
11047:
11024:
10912:
10869:
10682:
10559:
10525:
10446:{\displaystyle \left(b^{z}\right)^{t}\neq b^{zt},}
10445:
10369:
10324:
10251:
10222:
10121:
10082:
9996:
9927:
9808:
9778:
9731:
9678:
9633:
9592:
9521:
9479:
9256:
9182:
9138:
9041:
8989:
8946:
8920:
8840:
8811:
8774:
8536:
8463:
8281:
8161:
8010:
7954:
7896:
7827:
7639:
7595:
7493:
7446:
7413:
7362:
7333:
6536:
6501:
6449:
6423:
6403:
6367:
6347:
6323:
6278:
6252:
6232:
6208:
6179:
6150:
6098:
6078:
6042:
6022:
5998:
5953:
5933:
5907:
5887:
5867:
5837:
5817:
5791:
5484:
5405:
5369:
5280:The first power of a number is the number itself:
4760:
4544:
4477:
4370:
4162:
4095:
4066:
4004:
3980:
3889:
3860:
3783:
3731:
3666:
3590:
3525:
2789:
2744:
2709:
2677:
2631:
2604:
2584:
2545:
2499:
2479:
2439:
2402:
2382:
2355:
2304:
2271:
2227:
2176:
2149:
2065:
2032:
1999:
1882:
1846:
1785:
1758:
1686:
1666:
1640:
1611:
1401:
1381:
1361:
1334:
1259:
1054:
1029:
968:
943:
887:
862:
777:
712:
615:
589:
499:
474:
384:
359:
169:because any nonzero number raised to the power of
27660:
26944:
26326:
25636:
25607:
25605:
25534:
22072:is, if it exists, a right adjoint to the functor
19625:of elements. This number of elements is either a
19289:, which is a linear operator acting on functions
17393:. In such a monoid, exponentiation of an element
8474:where the limit is taken over rational values of
8194:
5143:are commonly used, and have special names, e.g.:
4635:
4622:
4478:{\displaystyle b^{p^{q}}=b^{\left(p^{q}\right)},}
29232:
26191:
25472:. Vol. 165 (3rd ed.). Providence, RI:
22681:through positive values, but not negative ones.
22484:Under this definition by continuity, we obtain:
21791:. It results that the set of the functions from
21141:Function (mathematics) § Set exponentiation
19393:th power of the differentiation operator is the
18994:
18478:of the integers. Otherwise, the cyclic group is
14884:So, the above described method gives the values
13332:these values are the same as those described in
12491:(that is, complex differentiable) elsewhere. If
10935:
9393:
9308:
9085:
8379:
8295:§ Failure of power and logarithm identities
6355:, but towards negative infinity with decreasing
5253:: the nonnegative exponents are the rank of the
5052:to denote large or small numbers. For instance,
3258:, since in those the exponents must be constant.
2685:. Equating the exponents on both sides, we have
500:{\displaystyle \scriptstyle {\text{difference}}}
27546:
27193:
26873:"BASCOM - A BASIC compiler for TRS-80 I and II"
26399:Journal für die reine und angewandte Mathematik
26200:(9th ed.). John Wiley & Sons. p.
25946:(3rd ed.). Industrial Press. p. 101.
24070:instead.). Many other languages followed suit:
21358:{\displaystyle (S^{T})^{U}\cong S^{T\times U},}
17368:denoted as a multiplication. The definition of
11853:. The unique primitive square root of unity is
11607:or algebraic solutions of algebraic equations (
9641:. It follows from the preceding equations that
8305:
7923:is even. In the latter case, whichever complex
4828:
3145:; there, the notation is introduced in Book I.
3057:in the 16th century. In the late 16th century,
2480:{\displaystyle {\sqrt {b}}\times {\sqrt {b}}=b}
1759:{\displaystyle b^{0}\times b^{n}=b^{0+n}=b^{n}}
1055:{\displaystyle \scriptstyle {\text{logarithm}}}
30:"Exponent" redirects here. For other uses, see
27340:
27326:
26881:. Software Reviews. Vol. 4, no. 31.
26864:
26726:
26060:National Institute of Standards and Technology
25999:
25602:
25408:
24593:Addition, subtraction, and multiplication only
23940:{\displaystyle 1/\sin(x)={\frac {1}{\sin x}}.}
22589:. This method does not permit a definition of
22573:These powers are obtained by taking limits of
21919:{\displaystyle (S^{T})^{U}\cong S^{T\times U}}
19108:{\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}
18922:is an ideal that equals its own radical. In a
18362:, and such that every element has an inverse.
9816:from the real numbers to any complex argument
8213:
4545:{\displaystyle \left(b^{p}\right)^{q}=b^{pq}.}
3110:
2976:In the 9th century, the Persian mathematician
2612:on the right-hand side can also be written as
27312:
27179:
26982:
26636:
26584:
25569:A Short Account of the History of Mathematics
25540:"Etymology of some common mathematical terms"
25015:Gamma function and factorial of a non-integer
24512:
24066:for exponentiation (the initial version used
23977:), intended for exponentiation, but this was
21826:in the preceding section can also be denoted
21559:{\displaystyle \mathbb {R} ^{(\mathbb {N} )}}
19254:can also be exponentiated. An example is the
19118:Matrix powers appear often in the context of
19048:is defined to be the identity matrix, and if
18570:Order of elements play a fundamental role in
16580:{\displaystyle \left(e^{x}\right)^{y}=e^{xy}}
12702:is real and positive, the principal value of
11664:{\displaystyle \omega =e^{\frac {2\pi i}{n}}}
8282:{\displaystyle \left(b^{r}\right)^{s}=b^{rs}}
5825:, are sometimes called power functions. When
4834:
1081:
616:{\displaystyle \scriptstyle {\text{product}}}
217:
29148:
27498:
26900:
26696:. Vol. 2 (3rd ed.). Chicago, USA:
26094:
23542:{\displaystyle f^{\circ 3}=f\circ f\circ f,}
22989:
22970:
22685:Efficient computation with integer exponents
22339:
22294:
22111:, if direct products exist, and the functor
21698:which can be identified with the set of the
21682:
21670:
21608:
21596:
21250:This generalizes to the following notation.
21234:
21216:
19954:) equals the set of the nonzero elements of
19941:
19865:
17359:
17326:
17314:
16783:{\displaystyle e^{-4\pi ^{2}n^{2}}=1\qquad }
12340:such that, for every nonzero complex number
12160:is used to define complex exponentiation as
12028:th root of unity with the smallest positive
10386:§ Non-integer powers of complex numbers
10129:can be used as an alternative definition of
9834:as the exponential function allows defining
8181:§ Non-integer powers of complex numbers
5457:. This is because there will be a remaining
4172:
3309:. (It is true that it could also be called "
26843:International Business Machines Corporation
26818:
26782:International Business Machines Corporation
26731:. Addison-Wesley Professional. p. 33.
26680:
26381:
26334:Difference Equations: From Rabbits to Chaos
25916:Abstract Algebra: an inquiry based approach
23949:
21518:{\displaystyle \mathbb {R} ^{\mathbb {N} }}
20511:, generated by the Frobenius automorphism.
19128:expresses a transition from a state vector
13675:{\displaystyle \rho ={\sqrt {a^{2}+b^{2}}}}
13486:can be computed from the canonical form of
10143:Complex exponents with a positive real base
8478:only. This limit exists for every positive
7381:, that is, the unique positive real number
5225:expresses any number as a sum of powers of
4118:, with an associative multiplication and a
2007:. Taking the cube root of both sides gives
888:{\displaystyle \scriptstyle {\text{power}}}
27613:Possessing a specific set of other numbers
27436:
27319:
27305:
27186:
27172:
26989:
26975:
26745:
26566:
25882:
25530:
25528:
24519:
24505:
24166:, Haskell (for floating-point exponents),
22150:
18559:(starting indifferently from the exponent
17950:may denote exponentiation with respect of
13791:{\displaystyle \log z=\ln \rho +i\theta ,}
13388:varies continuously along a circle around
11430:{\displaystyle -\pi <\theta \leq \pi ,}
11118:is the argument of a complex number, then
9848:function. Specifically, the fact that the
8208:
5199:give the number of possible values for an
3095:used the terms square, cube, zenzizenzic (
1088:
1074:
969:{\displaystyle \scriptstyle {\text{root}}}
224:
210:
29076:
28023:
26871:Daneliuk, Timothy "Tim" A. (1982-08-09).
26535:
26517:"A Survey of Fast Exponentiation Methods"
26224:
26218:
26045:
25804:, pour le multiplier encore une fois par
25764:
21547:
21538:
21509:
21503:
21102:
21076:
21048:
20539:
20483:
20434:
20358:
20297:
20282:
20233:
20089:
20030:
19963:
19823:
19785:
19706:
18464:
18032:
17593:{\displaystyle \left(x^{-1}\right)^{-n}.}
15846:
15756:
15478:
15297:Failure of power and logarithm identities
12729:equals its usual value defined above. If
12643:is the principal value of the logarithm.
12090:
11248:
10932:, and is therefore easier to understand.
10165:
8451:
8443:
8430:
8393:
5695:
5429:
4074:to negative exponents (consider the case
3981:{\displaystyle b^{-n}={\frac {1}{b^{n}}}}
3667:{\displaystyle b^{m+n}=b^{m}\cdot b^{n},}
3315:to the second power", but "the square of
3242:Variable exponents, non-integer exponents
3119:used in essence modern notation, when in
1168:, exponentiation corresponds to repeated
1025:
1021:
939:
935:
858:
854:
708:
704:
585:
581:
566:
562:
545:
541:
470:
466:
451:
447:
430:
426:
385:{\displaystyle \scriptstyle {\text{sum}}}
355:
351:
336:
332:
315:
311:
294:
290:
273:
269:
113:
26870:
26810:Brice Carnahan; James O. Wilkes (1968).
26670:
26642:
26590:
26416:
26229:. Jones and Bartlett. pp. 278–283.
25966:
25452:
25450:
23491:{\displaystyle f(x)^{2}=f(x)\cdot f(x).}
22642:is already meaningful for all values of
21659:, that is the set of the functions from
19564:, is one of the basic operations of the
19005:is a square matrix, then the product of
17870:. This includes, as specific instances,
17858:(which form a ring). They apply also to
11879:the primitive fourth roots of unity are
11572:
11461:th root is not real, although the usual
9823:
9593:{\displaystyle \exp(x)\exp(y)=\exp(x+y)}
9257:{\displaystyle \exp(x+y)=\exp(x)\exp(y)}
8309:
7243:
5739:
5725:
5257:on the left of the point (starting from
4873:). Such functions can be represented as
4781:), which is implied if they belong to a
4409:in superscript notation is top-down (or
4401:. Without parentheses, the conventional
4067:{\displaystyle b^{m+n}=b^{m}\cdot b^{n}}
3861:{\displaystyle b^{m+n}=b^{m}\cdot b^{n}}
3030:
1273:to the right of the base. In that case,
70:
26580:(in French). Vol. IV. p. 229.
26478:(3rd ed.). Brooks-Cole. Chapter 5.
26055:NIST Handbook of Mathematical Functions
25972:Trends in the Historiography of Science
25837:, in order to multiply it once more by
25647:MacTutor History of Mathematics Archive
25545:MacTutor History of Mathematics Archive
25525:
22710:to the exponent 100 written in binary:
22695:using iterated multiplication requires
22039:exist: in such a category, the functor
20767:, that allow identifying, for example,
18914:. The nilradical is the radical of the
18358:denoted as multiplication, that has an
12773:
11931:th roots of unity allow expressing all
9928:{\displaystyle b=\exp(\ln b)=e^{\ln b}}
8877:
6457:. Functions with this kind of symmetry
6106:increases. Functions with this kind of
3798:convention, which may be used in every
14:
29233:
29184:
26686:
26514:
26492:. American Mathematical Society, 1975.
26473:
26440:. Cambridge University Press. p.
26168:(14 ed.). Pearson. pp. 7–8.
25939:
25747:
25723:
25681:
25675:
25664:
25456:
23590:{\displaystyle f^{3}=f\cdot f\cdot f.}
22233:More precisely, consider the function
22031:This generalizes to the definition of
19801:{\displaystyle x\in \mathbb {F} _{q}.}
19677:is a positive integer. For every such
18871:), the commutative ring is said to be
18589:Superscript notation is also used for
18584:classification of finite simple groups
13345:
12017:{\displaystyle e^{\frac {2k\pi i}{n}}}
11978:with one vertex on the real number 1.
10526:{\displaystyle e^{iy}=\cos y+i\sin y,}
5472:are useful for expressing alternating
3214:
2235:, which can be more simply written as
29183:
29147:
29111:
29075:
29035:
28660:
28549:
28275:
28190:
28145:
28022:
27712:
27659:
27611:
27545:
27497:
27435:
27339:
27300:
27167:
26970:
26949:. Addison-Wesley. pp. 130, 324.
26572:
26431:
26363:
25888:
25777:Discourse de la méthode [...]
25496:
25447:
25409:
25258:There are three common notations for
24272:(for integer base, integer exponent).
24040:(for nonnegative integer exponents),
23191:{\displaystyle (g\circ f)(x)=g(f(x))}
23071:
21756:
21194:{\displaystyle (x_{1},\ldots ,x_{n})}
20990:{\displaystyle (x_{1},\ldots ,x_{n})}
20177:{\displaystyle (x+y)^{p}=x^{p}+y^{p}}
17866:to itself, which form a monoid under
14089:the principal value corresponding to
13732:for the definition of this function).
13098:is an integer. This results from the
11395:. The common choice is to choose the
10881:Non-integer powers of complex numbers
7239:
5708:
4488:which, in general, is different from
3922:
3591:{\displaystyle b^{n+1}=b^{n}\cdot b.}
3477:
3173:in multiplying it once more again by
2959:
2898:, here: "amplification") used by the
27713:
26669:(NB. Inhere, Herschel refers to his
25875:participating institution membership
25562:
25379:
25377:
25375:
25373:
25371:
23716:{\displaystyle \sin(x)\cdot \sin(x)}
22481:, which remain indeterminate forms.
22021:to the functor "direct product with
21134:
18804:If the nilradical is reduced to the
18730:{\displaystyle (gh)^{k}=g^{k}h^{k}.}
17374:requires further the existence of a
17282:(that is, not algebraic), except if
13554:being real), then its polar form is
9529:does not affect the limit, yielding
5485:§ nth roots of a complex number
5339:is undefined, because it must equal
4813:use a different notation (sometimes
3465:
3230:was used synonymously with the term
3025:
2948:, necessary to manipulate powers of
2923:
2906:for the square of a line, following
165:Each curve passes through the point
29112:
26996:
26755:; Beeber, R. J.; Best, Sheldon F.;
26700:. pp. 108, 176–179, 336, 346.
26693:A History of Mathematical Notations
26490:Functional Analysis and Semi-Groups
26476:Linear algebra and its applications
26249:
26031:, Leipzig). pp. 115–120, 802.
25895:Introductio in analysin infinitorum
25751:A History of Mathematical Notations
25688:A History of Mathematical Notations
25668:A History of Mathematical Notations
25611:
25500:The Beginnings of Greek Mathematics
24476:Unicode subscripts and superscripts
23883:fractions are generally used as in
22208:
21451:denotes the Cartesian product, and
21119:as well as their direct product as
19598:. Common examples are the field of
16797:but this is false when the integer
15567:
13392:, then, after a turn, the value of
10370:{\textstyle \left(b^{z}\right)^{t}}
10325:{\displaystyle b^{z+t}=b^{z}b^{t},}
10223:{\displaystyle b^{z}=e^{(z\ln b)},}
9997:{\displaystyle (e^{x})^{y}=e^{xy},}
9522:{\displaystyle {\frac {xy}{n^{2}}}}
8195:§ Limits of rational exponents
8174:
7640:{\displaystyle y=x^{\frac {1}{q}},}
6411:, flattening more in the middle as
6086:, flattening more in the middle as
5011:) number system, integer powers of
4985:
3794:This value is also obtained by the
3732:{\displaystyle (b^{m})^{n}=b^{mn}.}
2984:, "possessions", "property") for a
2878:, meaning "to put forth". The term
2790:{\displaystyle {\sqrt {b}}=b^{1/2}}
2546:{\displaystyle b^{r}\times b^{r}=b}
2279:, using the result from above that
1847:{\displaystyle b^{0}=b^{n}/b^{n}=1}
1269:The exponent is usually shown as a
24:
29036:
26339:Undergraduate Texts in Mathematics
26131:Springer-Handbuch der Mathematik I
25780:. Leiden: Jan Maire. p. 299.
25383:
23022:
22961:
21719:exponentiation of cardinal numbers
21706:, by mapping each function to the
19250:Apart from matrices, more general
18668:{\displaystyle (g^{h})^{k}=g^{hk}}
18547:and the cyclic group generated by
17335:{\displaystyle b\not \in \{0,1\},}
16724:{\displaystyle e^{x+y}=e^{x}e^{y}}
14745:are real, the principal one being
14169:{\displaystyle e^{x+y}=e^{x}e^{y}}
12610:{\displaystyle z^{w}=e^{w\log z},}
12304:
12208:{\displaystyle z^{w}=e^{w\log z},}
12126:that are real and nonpositive, or
9403:
9318:
9095:
8789:that have the same limit, denoted
8018:cannot be satisfied. For example,
8011:{\displaystyle (x^{a})^{b}=x^{ab}}
7897:{\displaystyle (x^{r})^{s}=x^{rs}}
5875:, two primary families exist: for
5715:
5491:
5229:, and denotes it as a sequence of
4901:are given in the following table:
4626:
4555:
3999:
3802:with a multiplication that has an
130:
98:
25:
29262:
26947:Concepts of Programming Languages
26029:B. G. Teubner Verlagsgesellschaft
26011:Semendjajew, Konstantin Adolfovič
25642:"Abu'l Hasan ibn Ali al Qalasadi"
25368:
25354:is sufficient for the definition.
23422:{\displaystyle f^{2}(x)=f(f(x)),}
20639:
20589:can be efficiently computed with
20555:{\displaystyle \mathbb {F} _{q},}
20374:{\displaystyle \mathbb {F} _{q},}
20249:{\displaystyle \mathbb {F} _{q},}
20105:{\displaystyle \mathbb {F} _{q},}
20046:{\displaystyle \mathbb {F} _{q},}
19979:{\displaystyle \mathbb {F} _{q}.}
19722:{\displaystyle \mathbb {F} _{q}.}
19204:is the state of the system after
19134:of some system to the next state
18875:. Reduced rings are important in
18793:, the nilpotent elements form an
17126:must be replaced by the identity
16199:. The identity holds, but saying
15337:of the complex logarithm one has
14221:{\displaystyle e^{y\ln x}=x^{y},}
11562:
11453:. This function equals the usual
11384:It is usual to choose one of the
10701:factor is one. This results from
9603:Euler's number can be defined as
8187:
7955:{\displaystyle x^{\frac {1}{n}},}
7370:denotes the unique positive real
6548:Table of powers of decimal digits
5293:
4924:-tuples of elements from the set
4413:-associative), not bottom-up (or
2745:{\displaystyle r={\tfrac {1}{2}}}
2440:{\displaystyle b^{r}={\sqrt {b}}}
29214:
28822:Perfect digit-to-digit invariant
28191:
26644:Herschel, John Frederick William
26592:Herschel, John Frederick William
25386:"Basic rules for exponentiation"
24409:
22317:
21525:denotes the vector space of the
21261:, the set of all functions from
21243:{\displaystyle \{1,\ldots ,n\}.}
21061:{\displaystyle \mathbb {R} ^{n}}
20496:{\displaystyle \mathbb {F} _{q}}
20447:{\displaystyle \mathbb {F} _{q}}
19836:{\displaystyle \mathbb {F} _{q}}
19590:and every nonzero element has a
19571:
19380:{\displaystyle (d/dx)f(x)=f'(x)}
18785:. Such an element is said to be
18297:{\displaystyle (f^{\circ n})(x)}
17893:whose valued can be multiplied,
17610:in the algebraic structure, and
15258:In both examples, all values of
13225:{\displaystyle w={\frac {m}{n}}}
11477:function that extends the usual
11381:th roots of the complex number.
10889:th roots, that is, of exponents
9840:for every positive real numbers
8347:with an arbitrary real exponent
6331:will also tend towards positive
6220:behavior reverses from positive
5584:Any power of one is always one:
5264:
5128:
4990:
3742:
26938:
26889:from the original on 2020-02-07
26852:from the original on 2022-03-29
26803:
26791:from the original on 2022-07-04
26720:
26658:from the original on 2020-08-04
26508:
26495:
26482:
26467:
26458:
26425:
26410:
26356:
26320:
26280:
26243:
26196:Calculus: Early Transcendentals
26157:
25960:
25933:
25906:
25846:
25758:
25741:
25717:
25703:
25658:
25630:
25470:Graduate Studies in Mathematics
25464:Advanced Modern Algebra, Part 1
25344:
24586:Elementary arithmetic operation
22621:are not accumulation points of
22366:(that is, the set of all pairs
22100:{\displaystyle Y\to T\times Y.}
20749:This operation is not properly
18890:More generally, given an ideal
17796:
17533:. In this case, the inverse of
17206:Irrationality and transcendence
16779:
16664:
16518:
16400:
15560:
13090:, that is, there is an integer
12905:{\displaystyle 2ik\pi +\log z,}
12831:at the negative real values of
10389:, below), one has, in general,
8921:{\displaystyle x\mapsto e^{x},}
8418:
7911:th root, which is negative, if
7841:is a positive rational number,
7427:is a positive real number, and
4106:The same definition applies to
3083:"Exponent"; "square" and "cube"
3020:Abu'l-Hasan ibn Ali al-Qalasadi
1311:th power", or most briefly as "
1130:. Exponentiation is written as
26225:Denlinger, Charles G. (2011).
25584:
25556:
25490:
25433:
25402:
25252:
24911:Inverse trigonometric function
23910:
23904:
23851:
23845:
23773:In this context, the exponent
23763:{\displaystyle \sin(\sin(x)),}
23754:
23751:
23745:
23736:
23710:
23704:
23692:
23686:
23660:
23654:
23482:
23476:
23467:
23461:
23446:
23439:
23413:
23410:
23404:
23398:
23389:
23383:
23332:
23329:
23326:
23320:
23314:
23308:
23285:
23279:
23185:
23182:
23176:
23170:
23161:
23155:
23152:
23140:
22809:
22806:
22794:
22775:
22309:
22297:
22255:
22243:
22143:has a right adjoint for every
22136:{\displaystyle Y\to X\times Y}
22121:
22082:
22049:
21991:
21973:
21961:
21942:
21888:
21874:
21851:
21839:
21551:
21543:
21324:
21310:
21188:
21156:
20984:
20952:
20885:
20867:
20844:
20835:
20823:
20820:
20798:
20795:
20783:
20774:
20681:
20669:
20315:
20292:
20139:
20126:
20008:
19996:
19520:
19514:
19509:
19503:
19492:
19486:
19448:
19442:
19374:
19368:
19354:
19348:
19342:
19325:
19305:
19299:
18967:
18935:
18692:
18682:
18640:
18626:
18540:{\displaystyle x^{n}=x^{0}=1,}
18327:
18321:
18291:
18285:
18282:
18266:
18237:
18230:
18207:
18201:
18198:
18185:
18159:
18156:
18150:
18147:
18141:
18135:
18126:
18120:
18111:
18105:
18102:
18086:
18057:
18051:
18042:
18036:
18029:
18023:
18008:
18004:
17998:
17992:
17986:
17980:
17977:
17964:
17760:
17750:
17717:
17703:
17486:for every nonnegative integer
17479:{\displaystyle x^{n+1}=xx^{n}}
17057:
17036:
17019:
16998:
16983:
16962:
16927:{\displaystyle \exp(y\log x),}
16918:
16903:
16847:
16841:
16439:
16418:
16138:
16128:
16087:
16077:
15999:
15989:
15975:
15965:
15942:
15923:
15912:are positive real numbers and
15577:
15561:
15469:
15442:
15427:
15418:
15394:
15385:
15373:
15364:
15354:
15351:
15333:is a real number. But for the
15329:is a positive real number and
15254:and different absolute values.
15202:
15199:
15184:
15169:
15154:
15145:
15140:
15122:
15085:
15082:
15079:
15061:
15040:
15025:
15022:
15004:
14983:
14974:
14969:
14951:
14905:
14895:
14877:{\displaystyle -2=2e^{i\pi }.}
14806:
14796:
14524:{\displaystyle i=e^{i\pi /2},}
14396:
14357:
14342:
14303:
14287:
14269:
14073:
14034:
14025:
13986:
13713:
13701:
13616:
13589:
13422:
12988:
12961:
12916:is any integer. Similarly, if
12684:{\displaystyle (z,w)\to z^{w}}
12668:
12665:
12653:
11143:{\displaystyle \theta +2k\pi }
11016:
10989:
10861:
10858:
10843:
10828:
10813:
10804:
10674:
10671:
10656:
10641:
10626:
10617:
10212:
10197:
9966:
9952:
9945:. For preserving the identity
9903:
9891:
9844:, in terms of exponential and
9773:
9767:
9713:
9707:
9660:
9654:
9628:
9622:
9587:
9575:
9563:
9557:
9548:
9542:
9400:
9315:
9301:
9295:
9286:
9280:
9251:
9245:
9236:
9230:
9218:
9206:
9168:
9162:
9092:
9078:
9072:
9042:{\displaystyle \exp(x)=e^{x}.}
9020:
9014:
8981:
8975:
8947:{\displaystyle e\approx 2.718}
8902:
8455:
8419:
8400:
8397:
8386:
8141:
8131:
8106:
8096:
8044:
8034:
7983:
7969:
7869:
7855:
7808:
7794:
7766:
7752:
7718:
7704:
7673:
7654:
7581:
7562:
7447:{\displaystyle {\frac {p}{q}}}
6496:
6490:
6478:
6469:
6302:
6296:
6203:
6197:
6145:
6139:
6130:
6121:
5977:
5971:
5770:
5764:
5195:. The positive integer powers
4823:non-commutative exponentiation
4720:
4708:
4586:
4573:
4325:
4312:
3701:
3687:
3262:
3002:mathematicians represented in
2356:{\displaystyle b^{-n}=1/b^{n}}
2228:{\displaystyle b^{-1}=1/b^{1}}
1911:
1897:
1018:
1010:
13:
1:
27661:Expressible via specific sums
27138:Conway chained arrow notation
26698:Open court publishing company
26140:Springer Fachmedien Wiesbaden
25693:Open Court Publishing Company
25474:American Mathematical Society
25361:
23052:addition-chain exponentiation
22358:can be viewed as a subset of
21112:{\displaystyle \mathbb {R} ,}
20458:automorphisms, which are the
20014:{\displaystyle \varphi (p-1)}
19210:time steps. The matrix power
18995:Matrices and linear operators
18898:, the set of the elements of
18345:
17104:{\displaystyle \log \exp z=z}
14835:Similarly, the polar form of
12529:{\displaystyle \log z=\ln z.}
12382:{\displaystyle e^{\log z}=z,}
11962:th roots of unity lie on the
11935:th roots of a complex number
10943:Every nonzero complex number
9732:{\displaystyle \exp(x)=e^{x}}
9679:{\displaystyle \exp(x)=e^{x}}
8198:, below), or in terms of the
5139:The first negative powers of
2944:proved the law of exponents,
2890:) is a mistranslation of the
2678:{\displaystyle b^{r+r}=b^{1}}
25980:10.1007/978-94-017-3596-4_20
25597:10.1017/CBO9780511695124.017
23973:included an uparrow symbol (
23666:{\displaystyle \sin ^{2}(x)}
23212:If the domain of a function
22652:obtained above for negative
22274:{\displaystyle f(x,y)=x^{y}}
22033:exponentiation in a category
21083:{\displaystyle \mathbb {R} }
19245:eigenvalues and eigenvectors
18739:
18471:{\displaystyle \mathbb {Z} }
13338:th roots of a complex number
12283:{\displaystyle e^{\log z}=z}
11366:th root, and provides a new
10939:th roots of a complex number
10270:This satisfies the identity
9863:of the exponential function
8306:Limits of rational exponents
8209:§ Powers via logarithms
5554:less than one tend to zero:
4829:Combinatorial interpretation
3184:René Descartes, La Géométrie
2918:
2888:potentia, potestas, dignitas
2854:
2831:, with applications such as
7:
28750:Multiplicative digital root
27195:Orders of magnitude of time
26436:Linear Algebra and Geometry
26264:10.1007/978-981-10-1789-6_6
26064:U.S. Department of Commerce
25145:Infinite continued fraction
24961:Inverse hyperbolic function
24426:Double exponential function
24402:
23341:{\displaystyle f(f(f(x))).}
22664:is odd, since in this case
20757:, but has these properties
20516:Diffie–Hellman key exchange
19633:; that is, it has the form
18867:for every positive integer
18860:{\displaystyle x^{n}\neq 0}
18449:are distinct, the group is
17943:{\displaystyle f^{\circ n}}
17228:is a rational number, then
16587:and expanding the exponent)
16445:{\displaystyle (1+2\pi in)}
15623:for the principal value of
14827:{\displaystyle (-2)^{3+4i}}
14447:
13135:{\displaystyle e^{a}=e^{b}}
12483:at negative real values of
10166:§ Exponential function
6502:{\displaystyle f(-x)=-f(x)}
6324:{\displaystyle f(x)=cx^{n}}
6006:will tend towards positive
5999:{\displaystyle f(x)=cx^{n}}
5792:{\displaystyle f(x)=cx^{n}}
5754:Real functions of the form
5696:§ Exponential function
5468:Because of this, powers of
5426:, or it is left undefined.
5269:Every power of one equals:
3913:
3391:3 = 3 · 3 · 3 · 3 · 3 = 243
3111:Modern exponential notation
2383:{\displaystyle {\sqrt {b}}}
1648:must be equal to 1 for any
1156:(raised) to the (power of)
10:
29267:
28146:
27090:Inverse for right argument
26945:Robert W. Sebesta (2010).
26488:E. Hille, R. S. Phillips:
26417:Bourbaki, Nicolas (1970).
26388:Steiner, J.; Clausen, T.;
26289:Introduction to Algorithms
26068:Cambridge University Press
26016:Taschenbuch der Mathematik
26007:Bronstein, Ilja Nikolaevič
25943:Technical Shop Mathematics
24461:List of exponential topics
24347:languages that prioritize
23627:{\displaystyle \sin ^{2}x}
23075:
23013:exponentiation by squaring
22226:has no limit at the point
22154:
22065:{\displaystyle X\to X^{T}}
21860:{\displaystyle \hom(X,Y).}
21760:
21138:
20853:{\displaystyle ((x,y),z),}
20807:{\displaystyle (x,(y,z)),}
20591:exponentiation by squaring
20530:is a primitive element in
19685:elements. The fields with
19575:
19120:discrete dynamical systems
18887:is always a reduced ring.
18213:{\displaystyle (f^{n})(x)}
17397:is defined inductively by
17209:
15303:as single-valued functions
13232:is a rational number with
13168:is an integer multiple of
11621:th roots of unity are the
11605:discrete Fourier transform
11592:complex numbers such that
11588:th roots of unity are the
11566:
11370:th root. This can be done
11087:. The argument is defined
10122:{\displaystyle e^{x\ln b}}
9490:and the second-order term
9183:{\displaystyle \exp(0)=1,}
8881:
7363:{\displaystyle {\sqrt{x}}}
6151:{\displaystyle f(-x)=f(x)}
5719:
5313:th power of zero is zero:
5132:
5000:
4994:
4832:
4176:
3336:Similarly, the expression
2927:
2913:
2841:chemical reaction kinetics
2487:. Therefore, the exponent
2272:{\displaystyle b^{-1}=1/b}
29:
29210:
29193:
29179:
29157:
29143:
29121:
29107:
29085:
29071:
29044:
29031:
29007:
28961:
28921:
28872:
28846:
28827:Perfect digital invariant
28779:
28763:
28742:
28709:
28674:
28670:
28656:
28564:
28545:
28514:
28481:
28438:
28415:
28402:Superior highly composite
28292:
28288:
28271:
28199:
28186:
28154:
28141:
28029:
28018:
27980:
27971:
27949:
27906:
27868:
27859:
27792:
27734:
27725:
27721:
27708:
27666:
27655:
27618:
27607:
27555:
27541:
27504:
27493:
27446:
27431:
27349:
27335:
27258:
27212:
27201:
27148:Knuth's up-arrow notation
27125:
27089:
27050:
27004:
26368:"Principal root of unity"
26227:Elements of Real Analysis
26021:Pocketbook of mathematics
25919:. CRC Press. p. 94.
25862:Oxford English Dictionary
25281:{\displaystyle x\times y}
24661:Finite continued fraction
24492:Zero to the power of zero
24384:as floating point numbers
24190:TRS-80 Level II/III BASIC
24036:(for integer exponents),
23971:original version of ASCII
23123:{\displaystyle g\circ f,}
22636:is an integer, the power
22392:extended real number line
22214:Zero to the power of zero
22179:Knuth's up-arrow notation
22109:Cartesian closed category
21763:Cartesian closed category
20903:This allows defining the
20187:is true for the exponent
19847:such that the set of the
19560:which, together with the
18989:Hilbert's Nullstellensatz
18336:{\displaystyle f^{n}(x).}
18252:{\displaystyle f(x)^{n},}
17872:geometric transformations
17360:Integer powers in algebra
17212:Gelfond–Schneider theorem
13542:is the canonical form of
13055:give different values of
12695:is real and nonpositive.
12040:, sometimes shortened as
11577:The three third roots of
11469:shows that the principal
9634:{\displaystyle e=\exp(1)}
8812:{\displaystyle b^{\pi }.}
8537:{\displaystyle b^{\pi }:}
8214:identities and properties
5440:is an even integer, then
5042:Exponentiation with base
5015:are written as the digit
4839:For nonnegative integers
4173:Identities and properties
3916:Zero to the power of zero
3908:is generally assigned to
3454:th", or most briefly as "
3415:3 raised to the 5th power
3049:. This was later used by
2952:. He then used powers of
2585:{\displaystyle b^{r+r}=b}
2312:. By a similar argument,
2157:. Dividing both sides by
1766:. Dividing both sides by
1291:(raised) to the power of
1152:; this is pronounced as "
987:
979:
909:
898:
797:
789:
635:
627:
518:
510:
403:
395:
246:
238:
57:
50:
41:
32:Exponent (disambiguation)
28440:Euler's totient function
28224:Euler–Jacobi pseudoprime
27499:Other polynomial numbers
27153:Steinhaus–Moser notation
26474:Strang, Gilbert (1988).
26432:Bloom, David M. (1979).
26331:; Robson, Robby (2005).
26104:; Schwarz, Hans Rudolf;
25843:, and thus to infinity).
25810:, & ainsi a l'infini
25748:Cajori, Florian (1928).
25665:Cajori, Florian (1928).
25652:University of St Andrews
25572:(6th ed.). London:
25550:University of St Andrews
25334:{\displaystyle x\cdot y}
25245:
24554:Mathematical expressions
24182:Algol Reference language
24046:computer algebra systems
23950:In programming languages
23352:pointwise multiplication
23291:{\displaystyle f^{3}(x)}
23031:{\displaystyle \sharp n}
22630:On the other hand, when
21691:{\displaystyle \{0,1\},}
21617:{\displaystyle \{0,1\}.}
20894:{\displaystyle (x,y,z).}
20418:{\displaystyle q=p^{k},}
20256:It follows that the map
20075:Euler's totient function
20066:{\displaystyle \varphi }
19681:, there are fields with
19662:{\displaystyle q=p^{k},}
18906:is an ideal, called the
17429:{\displaystyle x^{0}=1,}
16856:{\displaystyle \exp(y),}
16452:-th power of both sides)
16229:, discovered in 1827 by
16187:On the other hand, when
14443:for the principal value.
13884:{\displaystyle w\log z.}
13373:{\displaystyle z\neq 0,}
13346:§ Integer exponents
11518:(they are multiplied by
11374:times, and provides the
11355:{\displaystyle 2i\pi /n}
10569:real and imaginary parts
8990:{\displaystyle \exp(x),}
8339:can be expressed as the
7927:th root one chooses for
7414:{\displaystyle y^{n}=x.}
6404:{\displaystyle y=cx^{3}}
6079:{\displaystyle y=cx^{2}}
5550:Powers of a number with
5453:is an odd integer, then
5370:{\displaystyle 1/0^{-n}}
5176:, the set of all of its
4835:Exponentiation over sets
4811:computer algebra systems
4417:-associative). That is,
3784:{\displaystyle b^{0}=1.}
28254:Somer–Lucas pseudoprime
28244:Lucas–Carmichael number
28079:Lazy caterer's sequence
26883:Popular Computing, Inc.
26727:Richard Gillam (2003).
26606:Royal Society of London
26578:Formulaire mathématique
25940:Achatz, Thomas (2005).
25867:Oxford University Press
25825:, in order to multiply
25691:. Vol. 1. London:
24544:Closed-form expressions
24028:(and its derivatives),
23599:trigonometric functions
22453:by continuity whenever
22151:Repeated exponentiation
22107:A category is called a
21783:are the functions from
21589:can represents the set
21464:{\displaystyle \sqcup }
21444:{\displaystyle \times }
21205:can be considered as a
20742:{\displaystyle y\in T.}
20636:is sufficiently large.
20217:{\displaystyle x^{p}=x}
19760:{\displaystyle x^{q}=x}
19673:is a prime number, and
19318:to give a new function
18827:{\displaystyle x\neq 0}
18774:{\displaystyle x^{n}=0}
18445:. If all the powers of
17498:is a negative integer,
17376:multiplicative identity
17238:. This remains true if
16217:holds for real numbers
15247:{\displaystyle 4\ln 2,}
14120:. Using the identities
13957:{\displaystyle w\log z}
13844:{\displaystyle 2ik\pi }
13265:{\displaystyle n>0,}
13190:{\displaystyle 2\pi i.}
12538:The principal value of
12080:{\displaystyle 1^{1/n}}
11362:to the argument of the
11321:{\displaystyle \theta }
11111:{\displaystyle \theta }
11091:an integer multiple of
11076:{\displaystyle \theta }
9779:{\displaystyle \exp(z)}
8871:Well-defined expression
8501:non-terminating decimal
7904:to rational exponents.
7334:{\displaystyle x^{1/n}}
7306:is a positive integer,
5868:{\displaystyle n\geq 1}
5818:{\displaystyle c\neq 0}
5709:§ Limits of powers
5543:tends to infinity when
5273:. This is true even if
4871:cardinal exponentiation
4179:Laws of Indices (horse)
4163:{\displaystyle x^{-1}.}
4120:multiplicative identity
4005:{\displaystyle \infty }
3806:. This way the formula
3526:{\displaystyle b^{1}=b}
3236:its more common meaning
3179:, and thus to infinity.
2849:public-key cryptography
2305:{\displaystyle b^{1}=b}
2033:{\displaystyle b^{1}=b}
1883:{\displaystyle b^{1}=b}
1667:{\displaystyle b\neq 0}
28129:Wedderburn–Etherington
27529:Lucky numbers of Euler
26614:10.1098/rstl.1813.0005
26546:10.1006/jagm.1997.0913
26515:Gordon, D. M. (1998).
26341: ed.). Springer.
25335:
25305:
25282:
24886:Trigonometric function
24534:Polynomial expressions
24529:Arithmetic expressions
24466:Modular exponentiation
23941:
23881:multiplicative inverse
23873:
23790:
23764:
23717:
23667:
23628:
23591:
23543:
23492:
23423:
23366:the parentheses. Thus
23342:
23292:
23254:denotes generally the
23248:
23227:of the function. Thus
23192:
23124:
23038:denotes the number of
23032:
23005:
22816:
22346:
22275:
22137:
22101:
22066:
22001:
21920:
21861:
21820:
21747:is the cardinality of
21717:This fits in with the
21692:
21645:
21618:
21560:
21519:
21465:
21445:
21422:
21359:
21290:
21244:
21195:
21113:
21084:
21062:
21029:
20991:
20928:
20895:
20854:
20808:
20743:
20714:
20713:{\displaystyle x\in S}
20688:
20626:
20583:
20556:
20497:
20470:. In other words, the
20448:
20419:
20387:Frobenius automorphism
20375:
20336:
20250:
20218:
20178:
20106:
20067:
20047:
20021:primitive elements in
20015:
19980:
19948:
19837:
19802:
19761:
19723:
19663:
19592:multiplicative inverse
19530:
19381:
19312:
19283:
19258:operator of calculus,
19231:
19198:
19197:{\displaystyle A^{n}x}
19168:
19167:{\displaystyle A^{2}x}
19109:
19042:
18974:
18894:in a commutative ring
18861:
18828:
18775:
18731:
18669:
18541:
18472:
18416:
18415:{\displaystyle x\in G}
18390:
18337:
18298:
18253:
18214:
18169:
18067:
17944:
17914:
17880:mathematical structure
17833:
17594:
17531:multiplicative inverse
17519:
17480:
17430:
17336:
17194:
17111:for complex values of
17105:
17067:
16928:
16884:
16857:
16822:
16784:
16725:
16669:
16581:
16523:
16446:
16405:
16325:
16181:
16026:
15863:
15585:
15513:
15279:
15248:
15216:
14878:
14828:
14782:
14739:
14712:
14618:
14551:
14550:{\displaystyle \log i}
14525:
14473:
14437:
14411:
14222:
14170:
14109:
14083:
13958:
13921:
13920:{\displaystyle w=c+di}
13885:
13845:
13812:
13792:
13720:
13676:
13626:
13536:
13535:{\displaystyle z=a+ib}
13480:
13453:
13419:has changed of sheet.
13413:
13380:in the sense that its
13374:
13326:
13293:
13266:
13226:
13191:
13162:
13136:
13076:
13035:
12937:
12906:
12862:
12861:{\displaystyle \log z}
12825:
12798:
12797:{\displaystyle \log z}
12760:
12759:{\displaystyle w=1/n,}
12723:
12685:
12637:
12636:{\displaystyle \log z}
12611:
12559:
12530:
12473:
12441:
12383:
12334:
12333:{\displaystyle \log ,}
12284:
12238:
12237:{\displaystyle \log z}
12209:
12143:
12112:
12091:Complex exponentiation
12081:
12018:
11919:
11893:
11873:
11836:
11765:
11665:
11581:
11549:
11504:
11503:{\displaystyle 2\pi ,}
11475:complex differentiable
11473:th root is the unique
11431:
11356:
11322:
11302:
11276:
11164:
11144:
11112:
11098:; this means that, if
11077:
11049:
11026:
10914:
10871:
10684:
10561:
10536:allows expressing the
10527:
10447:
10377:is not defined, since
10371:
10326:
10253:
10224:
10135:for any positive real
10123:
10084:
9998:
9929:
9810:
9809:{\displaystyle e^{z},}
9780:
9733:
9680:
9635:
9594:
9523:
9481:
9258:
9184:
9140:
9043:
8991:
8948:
8922:
8842:
8813:
8776:
8538:
8465:
8332:
8331:tends to the infinity.
8283:
8163:
8012:
7956:
7919:, and no real root if
7898:
7829:
7641:
7597:
7495:
7448:
7415:
7364:
7335:
7291:
6538:
6537:{\displaystyle c<0}
6503:
6451:
6425:
6405:
6369:
6349:
6325:
6280:
6279:{\displaystyle c>0}
6254:
6234:
6210:
6181:
6152:
6100:
6080:
6044:
6024:
6000:
5955:
5935:
5934:{\displaystyle c>0}
5909:
5889:
5869:
5839:
5819:
5793:
5751:
5737:
5547:is greater than one".
5430:Powers of negative one
5407:
5371:
5221:different values. The
4889:-letter words from an
4762:
4690:
4618:
4546:
4479:
4372:
4164:
4137:is standardly denoted
4097:
4068:
4006:
3982:
3914:For more details, see
3891:
3862:
3785:
3733:
3668:
3592:
3527:
3385:When an exponent is a
3260:
3187:
2887:
2791:
2746:
2711:
2679:
2633:
2606:
2586:
2547:
2501:
2481:
2441:
2404:
2384:
2357:
2306:
2273:
2229:
2178:
2151:
2067:
2066:{\displaystyle b^{-1}}
2034:
2001:
1884:
1848:
1787:
1760:
1688:
1674:, as follows. For any
1668:
1642:
1613:
1403:
1383:
1363:
1336:
1261:
1172:of the base: that is,
1056:
1031:
970:
945:
889:
864:
779:
714:
617:
591:
501:
476:
386:
361:
195:
28417:Prime omega functions
28234:Frobenius pseudoprime
28024:Combinatorial numbers
27893:Centered dodecahedral
27686:Primary pseudoperfect
27286:terasecond and longer
27143:Grzegorczyk hierarchy
26675:Hans Heinrich Bürmann
26524:Journal of Algorithms
26252:"Limits of sequences"
26250:Tao, Terence (2016).
25497:Szabó, Árpád (1978).
25336:
25306:
25283:
25090:Infinite sum (series)
24539:Algebraic expressions
23979:replaced by the caret
23955:Programming languages
23942:
23874:
23791:
23765:
23718:
23668:
23629:
23592:
23544:
23493:
23424:
23343:
23293:
23249:
23247:{\displaystyle f^{n}}
23193:
23125:
23044:binary representation
23033:
23006:
22817:
22347:
22276:
22138:
22102:
22067:
22002:
21921:
21862:
21821:
21819:{\displaystyle Y^{X}}
21693:
21646:
21644:{\displaystyle 2^{S}}
21619:
21561:
21529:of real numbers, and
21520:
21466:
21446:
21423:
21360:
21291:
21289:{\displaystyle S^{T}}
21245:
21196:
21114:
21085:
21063:
21030:
21028:{\displaystyle S^{n}}
20992:
20929:
20927:{\displaystyle S^{n}}
20896:
20855:
20809:
20744:
20715:
20689:
20687:{\displaystyle (x,y)}
20627:
20625:{\displaystyle g^{e}}
20584:
20582:{\displaystyle g^{e}}
20557:
20520:secure communications
20498:
20449:
20420:
20376:
20337:
20251:
20219:
20179:
20107:
20068:
20048:
20016:
19981:
19949:
19838:
19803:
19762:
19724:
19664:
19558:fractional derivative
19531:
19382:
19313:
19284:
19232:
19230:{\displaystyle A^{n}}
19199:
19169:
19110:
19043:
19041:{\displaystyle A^{0}}
18975:
18902:that have a power in
18862:
18829:
18776:
18732:
18670:
18542:
18473:
18417:
18396:is defined for every
18391:
18389:{\displaystyle x^{n}}
18356:associative operation
18338:
18299:
18254:
18215:
18170:
18068:
17945:
17915:
17913:{\displaystyle f^{n}}
17834:
17595:
17520:
17518:{\displaystyle x^{n}}
17481:
17431:
17366:associative operation
17342:then at least one of
17337:
17195:
17106:
17068:
16929:
16885:
16883:{\displaystyle x^{y}}
16863:a true function, and
16858:
16823:
16821:{\displaystyle e^{y}}
16785:
16726:
16670:
16582:
16524:
16447:
16406:
16326:
16182:
16027:
15864:
15586:
15514:
15280:
15278:{\displaystyle z^{w}}
15249:
15217:
14879:
14829:
14783:
14740:
14738:{\displaystyle i^{i}}
14713:
14619:
14552:
14526:
14474:
14472:{\displaystyle i^{i}}
14438:
14412:
14223:
14171:
14110:
14084:
13959:
13922:
13886:
13846:
13813:
13793:
13748:of this logarithm is
13721:
13677:
13627:
13537:
13481:
13479:{\displaystyle z^{w}}
13454:
13414:
13412:{\displaystyle z^{w}}
13375:
13327:
13294:
13292:{\displaystyle z^{w}}
13267:
13227:
13192:
13163:
13137:
13077:
13075:{\displaystyle z^{w}}
13036:
12938:
12936:{\displaystyle z^{w}}
12907:
12863:
12837:multivalued functions
12826:
12824:{\displaystyle z^{w}}
12799:
12761:
12724:
12722:{\displaystyle z^{w}}
12686:
12638:
12612:
12560:
12558:{\displaystyle z^{w}}
12531:
12474:
12442:
12384:
12335:
12285:
12239:
12210:
12144:
12113:
12082:
12019:
11970:at the vertices of a
11920:
11894:
11874:
11837:
11783:; they have the form
11766:
11666:
11576:
11550:
11505:
11467:Analytic continuation
11432:
11357:
11323:
11303:
11301:{\displaystyle 2\pi }
11277:
11165:
11145:
11113:
11078:
11050:
11048:{\displaystyle \rho }
11027:
10915:
10872:
10685:
10562:
10560:{\displaystyle b^{z}}
10528:
10448:
10372:
10327:
10254:
10252:{\displaystyle \ln b}
10225:
10124:
10085:
9999:
9930:
9824:Powers via logarithms
9811:
9781:
9734:
9681:
9636:
9595:
9524:
9482:
9264:holds as well, since
9259:
9185:
9141:
9044:
8992:
8949:
8923:
8843:
8841:{\displaystyle b^{x}}
8814:
8777:
8539:
8466:
8313:
8284:
8175:§ Real exponents
8164:
8013:
7957:
7899:
7830:
7642:
7598:
7496:
7449:
7416:
7365:
7336:
7247:
6539:
6504:
6452:
6426:
6406:
6370:
6350:
6326:
6281:
6255:
6235:
6211:
6182:
6153:
6101:
6081:
6045:
6025:
6001:
5956:
5936:
5910:
5890:
5870:
5840:
5820:
5794:
5743:
5729:
5527:This can be read as "
5422:is either defined as
5408:
5372:
5245:indicates a power of
4865:elements to a set of
4763:
4670:
4598:
4547:
4480:
4407:serial exponentiation
4373:
4165:
4098:
4069:
4007:
3983:
3892:
3863:
3786:
3734:
3669:
3593:
3528:
3472:arithmetic operations
3430:can be expressed as "
3252:
3147:
3031:Introducing exponents
3004:mathematical notation
2980:used the terms مَال (
2972:("square" and "cube")
2792:
2747:
2712:
2710:{\displaystyle r+r=1}
2680:
2634:
2632:{\displaystyle b^{1}}
2607:
2587:
2548:
2502:
2482:
2442:
2405:
2385:
2358:
2307:
2274:
2230:
2179:
2177:{\displaystyle b^{1}}
2152:
2068:
2035:
2002:
1885:
1849:
1788:
1786:{\displaystyle b^{n}}
1761:
1689:
1669:
1643:
1641:{\displaystyle b^{0}}
1614:
1404:
1384:
1364:
1362:{\displaystyle b^{n}}
1337:
1262:
1057:
1032:
971:
946:
890:
865:
780:
715:
618:
592:
502:
477:
387:
362:
203:Arithmetic operations
194:is the number itself.
74:
28876:-composition related
28676:Arithmetic functions
28278:Arithmetic functions
28214:Elliptic pseudoprime
27898:Centered icosahedral
27878:Centered tetrahedral
26814:. pp. 2–2, 2–6.
26025:Verlag Harri Deutsch
25639:Robertson, Edmund F.
25537:Robertson, Edmund F.
25319:
25311:is most common when
25292:
25266:
24986:Root of a polynomial
24836:Exponential function
24549:Analytic expressions
23887:
23804:
23777:
23727:
23677:
23638:
23605:
23553:
23502:
23433:
23370:
23302:
23266:
23231:
23137:
23105:
23083:Function composition
23019:
22958:
22717:
22285:
22237:
22115:
22076:
22043:
21933:
21871:
21830:
21803:
21721:, in the sense that
21667:
21628:
21593:
21533:
21498:
21455:
21435:
21370:
21307:
21273:
21213:
21153:
21098:
21072:
21043:
21012:
20949:
20911:
20864:
20817:
20771:
20724:
20698:
20666:
20609:
20566:
20534:
20478:
20462:first powers (under
20429:
20393:
20353:
20263:
20228:
20195:
20123:
20084:
20057:
20025:
19990:
19958:
19862:
19818:
19774:
19738:
19701:
19637:
19550:Schrödinger equation
19406:
19322:
19311:{\displaystyle f(x)}
19293:
19282:{\displaystyle d/dx}
19262:
19214:
19178:
19148:
19144:, for example. Then
19058:
19054:is invertible, then
19025:
19017:times is called the
18929:
18885:affine algebraic set
18838:
18812:
18752:
18679:
18623:
18502:
18460:
18400:
18373:
18352:multiplicative group
18308:
18263:
18224:
18182:
18083:
17961:
17952:function composition
17924:
17897:
17868:function composition
17625:
17553:
17502:
17441:
17404:
17305:
17252:) are algebraic. If
17250:multivalued function
17236:algebraic extensions
17130:
17077:
16948:
16894:
16867:
16832:
16805:
16737:
16679:
16593:
16533:
16458:
16415:
16336:
16243:
16036:
15920:
15644:
15525:
15342:
15262:
15226:
14888:
14843:
14793:
14749:
14722:
14628:
14561:
14535:
14488:
14456:
14421:
14232:
14180:
14124:
14108:{\displaystyle k=0.}
14093:
13968:
13939:
13935:real, the values of
13896:
13863:
13826:
13811:{\displaystyle \ln }
13802:
13752:
13686:
13636:
13558:
13511:
13463:
13452:{\displaystyle x+iy}
13434:
13396:
13355:
13325:{\displaystyle m=1,}
13307:
13303:values. In the case
13276:
13247:
13203:
13172:
13146:
13106:
13059:
13051:Different values of
12950:
12920:
12872:
12846:
12808:
12782:
12774:Multivalued function
12733:
12706:
12650:
12621:
12569:
12542:
12499:
12472:{\displaystyle z=0,}
12454:
12407:
12351:
12321:
12299:domain of definition
12255:
12246:multivalued function
12222:
12167:
12151:multivalued function
12130:
12099:
12056:
12034:principal primitive
11985:
11943:products of a given
11903:
11883:
11857:
11787:
11675:
11629:
11522:
11488:
11403:
11332:
11312:
11289:
11192:
11154:
11122:
11102:
11067:
11039:
10958:
10913:{\displaystyle 1/n,}
10893:
10708:
10582:
10544:
10477:
10396:
10339:
10277:
10237:
10176:
10097:
10011:
9949:
9876:
9790:
9758:
9698:
9645:
9607:
9533:
9494:
9271:
9197:
9192:exponential identity
9153:
9063:
9056:, one of them being
9005:
8966:
8932:
8896:
8892:is often defined as
8890:exponential function
8884:Exponential function
8878:Exponential function
8825:
8793:
8550:
8518:
8362:
8300:multivalued function
8235:
8204:exponential function
8202:of the base and the
8025:
7966:
7931:
7852:
7651:
7610:
7508:
7494:{\textstyle x^{p/q}}
7470:
7431:
7389:
7345:
7310:
7248:From top to bottom:
6522:
6463:
6435:
6415:
6379:
6359:
6339:
6290:
6264:
6244:
6224:
6209:{\displaystyle f(x)}
6191:
6171:
6115:
6090:
6054:
6034:
6014:
5965:
5945:
5919:
5915:odd. In general for
5899:
5879:
5853:
5829:
5803:
5758:
5744:Power functions for
5730:Power functions for
5413:according to above.
5389:
5385:, and this would be
5343:
5223:binary number system
5121:, so a kilometre is
5071:) can be written as
4885:-element set (or as
4570:
4495:
4424:
4203:
4141:
4110:in a multiplicative
4096:{\displaystyle m=-n}
4078:
4019:
3996:
3942:
3875:
3813:
3762:
3684:
3616:
3547:
3504:
3222:introduced the term
2908:Hippocrates of Chios
2863:originates from the
2756:
2721:
2689:
2643:
2616:
2596:
2557:
2511:
2491:
2451:
2414:
2394:
2370:
2316:
2283:
2239:
2188:
2161:
2077:
2047:
2011:
1894:
1861:
1797:
1770:
1698:
1678:
1652:
1625:
1415:
1393:
1373:
1346:
1326:
1190:
1042:
993:
956:
915:
875:
803:
726:
641:
603:
524:
487:
409:
372:
252:
27:Arithmetic operation
28802:Kaprekar's constant
28322:Colossally abundant
28209:Catalan pseudoprime
28109:Schröder–Hipparchus
27888:Centered octahedral
27764:Centered heptagonal
27754:Centered pentagonal
27744:Centered triangular
27344:and related numbers
27117:Super-logarithm (4)
27076:Root extraction (3)
26920:: 5. October 1983.
26830:Backus, John Warner
26826:Backus, John Warner
26753:Backus, John Warner
26291:(second ed.).
26122:Gottwald, Siegfried
26106:Hackbusch, Wolfgang
25865:(Online ed.).
25730:Arithmetica integra
25637:O'Connor, John J.;
25535:O'Connor, John J.;
25352:power associativity
24936:Hyperbolic function
24811:Irrational exponent
24471:Scientific notation
23800:, if it exists. So
23796:denotes always the
23356:functional notation
23089:that is defined on
22423:accumulation points
22421:has a limit at all
22402:, endowed with the
19566:fractional calculus
19562:fractional integral
19122:, where the matrix
17525:is defined only if
17383:algebraic structure
17356:is transcendental.
17297:In other words, if
17220:is a positive real
14436:{\displaystyle k=0}
13161:{\displaystyle a-b}
12032:, it is called the
11958:Geometrically, the
11918:{\displaystyle -i.}
11872:{\displaystyle -1;}
11516:permuted circularly
11447:continuous function
9869:means that one has
8858:continuous function
8848:for every positive
8341:limit of a sequence
6450:{\displaystyle n=1}
5706:, are described in
5498:limit of a sequence
5406:{\displaystyle 1/0}
5166:, since a set with
5103:based on powers of
5050:scientific notation
4997:Scientific notation
4971:(1), (2), (3), (4)
4403:order of operations
4116:algebraic structure
4108:invertible elements
3890:{\displaystyle n=0}
3800:algebraic structure
3256:algebraic functions
3139:in his text titled
3051:Henricus Grammateus
29220:Mathematics portal
29162:Aronson's sequence
28908:Smarandache–Wellin
28665:-dependent numbers
28372:Primitive abundant
28259:Strong pseudoprime
28249:Perrin pseudoprime
28229:Fermat pseudoprime
28169:Wolstenholme prime
27993:Squared triangular
27779:Centered decagonal
27774:Centered nonagonal
27769:Centered octagonal
27759:Centered hexagonal
27133:Ackermann function
27027:Exponentiation (3)
27022:Multiplication (2)
26604:(Part 1). London:
26503:Topologie générale
26501:Nicolas Bourbaki,
26390:Abel, Niels Henrik
26365:Weisstein, Eric W.
26353:Defined on p. 351.
26313:2007-09-30 at the
25968:Knobloch, Eberhard
25798:par soy mesme; Et
25792:, pour multiplier
25619:. World Wide Words
25617:"Zenzizenzizenzic"
25411:Weisstein, Eric W.
25331:
25304:{\displaystyle xy}
25301:
25278:
24990:algebraic solution
24441:Exponential growth
24417:Mathematics portal
24248:formula language.
24228:. This is because
24224:is interpreted as
23937:
23869:
23789:{\displaystyle -1}
23786:
23760:
23713:
23663:
23624:
23587:
23539:
23488:
23419:
23338:
23288:
23244:
23188:
23120:
23072:Iterated functions
23066:Subset sum problem
23028:
23001:
22954:can be reduced to
22812:
22342:
22271:
22218:indeterminate form
22175:Ackermann function
22133:
22097:
22062:
21997:
21916:
21857:
21816:
21757:In category theory
21688:
21641:
21614:
21556:
21527:infinite sequences
21515:
21461:
21441:
21418:
21355:
21286:
21240:
21191:
21125:topological spaces
21109:
21080:
21058:
21025:
20987:
20938:as the set of all
20924:
20891:
20850:
20804:
20739:
20710:
20684:
20659:is the set of the
20622:
20579:
20552:
20524:discrete logarithm
20493:
20444:
20415:
20383:field automorphism
20371:
20332:
20330:
20246:
20214:
20174:
20102:
20063:
20043:
20011:
19976:
19944:
19833:
19798:
19757:
19719:
19697:elements, denoted
19659:
19621:is a field with a
19526:
19377:
19308:
19279:
19227:
19194:
19164:
19105:
19038:
18970:
18877:algebraic geometry
18857:
18824:
18771:
18727:
18665:
18537:
18490:. If the order of
18468:
18422:and every integer
18412:
18386:
18333:
18294:
18249:
18210:
18165:
18063:
17940:
17910:
17829:
17827:
17590:
17515:
17476:
17426:
17332:
17301:is irrational and
17190:
17101:
17063:
16924:
16890:is a notation for
16880:
16853:
16828:is a notation for
16818:
16780:
16721:
16665:
16577:
16519:
16442:
16401:
16321:
16233:: For any integer
16177:
16022:
15859:
15857:
15581:
15509:
15275:
15244:
15212:
15210:
14874:
14824:
14778:
14735:
14718:So, all values of
14708:
14614:
14547:
14531:and the values of
14521:
14480:The polar form of
14469:
14433:
14407:
14218:
14166:
14105:
14079:
13954:
13917:
13881:
13859:Canonical form of
13841:
13808:
13788:
13716:
13672:
13622:
13532:
13476:
13449:
13409:
13370:
13322:
13289:
13262:
13222:
13187:
13158:
13132:
13072:
13031:
12933:
12902:
12858:
12821:
12794:
12756:
12719:
12681:
12633:
12607:
12555:
12526:
12469:
12437:
12379:
12330:
12280:
12234:
12205:
12156:In all cases, the
12077:
12014:
11955:th root of unity.
11915:
11889:
11869:
11832:
11761:
11661:
11609:Lagrange resolvent
11582:
11500:
11427:
11399:th root for which
11352:
11318:
11298:
11272:
11160:
11140:
11108:
11073:
11045:
11022:
10947:may be written in
10930:complex logarithms
10910:
10867:
10680:
10557:
10523:
10443:
10367:
10322:
10249:
10220:
10119:
10080:
9994:
9925:
9828:The definition of
9806:
9776:
9729:
9676:
9631:
9590:
9519:
9477:
9407:
9322:
9254:
9180:
9136:
9099:
9039:
8987:
8960:circular reasoning
8944:
8918:
8838:
8809:
8772:
8534:
8461:
8407:
8351:can be defined by
8333:
8279:
8159:
8008:
7952:
7894:
7825:
7637:
7593:
7491:
7444:
7411:
7360:
7331:
7292:
7240:Rational exponents
6534:
6499:
6447:
6421:
6401:
6365:
6345:
6321:
6276:
6250:
6230:
6206:
6177:
6148:
6096:
6076:
6040:
6020:
5996:
5951:
5931:
5905:
5885:
5865:
5835:
5815:
5789:
5752:
5738:
5704:indeterminate form
5614:alternate between
5403:
5367:
5187:Integer powers of
4758:
4542:
4475:
4368:
4366:
4160:
4132:invertible element
4126:(for example, the
4093:
4064:
4002:
3978:
3923:Negative exponents
3887:
3858:
3781:
3729:
3664:
3588:
3523:
3478:Positive exponents
3226:in 1696. The term
3193:, for example, as
3121:L'algèbre de Viète
2960:Islamic Golden Age
2872:present participle
2787:
2742:
2740:
2707:
2675:
2629:
2602:
2582:
2543:
2507:must be such that
2497:
2477:
2437:
2400:
2380:
2353:
2302:
2269:
2225:
2174:
2147:
2063:
2030:
1997:
1880:
1844:
1783:
1756:
1684:
1664:
1638:
1609:
1607:
1571:
1559:
1532:
1520:
1486:
1468:
1399:
1379:
1359:
1332:
1257:
1253:
1241:
1052:
1051:
1027:
1026:
966:
965:
941:
940:
927:
885:
884:
860:
859:
848:
845:
827:
775:
774:
769:
766:
755:
744:
710:
709:
698:
695:
692:
685:
671:
668:
661:
613:
612:
587:
586:
575:
572:
551:
497:
496:
472:
471:
460:
457:
436:
382:
381:
357:
356:
345:
342:
321:
300:
279:
196:
85:for various bases
29246:Binary operations
29228:
29227:
29206:
29205:
29175:
29174:
29139:
29138:
29103:
29102:
29067:
29066:
29027:
29026:
29023:
29022:
28842:
28841:
28652:
28651:
28541:
28540:
28537:
28536:
28483:Aliquot sequences
28294:Divisor functions
28267:
28266:
28239:Lucas pseudoprime
28219:Euler pseudoprime
28204:Carmichael number
28182:
28181:
28137:
28136:
28014:
28013:
28010:
28009:
28006:
28005:
27967:
27966:
27855:
27854:
27812:Square triangular
27704:
27703:
27651:
27650:
27603:
27602:
27537:
27536:
27489:
27488:
27427:
27426:
27294:
27293:
27161:
27160:
27054:for left argument
26845:. pp. 4, 6.
26757:Goldberg, Richard
26707:978-1-60206-714-1
26451:978-0-521-29324-2
26348:978-0-387-23234-8
26302:978-0-262-03293-3
26273:978-981-10-1789-6
26236:978-0-7637-7947-4
26149:978-3-658-00284-8
26136:Springer Spektrum
26126:Zeidler, Eberhard
26102:Zeidler, Eberhard
26077:978-0-521-19225-5
25953:978-0-8311-3086-2
25926:978-1-4665-6706-1
25873:(Subscription or
25735:Johannes Petreius
25564:Ball, W. W. Rouse
25483:978-1-4704-1554-9
25476:. p. 130, fn. 4.
25458:Rotman, Joseph J.
25243:
25242:
24786:Integer factorial
24761:Rational exponent
24436:Exponential field
24431:Exponential decay
24218:right-associative
23932:
23205:in the domain of
23078:Iterated function
22943:
22942:
22658:problematic when
22390:belonging to the
21926:can be rewritten
21585:In this context,
21135:Sets as exponents
20646:Cartesian product
19812:primitive element
19689:elements are all
19481:
19427:
18973:{\displaystyle k}
18781:for some integer
18354:is a set with as
17823:
17800:
16201:{1} = {(−1 ⋅ −1)}
16166:
16153:
16149:
16124:
16098:
16071:
16057:
16010:
15986:
15953:
15601:contain those of
15495:
14768:
14682:
14592:
13820:natural logarithm
13670:
13220:
12315:complex logarithm
12158:complex logarithm
12024:is the primitive
12011:
11892:{\displaystyle i}
11826:
11781:th roots of unity
11658:
11465:th root is real.
11266:
11246:
11230:
11163:{\displaystyle k}
10261:natural logarithm
9850:natural logarithm
9517:
9461:
9436:
9392:
9376:
9343:
9307:
9120:
9084:
8378:
8337:irrational number
8124:
8084:
8066:
7946:
7845:, by definition.
7819:
7788:
7740:
7669:
7631:
7577:
7556:
7523:
7442:
7358:
7298:is a nonnegative
7237:
7236:
6424:{\displaystyle n}
6368:{\displaystyle x}
6348:{\displaystyle x}
6253:{\displaystyle x}
6233:{\displaystyle x}
6180:{\displaystyle n}
6099:{\displaystyle n}
6043:{\displaystyle x}
6023:{\displaystyle x}
5954:{\displaystyle n}
5908:{\displaystyle n}
5888:{\displaystyle n}
5838:{\displaystyle n}
5335:th power of zero
5237:, separated by a
5213:; for example, a
5191:are important in
5069:metres per second
5007:In the base ten (
4983:
4982:
4853:is the number of
4727:
4633:
3976:
3497:The base case is
3466:Integer exponents
3026:15th–18th century
2937:The Sand Reckoner
2930:The Sand Reckoner
2924:The Sand Reckoner
2837:population growth
2833:compound interest
2764:
2739:
2605:{\displaystyle b}
2500:{\displaystyle r}
2469:
2459:
2435:
2403:{\displaystyle r}
2378:
1687:{\displaystyle n}
1568:
1538:
1536:
1529:
1499:
1497:
1483:
1447:
1445:
1402:{\displaystyle b}
1382:{\displaystyle n}
1335:{\displaystyle n}
1250:
1208:
1206:
1098:
1097:
1065:
1064:
1049:
1016:
1004:
963:
933:
931:
925:
882:
842:
837:
824:
819:
764:
753:
742:
693:
690:
683:
669:
666:
659:
610:
570:
560:
549:
539:
494:
455:
445:
434:
424:
379:
340:
330:
319:
309:
298:
288:
277:
267:
69:
68:
18:Pow function in C
16:(Redirected from
29258:
29251:Unary operations
29218:
29181:
29180:
29150:Natural language
29145:
29144:
29109:
29108:
29077:Generated via a
29073:
29072:
29033:
29032:
28938:Digit-reassembly
28903:Self-descriptive
28707:
28706:
28672:
28671:
28658:
28657:
28609:Lucas–Carmichael
28599:Harmonic divisor
28547:
28546:
28473:Sparsely totient
28448:Highly cototient
28357:Multiply perfect
28347:Highly composite
28290:
28289:
28273:
28272:
28188:
28187:
28143:
28142:
28124:Telephone number
28020:
28019:
27978:
27977:
27959:Square pyramidal
27941:Stella octangula
27866:
27865:
27732:
27731:
27723:
27722:
27715:Figurate numbers
27710:
27709:
27657:
27656:
27609:
27608:
27543:
27542:
27495:
27494:
27433:
27432:
27337:
27336:
27321:
27314:
27307:
27298:
27297:
27220:<1 attosecond
27188:
27181:
27174:
27165:
27164:
27126:Related articles
26991:
26984:
26977:
26968:
26967:
26961:
26960:
26942:
26936:
26935:
26933:
26932:
26904:
26898:
26897:
26895:
26894:
26885:pp. 41–42.
26868:
26862:
26860:
26858:
26857:
26851:
26840:
26822:
26816:
26815:
26807:
26801:
26799:
26797:
26796:
26790:
26779:
26749:
26743:
26742:
26724:
26718:
26717:
26715:
26714:
26684:
26678:
26666:
26664:
26663:
26640:
26634:
26633:
26588:
26582:
26581:
26570:
26564:
26563:
26561:
26560:
26554:
26548:. Archived from
26539:
26521:
26512:
26506:
26499:
26493:
26486:
26480:
26479:
26471:
26465:
26462:
26456:
26455:
26439:
26429:
26423:
26422:
26421:. Springer. I.2.
26414:
26408:
26407:
26385:
26379:
26378:
26377:
26360:
26354:
26352:
26324:
26318:
26306:
26284:
26278:
26277:
26247:
26241:
26240:
26222:
26216:
26215:
26199:
26189:
26180:
26179:
26166:Thomas' Calculus
26161:
26155:
26153:
26118:Hromkovič, Juraj
26098:
26092:
26089:
26049:
26043:
26042:
26003:
25997:
25996:
25964:
25958:
25957:
25937:
25931:
25930:
25910:
25904:
25903:
25886:
25880:
25878:
25870:
25858:
25850:
25844:
25842:
25836:
25830:
25824:
25818:
25812:
25809:
25803:
25797:
25791:
25762:
25756:
25755:
25745:
25739:
25738:
25721:
25715:
25714:
25707:
25701:
25700:
25679:
25673:
25672:
25662:
25656:
25655:
25634:
25628:
25627:
25625:
25624:
25613:Quinion, Michael
25609:
25600:
25588:
25582:
25581:
25560:
25554:
25553:
25532:
25523:
25522:
25494:
25488:
25487:
25467:
25454:
25445:
25444:
25437:
25431:
25430:
25429:
25427:
25426:
25406:
25400:
25399:
25397:
25396:
25381:
25355:
25350:More generally,
25348:
25342:
25340:
25338:
25337:
25332:
25310:
25308:
25307:
25302:
25287:
25285:
25284:
25279:
25256:
25120:Infinite product
25065:Special function
24736:Integer nth root
24711:Integer exponent
24521:
24514:
24507:
24498:
24497:
24419:
24414:
24413:
24397:
24393:
24389:
24383:
24379:
24375:
24369:
24365:
24361:
24345:statically typed
24334:
24324:
24314:
24304:
24298:
24286:
24267:
24257:
24235:
24231:
24227:
24223:
24207:
24197:
24179:
24069:
24065:
24061:
24053:
24006:Wolfram Language
23987:
23976:
23968:
23946:
23944:
23943:
23938:
23933:
23931:
23917:
23897:
23878:
23876:
23875:
23870:
23841:
23840:
23819:
23818:
23798:inverse function
23795:
23793:
23792:
23787:
23769:
23767:
23766:
23761:
23722:
23720:
23719:
23714:
23672:
23670:
23669:
23664:
23650:
23649:
23633:
23631:
23630:
23625:
23617:
23616:
23596:
23594:
23593:
23588:
23565:
23564:
23548:
23546:
23545:
23540:
23517:
23516:
23497:
23495:
23494:
23489:
23454:
23453:
23428:
23426:
23425:
23420:
23382:
23381:
23347:
23345:
23344:
23339:
23297:
23295:
23294:
23289:
23278:
23277:
23261:
23257:
23253:
23251:
23250:
23245:
23243:
23242:
23224:
23219:
23215:
23208:
23204:
23197:
23195:
23194:
23189:
23129:
23127:
23126:
23121:
23087:binary operation
23063:
23049:
23041:
23037:
23035:
23034:
23029:
23010:
23008:
23007:
23002:
22982:
22981:
22953:
22939:
22938:
22935:
22932:
22929:
22926:
22923:
22920:
22917:
22914:
22911:
22901:
22900:
22897:
22894:
22891:
22888:
22878:
22877:
22874:
22864:
22863:
22860:
22850:
22829:
22828:
22821:
22819:
22818:
22813:
22793:
22792:
22774:
22773:
22761:
22760:
22748:
22747:
22735:
22734:
22705:
22701:
22694:
22680:
22676:
22670:
22663:
22657:
22651:
22647:
22641:
22635:
22626:
22620:
22613:
22601:
22594:
22588:
22578:
22568:
22560:
22556:
22550:
22542:
22538:
22532:
22524:
22517:
22508:
22500:
22493:
22480:
22476:
22472:
22468:
22464:
22460:
22452:
22446:
22442:
22438:
22434:
22430:
22420:
22411:
22404:product topology
22401:
22399:
22389:
22383:
22377:
22365:
22364:
22357:
22351:
22349:
22348:
22343:
22326:
22325:
22320:
22280:
22278:
22277:
22272:
22270:
22269:
22229:
22225:
22209:Limits of powers
22205:) respectively.
22204:
22200:
22199:
22196:
22193:
22190:
22184:
22146:
22142:
22140:
22139:
22134:
22106:
22104:
22103:
22098:
22071:
22069:
22068:
22063:
22061:
22060:
22035:in which finite
22027:
22026:
22016:
22015:
22006:
22004:
22003:
21998:
21960:
21959:
21925:
21923:
21922:
21917:
21915:
21914:
21896:
21895:
21886:
21885:
21867:The isomorphism
21866:
21864:
21863:
21858:
21825:
21823:
21822:
21817:
21815:
21814:
21799:that is denoted
21798:
21794:
21790:
21786:
21782:
21778:
21769:category of sets
21752:
21746:
21744:
21736:
21734:
21728:
21713:
21705:
21697:
21695:
21694:
21689:
21662:
21658:
21650:
21648:
21647:
21642:
21640:
21639:
21623:
21621:
21620:
21615:
21588:
21573:
21565:
21563:
21562:
21557:
21555:
21554:
21550:
21541:
21524:
21522:
21521:
21516:
21514:
21513:
21512:
21506:
21470:
21468:
21467:
21462:
21450:
21448:
21447:
21442:
21427:
21425:
21424:
21419:
21414:
21413:
21401:
21400:
21388:
21387:
21364:
21362:
21361:
21356:
21351:
21350:
21332:
21331:
21322:
21321:
21295:
21293:
21292:
21287:
21285:
21284:
21268:
21264:
21260:
21256:
21249:
21247:
21246:
21241:
21204:
21200:
21198:
21197:
21192:
21187:
21186:
21168:
21167:
21148:
21118:
21116:
21115:
21110:
21105:
21093:
21089:
21087:
21086:
21081:
21079:
21067:
21065:
21064:
21059:
21057:
21056:
21051:
21034:
21032:
21031:
21026:
21024:
21023:
21007:
21000:
20996:
20994:
20993:
20988:
20983:
20982:
20964:
20963:
20941:
20937:
20933:
20931:
20930:
20925:
20923:
20922:
20906:
20900:
20898:
20897:
20892:
20859:
20857:
20856:
20851:
20813:
20811:
20810:
20805:
20748:
20746:
20745:
20740:
20719:
20717:
20716:
20711:
20693:
20691:
20690:
20685:
20658:
20654:
20635:
20631:
20629:
20628:
20623:
20621:
20620:
20604:
20600:
20596:
20588:
20586:
20585:
20580:
20578:
20577:
20561:
20559:
20558:
20553:
20548:
20547:
20542:
20529:
20510:
20502:
20500:
20499:
20494:
20492:
20491:
20486:
20469:
20461:
20457:
20453:
20451:
20450:
20445:
20443:
20442:
20437:
20424:
20422:
20421:
20416:
20411:
20410:
20380:
20378:
20377:
20372:
20367:
20366:
20361:
20341:
20339:
20338:
20333:
20331:
20327:
20326:
20310:
20306:
20305:
20300:
20291:
20290:
20285:
20277:
20255:
20253:
20252:
20247:
20242:
20241:
20236:
20223:
20221:
20220:
20215:
20207:
20206:
20190:
20183:
20181:
20180:
20175:
20173:
20172:
20160:
20159:
20147:
20146:
20114:freshman's dream
20111:
20109:
20108:
20103:
20098:
20097:
20092:
20072:
20070:
20069:
20064:
20052:
20050:
20049:
20044:
20039:
20038:
20033:
20020:
20018:
20017:
20012:
19985:
19983:
19982:
19977:
19972:
19971:
19966:
19953:
19951:
19950:
19945:
19934:
19933:
19921:
19920:
19896:
19895:
19877:
19876:
19857:
19854:first powers of
19853:
19846:
19842:
19840:
19839:
19834:
19832:
19831:
19826:
19807:
19805:
19804:
19799:
19794:
19793:
19788:
19766:
19764:
19763:
19758:
19750:
19749:
19728:
19726:
19725:
19720:
19715:
19714:
19709:
19696:
19688:
19684:
19680:
19676:
19672:
19668:
19666:
19665:
19660:
19655:
19654:
19608:rational numbers
19597:
19535:
19533:
19532:
19527:
19513:
19512:
19482:
19480:
19479:
19478:
19465:
19464:
19455:
19438:
19437:
19432:
19428:
19426:
19415:
19398:
19392:
19386:
19384:
19383:
19378:
19367:
19335:
19317:
19315:
19314:
19309:
19288:
19286:
19285:
19280:
19272:
19252:linear operators
19242:
19236:
19234:
19233:
19228:
19226:
19225:
19209:
19203:
19201:
19200:
19195:
19190:
19189:
19173:
19171:
19170:
19165:
19160:
19159:
19139:
19133:
19127:
19114:
19112:
19111:
19106:
19104:
19103:
19098:
19094:
19093:
19073:
19072:
19053:
19047:
19045:
19044:
19039:
19037:
19036:
19016:
19010:
19004:
18986:
18979:
18977:
18976:
18971:
18966:
18965:
18947:
18946:
18913:
18905:
18901:
18897:
18893:
18870:
18866:
18864:
18863:
18858:
18850:
18849:
18833:
18831:
18830:
18825:
18791:commutative ring
18784:
18780:
18778:
18777:
18772:
18764:
18763:
18736:
18734:
18733:
18728:
18723:
18722:
18713:
18712:
18700:
18699:
18674:
18672:
18671:
18666:
18664:
18663:
18648:
18647:
18638:
18637:
18617:
18611:
18605:
18566:
18562:
18558:
18555:first powers of
18554:
18551:consists of the
18550:
18546:
18544:
18543:
18538:
18527:
18526:
18514:
18513:
18497:
18493:
18489:
18477:
18475:
18474:
18469:
18467:
18448:
18444:
18436:
18425:
18421:
18419:
18418:
18413:
18395:
18393:
18392:
18387:
18385:
18384:
18368:
18360:identity element
18342:
18340:
18339:
18334:
18320:
18319:
18303:
18301:
18300:
18295:
18281:
18280:
18258:
18256:
18255:
18250:
18245:
18244:
18219:
18217:
18216:
18211:
18197:
18196:
18174:
18172:
18171:
18166:
18101:
18100:
18072:
18070:
18069:
18064:
18016:
18015:
17976:
17975:
17949:
17947:
17946:
17941:
17939:
17938:
17919:
17917:
17916:
17911:
17909:
17908:
17888:
17838:
17836:
17835:
17830:
17828:
17824:
17821:
17801:
17798:
17795:
17794:
17785:
17784:
17768:
17767:
17745:
17744:
17725:
17724:
17715:
17714:
17698:
17697:
17688:
17687:
17671:
17670:
17641:
17640:
17617:
17613:
17609:
17605:
17599:
17597:
17596:
17591:
17586:
17585:
17577:
17573:
17572:
17548:
17542:
17536:
17528:
17524:
17522:
17521:
17516:
17514:
17513:
17497:
17489:
17485:
17483:
17482:
17477:
17475:
17474:
17459:
17458:
17435:
17433:
17432:
17427:
17416:
17415:
17396:
17388:
17373:
17355:
17349:
17345:
17341:
17339:
17338:
17333:
17300:
17293:
17289:
17285:
17277:
17271:
17267:
17255:
17247:
17241:
17233:
17227:
17222:algebraic number
17219:
17199:
17197:
17196:
17191:
17186:
17185:
17184:
17183:
17157:
17156:
17151:
17147:
17146:
17125:
17114:
17110:
17108:
17107:
17102:
17072:
17070:
17069:
17064:
17026:
17022:
16943:
16933:
16931:
16930:
16925:
16889:
16887:
16886:
16881:
16879:
16878:
16862:
16860:
16859:
16854:
16827:
16825:
16824:
16819:
16817:
16816:
16800:
16793:
16789:
16787:
16786:
16781:
16772:
16771:
16770:
16769:
16760:
16759:
16730:
16728:
16727:
16722:
16720:
16719:
16710:
16709:
16697:
16696:
16674:
16672:
16671:
16666:
16657:
16656:
16655:
16654:
16645:
16644:
16624:
16623:
16605:
16604:
16586:
16584:
16583:
16578:
16576:
16575:
16560:
16559:
16554:
16550:
16549:
16528:
16526:
16525:
16520:
16511:
16510:
16509:
16508:
16499:
16498:
16451:
16449:
16448:
16443:
16410:
16408:
16407:
16402:
16393:
16392:
16372:
16368:
16367:
16330:
16328:
16327:
16322:
16302:
16301:
16283:
16282:
16270:
16269:
16236:
16224:
16220:
16216:
16202:
16198:
16194:
16190:
16186:
16184:
16183:
16178:
16167:
16159:
16154:
16152:
16151:
16150:
16142:
16126:
16125:
16117:
16111:
16100:
16099:
16091:
16073:
16072:
16064:
16062:
16058:
16056:
16045:
16031:
16029:
16028:
16023:
16012:
16011:
16003:
15988:
15987:
15979:
15955:
15954:
15946:
15915:
15911:
15907:
15903:
15884:
15868:
15866:
15865:
15860:
15858:
15854:
15850:
15849:
15790:
15786:
15764:
15760:
15759:
15676:
15672:
15671:
15670:
15638:
15634:
15630:
15622:
15610:
15600:
15590:
15588:
15587:
15582:
15580:
15543:
15542:
15518:
15516:
15515:
15510:
15496:
15491:
15480:
15468:
15467:
15463:
15372:
15371:
15335:principal branch
15332:
15328:
15324:
15292:
15284:
15282:
15281:
15276:
15274:
15273:
15253:
15251:
15250:
15245:
15221:
15219:
15218:
15213:
15211:
15144:
15143:
15110:
15109:
15091:
14973:
14972:
14939:
14938:
14922:
14921:
14883:
14881:
14880:
14875:
14870:
14869:
14838:
14833:
14831:
14830:
14825:
14823:
14822:
14787:
14785:
14784:
14779:
14771:
14770:
14769:
14761:
14744:
14742:
14741:
14736:
14734:
14733:
14717:
14715:
14714:
14709:
14704:
14703:
14685:
14684:
14683:
14675:
14662:
14661:
14640:
14639:
14624:It follows that
14623:
14621:
14620:
14615:
14610:
14606:
14593:
14585:
14556:
14554:
14553:
14548:
14530:
14528:
14527:
14522:
14517:
14516:
14512:
14483:
14478:
14476:
14475:
14470:
14468:
14467:
14442:
14440:
14439:
14434:
14416:
14414:
14413:
14408:
14403:
14399:
14291:
14290:
14257:
14256:
14244:
14243:
14227:
14225:
14224:
14219:
14214:
14213:
14201:
14200:
14175:
14173:
14172:
14167:
14165:
14164:
14155:
14154:
14142:
14141:
14114:
14112:
14111:
14106:
14088:
14086:
14085:
14080:
13963:
13961:
13960:
13955:
13934:
13930:
13926:
13924:
13923:
13918:
13890:
13888:
13887:
13882:
13854:
13851:for any integer
13850:
13848:
13847:
13842:
13817:
13815:
13814:
13809:
13797:
13795:
13794:
13789:
13742:
13725:
13723:
13722:
13717:
13681:
13679:
13678:
13673:
13671:
13669:
13668:
13656:
13655:
13646:
13631:
13629:
13628:
13623:
13582:
13581:
13553:
13549:
13545:
13541:
13539:
13538:
13533:
13505:
13493:
13489:
13485:
13483:
13482:
13477:
13475:
13474:
13458:
13456:
13455:
13450:
13418:
13416:
13415:
13410:
13408:
13407:
13391:
13387:
13379:
13377:
13376:
13371:
13343:
13337:
13331:
13329:
13328:
13323:
13302:
13298:
13296:
13295:
13290:
13288:
13287:
13271:
13269:
13268:
13263:
13241:coprime integers
13239:
13235:
13231:
13229:
13228:
13223:
13221:
13213:
13196:
13194:
13193:
13188:
13167:
13165:
13164:
13159:
13141:
13139:
13138:
13133:
13131:
13130:
13118:
13117:
13097:
13093:
13085:
13081:
13079:
13078:
13073:
13071:
13070:
13054:
13048:is any integer.
13047:
13040:
13038:
13037:
13032:
13027:
13026:
13005:
13004:
12992:
12991:
12942:
12940:
12939:
12934:
12932:
12931:
12915:
12911:
12909:
12908:
12903:
12867:
12865:
12864:
12859:
12834:
12830:
12828:
12827:
12822:
12820:
12819:
12803:
12801:
12800:
12795:
12769:
12765:
12763:
12762:
12757:
12749:
12728:
12726:
12725:
12720:
12718:
12717:
12701:
12694:
12690:
12688:
12687:
12682:
12680:
12679:
12642:
12640:
12639:
12634:
12616:
12614:
12613:
12608:
12603:
12602:
12581:
12580:
12564:
12562:
12561:
12556:
12554:
12553:
12535:
12533:
12532:
12527:
12494:
12486:
12478:
12476:
12475:
12470:
12446:
12444:
12443:
12438:
12399:
12388:
12386:
12385:
12380:
12369:
12368:
12343:
12339:
12337:
12336:
12331:
12296:
12289:
12287:
12286:
12281:
12273:
12272:
12243:
12241:
12240:
12235:
12214:
12212:
12211:
12206:
12201:
12200:
12179:
12178:
12149:is defined as a
12148:
12146:
12145:
12140:
12139:
12125:
12117:
12115:
12114:
12109:
12108:
12086:
12084:
12083:
12078:
12076:
12075:
12071:
12046:th root of unity
12045:
12038:th root of unity
12037:
12027:
12023:
12021:
12020:
12015:
12013:
12012:
12007:
11993:
11975:
11961:
11954:
11950:
11946:
11942:
11938:
11934:
11930:
11924:
11922:
11921:
11916:
11898:
11896:
11895:
11890:
11878:
11876:
11875:
11870:
11852:
11845:
11841:
11839:
11838:
11833:
11828:
11827:
11822:
11808:
11799:
11798:
11780:
11774:
11770:
11768:
11767:
11762:
11757:
11756:
11738:
11737:
11725:
11724:
11706:
11705:
11693:
11692:
11670:
11668:
11667:
11662:
11660:
11659:
11654:
11643:
11625:first powers of
11624:
11620:
11617:
11602:
11598:
11591:
11587:
11580:
11558:
11554:
11552:
11551:
11546:
11545:
11541:
11513:
11509:
11507:
11506:
11501:
11480:
11472:
11464:
11460:
11456:
11444:
11440:
11436:
11434:
11433:
11428:
11398:
11390:
11387:
11380:
11377:
11373:
11369:
11365:
11361:
11359:
11358:
11353:
11348:
11327:
11325:
11324:
11319:
11307:
11305:
11304:
11299:
11281:
11279:
11278:
11273:
11268:
11267:
11262:
11254:
11247:
11245:
11237:
11232:
11231:
11223:
11221:
11217:
11216:
11215:
11184:
11180:
11176:
11169:
11167:
11166:
11161:
11149:
11147:
11146:
11141:
11117:
11115:
11114:
11109:
11097:
11096:
11082:
11080:
11079:
11074:
11062:
11054:
11052:
11051:
11046:
11031:
11029:
11028:
11023:
10982:
10981:
10946:
10938:
10927:
10923:
10919:
10917:
10916:
10911:
10903:
10888:
10876:
10874:
10873:
10868:
10803:
10802:
10790:
10789:
10768:
10767:
10755:
10754:
10742:
10741:
10729:
10728:
10689:
10687:
10686:
10681:
10616:
10615:
10603:
10602:
10574:
10567:in terms of the
10566:
10564:
10563:
10558:
10556:
10555:
10532:
10530:
10529:
10524:
10492:
10491:
10463:
10459:
10452:
10450:
10449:
10444:
10439:
10438:
10423:
10422:
10417:
10413:
10412:
10382:
10376:
10374:
10373:
10368:
10366:
10365:
10360:
10356:
10355:
10331:
10329:
10328:
10323:
10318:
10317:
10308:
10307:
10295:
10294:
10266:
10258:
10256:
10255:
10250:
10229:
10227:
10226:
10221:
10216:
10215:
10188:
10187:
10162:
10154:
10150:
10138:
10134:
10128:
10126:
10125:
10120:
10118:
10117:
10089:
10087:
10086:
10081:
10079:
10078:
10057:
10056:
10051:
10047:
10046:
10023:
10022:
10003:
10001:
10000:
9995:
9990:
9989:
9974:
9973:
9964:
9963:
9944:
9934:
9932:
9931:
9926:
9924:
9923:
9868:
9858:
9843:
9839:
9833:
9819:
9815:
9813:
9812:
9807:
9802:
9801:
9785:
9783:
9782:
9777:
9753:
9742:
9738:
9736:
9735:
9730:
9728:
9727:
9693:
9689:
9685:
9683:
9682:
9677:
9675:
9674:
9640:
9638:
9637:
9632:
9599:
9597:
9596:
9591:
9528:
9526:
9525:
9520:
9518:
9516:
9515:
9506:
9498:
9486:
9484:
9483:
9478:
9473:
9472:
9467:
9463:
9462:
9460:
9459:
9450:
9442:
9437:
9432:
9421:
9406:
9388:
9387:
9382:
9378:
9377:
9369:
9355:
9354:
9349:
9345:
9344:
9336:
9321:
9263:
9261:
9260:
9255:
9189:
9187:
9186:
9181:
9145:
9143:
9142:
9137:
9132:
9131:
9126:
9122:
9121:
9113:
9098:
9048:
9046:
9045:
9040:
9035:
9034:
8996:
8994:
8993:
8988:
8953:
8951:
8950:
8945:
8927:
8925:
8924:
8919:
8914:
8913:
8867:
8863:
8855:
8851:
8847:
8845:
8844:
8839:
8837:
8836:
8818:
8816:
8815:
8810:
8805:
8804:
8781:
8779:
8778:
8773:
8765:
8761:
8760:
8759:
8747:
8746:
8729:
8725:
8724:
8723:
8711:
8710:
8693:
8689:
8688:
8687:
8675:
8674:
8657:
8653:
8652:
8651:
8639:
8638:
8621:
8617:
8616:
8615:
8603:
8602:
8585:
8581:
8580:
8579:
8567:
8566:
8543:
8541:
8540:
8535:
8530:
8529:
8509:
8498:
8497:
8489:For example, if
8485:
8481:
8477:
8470:
8468:
8467:
8462:
8454:
8439:
8438:
8433:
8417:
8416:
8406:
8396:
8374:
8373:
8350:
8346:
8330:
8326:
8319:
8288:
8286:
8285:
8280:
8278:
8277:
8262:
8261:
8256:
8252:
8251:
8168:
8166:
8165:
8160:
8149:
8148:
8127:
8126:
8125:
8117:
8086:
8085:
8077:
8068:
8067:
8059:
8057:
8053:
8052:
8051:
8017:
8015:
8014:
8009:
8007:
8006:
7991:
7990:
7981:
7980:
7961:
7959:
7958:
7953:
7948:
7947:
7939:
7926:
7922:
7914:
7910:
7903:
7901:
7900:
7895:
7893:
7892:
7877:
7876:
7867:
7866:
7844:
7840:
7834:
7832:
7831:
7826:
7821:
7820:
7812:
7806:
7805:
7790:
7789:
7781:
7779:
7775:
7774:
7773:
7764:
7763:
7742:
7741:
7733:
7731:
7727:
7726:
7725:
7716:
7715:
7694:
7693:
7681:
7680:
7671:
7670:
7662:
7646:
7644:
7643:
7638:
7633:
7632:
7624:
7602:
7600:
7599:
7594:
7589:
7588:
7579:
7578:
7570:
7558:
7557:
7549:
7547:
7543:
7542:
7525:
7524:
7516:
7500:
7498:
7497:
7492:
7490:
7489:
7485:
7465:
7461:
7453:
7451:
7450:
7445:
7443:
7435:
7426:
7420:
7418:
7417:
7412:
7401:
7400:
7384:
7380:
7374:
7369:
7367:
7366:
7361:
7359:
7357:
7349:
7340:
7338:
7337:
7332:
7330:
7329:
7325:
7305:
7297:
7289:
7283:
7277:
7271:
7265:
7259:
7253:
7233:
7232:
7229:
7226:
7219:
7218:
7215:
7212:
7205:
7204:
7201:
7194:
7193:
7190:
7183:
7182:
7179:
7172:
7171:
7164:
7163:
7143:
7142:
7139:
7136:
7129:
7128:
7125:
7118:
7117:
7114:
7107:
7106:
7103:
7096:
7095:
7088:
7087:
7064:
7063:
7060:
7057:
7050:
7049:
7046:
7039:
7038:
7035:
7028:
7027:
7024:
7017:
7016:
7009:
7008:
6985:
6984:
6981:
6974:
6973:
6970:
6963:
6962:
6959:
6952:
6951:
6944:
6943:
6936:
6935:
6912:
6911:
6908:
6901:
6900:
6897:
6890:
6889:
6886:
6879:
6878:
6871:
6870:
6863:
6842:
6841:
6838:
6831:
6830:
6827:
6820:
6819:
6812:
6811:
6804:
6803:
6777:
6776:
6773:
6766:
6765:
6758:
6757:
6750:
6749:
6742:
6718:
6717:
6710:
6709:
6702:
6697:
6552:
6551:
6543:
6541:
6540:
6535:
6510:
6508:
6506:
6505:
6500:
6456:
6454:
6453:
6448:
6430:
6428:
6427:
6422:
6410:
6408:
6407:
6402:
6400:
6399:
6374:
6372:
6371:
6366:
6354:
6352:
6351:
6346:
6335:with increasing
6330:
6328:
6327:
6322:
6320:
6319:
6285:
6283:
6282:
6277:
6259:
6257:
6256:
6251:
6239:
6237:
6236:
6231:
6215:
6213:
6212:
6207:
6186:
6184:
6183:
6178:
6159:
6157:
6155:
6154:
6149:
6105:
6103:
6102:
6097:
6085:
6083:
6082:
6077:
6075:
6074:
6049:
6047:
6046:
6041:
6029:
6027:
6026:
6021:
6010:with increasing
6005:
6003:
6002:
5997:
5995:
5994:
5960:
5958:
5957:
5952:
5940:
5938:
5937:
5932:
5914:
5912:
5911:
5906:
5894:
5892:
5891:
5886:
5874:
5872:
5871:
5866:
5844:
5842:
5841:
5836:
5824:
5822:
5821:
5816:
5798:
5796:
5795:
5790:
5788:
5787:
5750:
5736:
5689:
5682:
5668:
5661:
5655:
5649:
5643:
5633:
5627:
5621:
5617:
5613:
5606:
5599:
5593:
5580:
5578:
5570:
5563:
5531:to the power of
5523:
5516:
5509:
5481:
5471:
5464:
5460:
5456:
5452:
5443:
5439:
5425:
5420:
5412:
5410:
5409:
5404:
5399:
5384:
5376:
5374:
5373:
5368:
5366:
5365:
5353:
5338:
5334:
5330:
5323:
5320:If the exponent
5316:
5312:
5308:
5301:
5298:If the exponent
5289:
5276:
5272:
5260:
5256:
5252:
5248:
5244:
5236:
5232:
5228:
5220:
5204:
5198:
5193:computer science
5190:
5183:
5171:
5161:
5142:
5124:
5120:
5119:
5115:
5106:
5096:
5094:
5084:
5082:
5079:
5076:
5062:
5060:
5057:
5047:
5038:
5037:
5033:
5028:
5027:
5023:
5018:
5014:
4986:Particular bases
4931:
4923:
4919:
4911:
4904:
4903:
4900:
4896:
4892:
4888:
4884:
4876:
4868:
4864:
4852:
4846:
4842:
4820:
4816:
4803:computer algebra
4796:
4792:
4789:. Otherwise, if
4780:
4767:
4765:
4764:
4759:
4754:
4753:
4738:
4737:
4728:
4726:
4700:
4692:
4689:
4684:
4666:
4665:
4650:
4649:
4640:
4639:
4638:
4625:
4617:
4612:
4594:
4593:
4562:binomial formula
4551:
4549:
4548:
4543:
4538:
4537:
4522:
4521:
4516:
4512:
4511:
4484:
4482:
4481:
4476:
4471:
4470:
4469:
4465:
4464:
4443:
4442:
4441:
4440:
4400:
4396:
4388:
4377:
4375:
4374:
4369:
4367:
4363:
4362:
4350:
4349:
4333:
4332:
4307:
4306:
4284:
4283:
4278:
4274:
4273:
4255:
4254:
4242:
4241:
4225:
4224:
4194:
4193:
4169:
4167:
4166:
4161:
4156:
4155:
4136:
4125:
4102:
4100:
4099:
4094:
4073:
4071:
4070:
4065:
4063:
4062:
4050:
4049:
4037:
4036:
4011:
4009:
4008:
4003:
3987:
3985:
3984:
3979:
3977:
3975:
3974:
3962:
3957:
3956:
3934:
3930:
3919:
3911:
3907:
3903:
3896:
3894:
3893:
3888:
3867:
3865:
3864:
3859:
3857:
3856:
3844:
3843:
3831:
3830:
3790:
3788:
3787:
3782:
3774:
3773:
3754:
3750:
3738:
3736:
3735:
3730:
3725:
3724:
3709:
3708:
3699:
3698:
3673:
3671:
3670:
3665:
3660:
3659:
3647:
3646:
3634:
3633:
3608:
3604:
3597:
3595:
3594:
3589:
3578:
3577:
3565:
3564:
3532:
3530:
3529:
3524:
3516:
3515:
3494:multiplication:
3434:to the power of
3429:
3423:
3408:
3404:
3400:
3396:
3392:
3387:positive integer
3381:
3375:
3369:
3363:
3353:
3332:
3326:
3320:
3314:
3308:
3302:
3296:
3290:
3280:
3210:
3185:
3178:
3172:
3166:
3160:
3154:
3149:I designate ...
3134:
3128:
3101:zenzizenzizenzic
3078:
3071:
3070:
3069:
3066:
3048:
3041:
2994:, "cube") for a
2955:
2951:
2947:
2829:computer science
2796:
2794:
2793:
2788:
2786:
2785:
2781:
2765:
2760:
2751:
2749:
2748:
2743:
2741:
2732:
2716:
2714:
2713:
2708:
2684:
2682:
2681:
2676:
2674:
2673:
2661:
2660:
2638:
2636:
2635:
2630:
2628:
2627:
2611:
2609:
2608:
2603:
2591:
2589:
2588:
2583:
2575:
2574:
2552:
2550:
2549:
2544:
2536:
2535:
2523:
2522:
2506:
2504:
2503:
2498:
2486:
2484:
2483:
2478:
2470:
2465:
2460:
2455:
2446:
2444:
2443:
2438:
2436:
2431:
2426:
2425:
2409:
2407:
2406:
2401:
2389:
2387:
2386:
2381:
2379:
2374:
2362:
2360:
2359:
2354:
2352:
2351:
2342:
2331:
2330:
2311:
2309:
2308:
2303:
2295:
2294:
2278:
2276:
2275:
2270:
2265:
2254:
2253:
2234:
2232:
2231:
2226:
2224:
2223:
2214:
2203:
2202:
2183:
2181:
2180:
2175:
2173:
2172:
2156:
2154:
2153:
2148:
2140:
2139:
2127:
2126:
2105:
2104:
2092:
2091:
2072:
2070:
2069:
2064:
2062:
2061:
2039:
2037:
2036:
2031:
2023:
2022:
2006:
2004:
2003:
1998:
1996:
1995:
1983:
1982:
1958:
1957:
1945:
1944:
1932:
1931:
1919:
1918:
1909:
1908:
1889:
1887:
1886:
1881:
1873:
1872:
1853:
1851:
1850:
1845:
1837:
1836:
1827:
1822:
1821:
1809:
1808:
1792:
1790:
1789:
1784:
1782:
1781:
1765:
1763:
1762:
1757:
1755:
1754:
1742:
1741:
1723:
1722:
1710:
1709:
1693:
1691:
1690:
1685:
1673:
1671:
1670:
1665:
1647:
1645:
1644:
1639:
1637:
1636:
1618:
1616:
1615:
1610:
1608:
1604:
1603:
1591:
1590:
1575:
1570:
1569:
1566:
1560:
1555:
1531:
1530:
1527:
1521:
1516:
1490:
1485:
1484:
1481:
1469:
1464:
1437:
1436:
1408:
1406:
1405:
1400:
1388:
1386:
1385:
1380:
1368:
1366:
1365:
1360:
1358:
1357:
1341:
1339:
1338:
1333:
1278:
1266:
1264:
1263:
1258:
1252:
1251:
1248:
1242:
1237:
1202:
1201:
1185:
1177:
1163:
1159:
1155:
1147:
1139:
1135:
1090:
1083:
1076:
1061:
1059:
1058:
1053:
1050:
1047:
1036:
1034:
1033:
1028:
1017:
1014:
1006:
1005:
1002:
975:
973:
972:
967:
964:
961:
950:
948:
947:
942:
934:
932:
929:
926:
923:
920:
894:
892:
891:
886:
883:
880:
869:
867:
866:
861:
853:
849:
844:
843:
840:
838:
835:
826:
825:
822:
820:
817:
784:
782:
781:
776:
773:
770:
765:
762:
754:
751:
743:
740:
719:
717:
716:
711:
703:
699:
694:
691:
688:
684:
681:
678:
670:
667:
664:
660:
657:
654:
622:
620:
619:
614:
611:
608:
596:
594:
593:
588:
580:
576:
571:
568:
561:
558:
550:
547:
540:
537:
506:
504:
503:
498:
495:
492:
481:
479:
478:
473:
465:
461:
456:
453:
446:
443:
435:
432:
425:
422:
391:
389:
388:
383:
380:
377:
366:
364:
363:
358:
350:
346:
341:
338:
331:
328:
320:
317:
310:
307:
299:
296:
289:
286:
278:
275:
268:
265:
236:
235:
226:
219:
212:
205:
198:
197:
193:
189:
183:
176:
172:
168:
164:
163:
161:
160:
158:
157:
154:
151:
144: base
143:
138:
137:
134:
128:
123:
122:
119:
111:
106:
105:
102:
96:
90:
84:
46:
39:
38:
21:
29266:
29265:
29261:
29260:
29259:
29257:
29256:
29255:
29231:
29230:
29229:
29224:
29202:
29198:Strobogrammatic
29189:
29171:
29153:
29135:
29117:
29099:
29081:
29063:
29040:
29019:
29003:
28962:Divisor-related
28957:
28917:
28868:
28838:
28775:
28759:
28738:
28705:
28678:
28666:
28648:
28560:
28559:related numbers
28533:
28510:
28477:
28468:Perfect totient
28434:
28411:
28342:Highly abundant
28284:
28263:
28195:
28178:
28150:
28133:
28119:Stirling second
28025:
28002:
27963:
27945:
27902:
27851:
27788:
27749:Centered square
27717:
27700:
27662:
27647:
27614:
27599:
27551:
27550:defined numbers
27533:
27500:
27485:
27456:Double Mersenne
27442:
27423:
27345:
27331:
27329:natural numbers
27325:
27295:
27290:
27259:Positive powers
27254:
27213:Negative powers
27208:
27197:
27192:
27162:
27157:
27121:
27102:Subtraction (1)
27097:Predecessor (0)
27085:
27066:Subtraction (1)
27061:Predecessor (0)
27046:
27000:
26998:Hyperoperations
26995:
26965:
26964:
26957:
26943:
26939:
26930:
26928:
26906:
26905:
26901:
26892:
26890:
26869:
26865:
26855:
26853:
26849:
26838:
26823:
26819:
26808:
26804:
26794:
26792:
26788:
26777:
26750:
26746:
26739:
26725:
26721:
26712:
26710:
26708:
26688:Cajori, Florian
26685:
26681:
26677:'s older work.)
26661:
26659:
26641:
26637:
26589:
26585:
26574:Peano, Giuseppe
26571:
26567:
26558:
26556:
26552:
26519:
26513:
26509:
26500:
26496:
26487:
26483:
26472:
26468:
26463:
26459:
26452:
26430:
26426:
26415:
26411:
26386:
26382:
26361:
26357:
26349:
26325:
26321:
26315:Wayback Machine
26308:Online resource
26303:
26285:
26281:
26274:
26248:
26244:
26237:
26223:
26219:
26212:
26190:
26183:
26176:
26162:
26158:
26154:(xii+635 pages)
26150:
26142:. p. 590.
26099:
26095:
26078:
26050:
26046:
26039:
26004:
26000:
25990:
25965:
25961:
25954:
25938:
25934:
25927:
25911:
25907:
25890:Euler, Leonhard
25887:
25883:
25872:
25851:
25847:
25838:
25832:
25831:by itself; and
25826:
25820:
25814:
25805:
25799:
25793:
25787:
25766:Descartes, René
25763:
25759:
25746:
25742:
25737:. p. 235v.
25725:Stifel, Michael
25722:
25718:
25709:
25708:
25704:
25683:Cajori, Florian
25680:
25676:
25663:
25659:
25635:
25631:
25622:
25620:
25610:
25603:
25589:
25585:
25561:
25557:
25533:
25526:
25519:
25495:
25491:
25484:
25455:
25448:
25439:
25438:
25434:
25424:
25422:
25407:
25403:
25394:
25392:
25384:Nykamp, Duane.
25382:
25369:
25364:
25359:
25358:
25349:
25345:
25320:
25317:
25316:
25293:
25290:
25289:
25267:
25264:
25263:
25257:
25253:
25248:
25161:Convergent only
25136:Convergent only
25111:Convergent only
25040:Bessel function
24988:that is not an
24525:
24496:
24415:
24408:
24405:
24395:
24394:as a float and
24391:
24387:
24381:
24377:
24373:
24367:
24363:
24359:
24332:
24322:
24312:
24302:
24296:
24284:
24265:
24255:
24246:Microsoft Excel
24233:
24229:
24225:
24221:
24205:
24195:
24186:Commodore BASIC
24177:
24067:
24063:
24060:+-*/()&=.,'
24059:
24051:
24018:Microsoft Excel
23985:
23974:
23966:
23952:
23921:
23916:
23893:
23888:
23885:
23884:
23833:
23829:
23811:
23807:
23805:
23802:
23801:
23778:
23775:
23774:
23728:
23725:
23724:
23678:
23675:
23674:
23645:
23641:
23639:
23636:
23635:
23612:
23608:
23606:
23603:
23602:
23560:
23556:
23554:
23551:
23550:
23509:
23505:
23503:
23500:
23499:
23449:
23445:
23434:
23431:
23430:
23377:
23373:
23371:
23368:
23367:
23303:
23300:
23299:
23273:
23269:
23267:
23264:
23263:
23262:; for example,
23259:
23255:
23238:
23234:
23232:
23229:
23228:
23222:
23217:
23213:
23206:
23202:
23138:
23135:
23134:
23130:and defined as
23106:
23103:
23102:
23080:
23074:
23059:
23047:
23039:
23020:
23017:
23016:
22977:
22973:
22959:
22956:
22955:
22949:
22936:
22933:
22930:
22927:
22924:
22921:
22918:
22915:
22912:
22909:
22907:
22898:
22895:
22892:
22889:
22886:
22884:
22875:
22872:
22870:
22861:
22858:
22856:
22848:
22788:
22784:
22769:
22765:
22756:
22752:
22743:
22739:
22730:
22726:
22718:
22715:
22714:
22703:
22696:
22690:
22687:
22678:
22672:
22665:
22659:
22653:
22649:
22643:
22637:
22631:
22622:
22615:
22603:
22596:
22590:
22584:
22574:
22562:
22558:
22554:
22544:
22540:
22536:
22526:
22519:
22512:
22502:
22495:
22488:
22478:
22474:
22470:
22466:
22462:
22454:
22448:
22444:
22440:
22436:
22432:
22426:
22416:
22407:
22395:
22394:
22385:
22379:
22367:
22360:
22359:
22353:
22321:
22316:
22315:
22286:
22283:
22282:
22265:
22261:
22238:
22235:
22234:
22227:
22221:
22211:
22202:
22197:
22194:
22191:
22188:
22186:
22182:
22163:
22155:Main articles:
22153:
22144:
22116:
22113:
22112:
22077:
22074:
22073:
22056:
22052:
22044:
22041:
22040:
22037:direct products
22024:
22022:
22013:
22011:
21955:
21951:
21934:
21931:
21930:
21904:
21900:
21891:
21887:
21881:
21877:
21872:
21869:
21868:
21831:
21828:
21827:
21810:
21806:
21804:
21801:
21800:
21796:
21792:
21788:
21784:
21780:
21776:
21765:
21759:
21748:
21740:
21738:
21730:
21729:| = |
21724:
21722:
21711:
21703:
21668:
21665:
21664:
21660:
21656:
21635:
21631:
21629:
21626:
21625:
21594:
21591:
21590:
21586:
21571:
21546:
21542:
21537:
21536:
21534:
21531:
21530:
21508:
21507:
21502:
21501:
21499:
21496:
21495:
21456:
21453:
21452:
21436:
21433:
21432:
21409:
21405:
21396:
21392:
21377:
21373:
21371:
21368:
21367:
21340:
21336:
21327:
21323:
21317:
21313:
21308:
21305:
21304:
21280:
21276:
21274:
21271:
21270:
21266:
21262:
21258:
21254:
21253:Given two sets
21214:
21211:
21210:
21202:
21201:of elements of
21182:
21178:
21163:
21159:
21154:
21151:
21150:
21146:
21143:
21137:
21101:
21099:
21096:
21095:
21091:
21075:
21073:
21070:
21069:
21052:
21047:
21046:
21044:
21041:
21040:
21019:
21015:
21013:
21010:
21009:
21005:
20998:
20997:of elements of
20978:
20974:
20959:
20955:
20950:
20947:
20946:
20939:
20935:
20918:
20914:
20912:
20909:
20908:
20904:
20865:
20862:
20861:
20818:
20815:
20814:
20772:
20769:
20768:
20725:
20722:
20721:
20699:
20696:
20695:
20667:
20664:
20663:
20656:
20652:
20642:
20640:Powers of sets
20633:
20616:
20612:
20610:
20607:
20606:
20602:
20598:
20594:
20573:
20569:
20567:
20564:
20563:
20543:
20538:
20537:
20535:
20532:
20531:
20527:
20508:
20487:
20482:
20481:
20479:
20476:
20475:
20467:
20459:
20455:
20438:
20433:
20432:
20430:
20427:
20426:
20406:
20402:
20394:
20391:
20390:
20362:
20357:
20356:
20354:
20351:
20350:
20329:
20328:
20322:
20318:
20308:
20307:
20301:
20296:
20295:
20286:
20281:
20280:
20278:
20276:
20266:
20264:
20261:
20260:
20237:
20232:
20231:
20229:
20226:
20225:
20202:
20198:
20196:
20193:
20192:
20188:
20168:
20164:
20155:
20151:
20142:
20138:
20124:
20121:
20120:
20093:
20088:
20087:
20085:
20082:
20081:
20058:
20055:
20054:
20034:
20029:
20028:
20026:
20023:
20022:
19991:
19988:
19987:
19967:
19962:
19961:
19959:
19956:
19955:
19929:
19925:
19910:
19906:
19891:
19887:
19872:
19868:
19863:
19860:
19859:
19855:
19848:
19844:
19827:
19822:
19821:
19819:
19816:
19815:
19789:
19784:
19783:
19775:
19772:
19771:
19745:
19741:
19739:
19736:
19735:
19710:
19705:
19704:
19702:
19699:
19698:
19694:
19686:
19682:
19678:
19674:
19670:
19650:
19646:
19638:
19635:
19634:
19600:complex numbers
19595:
19580:
19574:
19502:
19498:
19474:
19470:
19466:
19460:
19456:
19454:
19433:
19419:
19414:
19410:
19409:
19407:
19404:
19403:
19399:th derivative:
19394:
19388:
19360:
19331:
19323:
19320:
19319:
19294:
19291:
19290:
19268:
19263:
19260:
19259:
19238:
19221:
19217:
19215:
19212:
19211:
19205:
19185:
19181:
19179:
19176:
19175:
19155:
19151:
19149:
19146:
19145:
19135:
19129:
19123:
19099:
19086:
19082:
19078:
19077:
19065:
19061:
19059:
19056:
19055:
19049:
19032:
19028:
19026:
19023:
19022:
19012:
19006:
19000:
18997:
18984:
18961:
18957:
18942:
18938:
18930:
18927:
18926:
18924:polynomial ring
18911:
18903:
18899:
18895:
18891:
18881:coordinate ring
18868:
18845:
18841:
18839:
18836:
18835:
18813:
18810:
18809:
18782:
18759:
18755:
18753:
18750:
18749:
18742:
18718:
18714:
18708:
18704:
18695:
18691:
18680:
18677:
18676:
18656:
18652:
18643:
18639:
18633:
18629:
18624:
18621:
18620:
18613:
18607:
18594:
18564:
18560:
18556:
18552:
18548:
18522:
18518:
18509:
18505:
18503:
18500:
18499:
18495:
18491:
18487:
18463:
18461:
18458:
18457:
18446:
18442:
18434:
18423:
18401:
18398:
18397:
18380:
18376:
18374:
18371:
18370:
18366:
18348:
18315:
18311:
18309:
18306:
18305:
18273:
18269:
18264:
18261:
18260:
18240:
18236:
18225:
18222:
18221:
18192:
18188:
18183:
18180:
18179:
18093:
18089:
18084:
18081:
18080:
18011:
18007:
17971:
17967:
17962:
17959:
17958:
17931:
17927:
17925:
17922:
17921:
17904:
17900:
17898:
17895:
17894:
17886:
17856:square matrices
17826:
17825:
17820:
17797:
17790:
17786:
17780:
17776:
17769:
17763:
17759:
17747:
17746:
17737:
17733:
17726:
17720:
17716:
17710:
17706:
17700:
17699:
17693:
17689:
17683:
17679:
17672:
17660:
17656:
17653:
17652:
17642:
17636:
17632:
17628:
17626:
17623:
17622:
17615:
17611:
17607:
17603:
17578:
17565:
17561:
17557:
17556:
17554:
17551:
17550:
17544:
17538:
17534:
17526:
17509:
17505:
17503:
17500:
17499:
17495:
17487:
17470:
17466:
17448:
17444:
17442:
17439:
17438:
17411:
17407:
17405:
17402:
17401:
17394:
17386:
17369:
17362:
17351:
17347:
17343:
17306:
17303:
17302:
17298:
17291:
17287:
17283:
17273:
17269:
17265:
17253:
17243:
17239:
17229:
17225:
17217:
17214:
17208:
17203:
17179:
17175:
17165:
17161:
17152:
17142:
17138:
17134:
17133:
17131:
17128:
17127:
17116:
17112:
17078:
17075:
17074:
16961:
16957:
16949:
16946:
16945:
16935:
16895:
16892:
16891:
16874:
16870:
16868:
16865:
16864:
16833:
16830:
16829:
16812:
16808:
16806:
16803:
16802:
16798:
16791:
16765:
16761:
16755:
16751:
16744:
16740:
16738:
16735:
16734:
16715:
16711:
16705:
16701:
16686:
16682:
16680:
16677:
16676:
16650:
16646:
16640:
16636:
16629:
16625:
16610:
16606:
16600:
16596:
16594:
16591:
16590:
16568:
16564:
16555:
16545:
16541:
16537:
16536:
16534:
16531:
16530:
16504:
16500:
16494:
16490:
16465:
16461:
16459:
16456:
16455:
16416:
16413:
16412:
16373:
16348:
16344:
16340:
16339:
16337:
16334:
16333:
16288:
16284:
16278:
16274:
16250:
16246:
16244:
16241:
16240:
16234:
16222:
16218:
16207:
16200:
16196:
16192:
16188:
16158:
16141:
16137:
16127:
16116:
16112:
16110:
16090:
16086:
16063:
16049:
16044:
16040:
16039:
16037:
16034:
16033:
16002:
15998:
15978:
15974:
15945:
15941:
15921:
15918:
15917:
15913:
15909:
15905:
15904:are valid when
15886:
15872:
15871:The identities
15856:
15855:
15845:
15802:
15798:
15791:
15773:
15769:
15766:
15765:
15755:
15688:
15684:
15677:
15666:
15662:
15655:
15651:
15647:
15645:
15642:
15641:
15636:
15632:
15624:
15616:
15602:
15594:
15559:
15538:
15534:
15526:
15523:
15522:
15481:
15479:
15459:
15449:
15445:
15367:
15363:
15343:
15340:
15339:
15330:
15326:
15325:holds whenever
15311:
15305:. For example:
15299:
15293:is an integer.
15290:
15269:
15265:
15263:
15260:
15259:
15227:
15224:
15223:
15209:
15208:
15115:
15111:
15105:
15101:
15089:
15088:
14944:
14940:
14934:
14930:
14923:
14908:
14904:
14891:
14889:
14886:
14885:
14862:
14858:
14844:
14841:
14840:
14836:
14834:
14809:
14805:
14794:
14791:
14790:
14760:
14756:
14752:
14750:
14747:
14746:
14729:
14725:
14723:
14720:
14719:
14690:
14686:
14674:
14670:
14666:
14648:
14644:
14635:
14631:
14629:
14626:
14625:
14584:
14583:
14579:
14562:
14559:
14558:
14536:
14533:
14532:
14508:
14501:
14497:
14489:
14486:
14485:
14481:
14479:
14463:
14459:
14457:
14454:
14453:
14450:
14422:
14419:
14418:
14296:
14292:
14262:
14258:
14252:
14248:
14239:
14235:
14233:
14230:
14229:
14209:
14205:
14187:
14183:
14181:
14178:
14177:
14160:
14156:
14150:
14146:
14131:
14127:
14125:
14122:
14121:
14094:
14091:
14090:
13969:
13966:
13965:
13940:
13937:
13936:
13932:
13928:
13897:
13894:
13893:
13864:
13861:
13860:
13852:
13827:
13824:
13823:
13803:
13800:
13799:
13753:
13750:
13749:
13746:principal value
13740:
13687:
13684:
13683:
13664:
13660:
13651:
13647:
13645:
13637:
13634:
13633:
13574:
13570:
13559:
13556:
13555:
13551:
13547:
13543:
13512:
13509:
13508:
13503:
13491:
13487:
13470:
13466:
13464:
13461:
13460:
13435:
13432:
13431:
13425:
13403:
13399:
13397:
13394:
13393:
13389:
13385:
13356:
13353:
13352:
13341:
13335:
13308:
13305:
13304:
13300:
13283:
13279:
13277:
13274:
13273:
13248:
13245:
13244:
13237:
13233:
13212:
13204:
13201:
13200:
13173:
13170:
13169:
13147:
13144:
13143:
13142:if and only if
13126:
13122:
13113:
13109:
13107:
13104:
13103:
13095:
13091:
13088:rational number
13083:
13066:
13062:
13060:
13057:
13056:
13052:
13045:
13010:
13006:
13000:
12996:
12957:
12953:
12951:
12948:
12947:
12927:
12923:
12921:
12918:
12917:
12913:
12873:
12870:
12869:
12847:
12844:
12843:
12832:
12815:
12811:
12809:
12806:
12805:
12783:
12780:
12779:
12776:
12767:
12745:
12734:
12731:
12730:
12713:
12709:
12707:
12704:
12703:
12699:
12692:
12675:
12671:
12651:
12648:
12647:
12622:
12619:
12618:
12589:
12585:
12576:
12572:
12570:
12567:
12566:
12549:
12545:
12543:
12540:
12539:
12500:
12497:
12496:
12492:
12484:
12455:
12452:
12451:
12408:
12405:
12404:
12397:
12358:
12354:
12352:
12349:
12348:
12341:
12322:
12319:
12318:
12311:principal value
12307:
12305:Principal value
12294:
12262:
12258:
12256:
12253:
12252:
12223:
12220:
12219:
12187:
12183:
12174:
12170:
12168:
12165:
12164:
12135:
12131:
12128:
12127:
12123:
12120:principal value
12118:. So, either a
12104:
12100:
12097:
12096:
12093:
12067:
12063:
12059:
12057:
12054:
12053:
12050:principal value
12043:
12035:
12025:
11994:
11992:
11988:
11986:
11983:
11982:
11973:
11959:
11952:
11948:
11944:
11940:
11936:
11932:
11928:
11904:
11901:
11900:
11884:
11881:
11880:
11858:
11855:
11854:
11850:
11843:
11809:
11807:
11803:
11794:
11790:
11788:
11785:
11784:
11778:
11772:
11746:
11742:
11733:
11729:
11720:
11716:
11701:
11697:
11688:
11684:
11676:
11673:
11672:
11644:
11642:
11638:
11630:
11627:
11626:
11622:
11618:
11615:
11600:
11593:
11589:
11585:
11578:
11571:
11565:
11556:
11537:
11527:
11523:
11520:
11519:
11511:
11489:
11486:
11485:
11478:
11470:
11462:
11458:
11454:
11442:
11438:
11404:
11401:
11400:
11396:
11391:th root as the
11388:
11385:
11378:
11375:
11371:
11367:
11363:
11344:
11333:
11330:
11329:
11313:
11310:
11309:
11290:
11287:
11286:
11255:
11253:
11249:
11241:
11236:
11222:
11208:
11204:
11200:
11196:
11195:
11193:
11190:
11189:
11182:
11178:
11174:
11155:
11152:
11151:
11123:
11120:
11119:
11103:
11100:
11099:
11094:
11092:
11068:
11065:
11064:
11060:
11040:
11037:
11036:
10974:
10970:
10959:
10956:
10955:
10944:
10941:
10936:
10925:
10921:
10899:
10894:
10891:
10890:
10886:
10883:
10798:
10794:
10773:
10769:
10763:
10759:
10747:
10743:
10737:
10733:
10715:
10711:
10709:
10706:
10705:
10611:
10607:
10589:
10585:
10583:
10580:
10579:
10572:
10551:
10547:
10545:
10542:
10541:
10484:
10480:
10478:
10475:
10474:
10468:Euler's formula
10464:is an integer.
10461:
10457:
10431:
10427:
10418:
10408:
10404:
10400:
10399:
10397:
10394:
10393:
10378:
10361:
10351:
10347:
10343:
10342:
10340:
10337:
10336:
10313:
10309:
10303:
10299:
10284:
10280:
10278:
10275:
10274:
10264:
10238:
10235:
10234:
10196:
10192:
10183:
10179:
10177:
10174:
10173:
10160:
10152:
10148:
10145:
10136:
10130:
10104:
10100:
10098:
10095:
10094:
10065:
10061:
10052:
10036:
10032:
10028:
10027:
10018:
10014:
10012:
10009:
10008:
9982:
9978:
9969:
9965:
9959:
9955:
9950:
9947:
9946:
9939:
9913:
9909:
9877:
9874:
9873:
9864:
9852:
9841:
9835:
9829:
9826:
9817:
9797:
9793:
9791:
9788:
9787:
9759:
9756:
9755:
9751:
9740:
9723:
9719:
9699:
9696:
9695:
9691:
9687:
9670:
9666:
9646:
9643:
9642:
9608:
9605:
9604:
9534:
9531:
9530:
9511:
9507:
9499:
9497:
9495:
9492:
9491:
9468:
9455:
9451:
9443:
9441:
9422:
9420:
9413:
9409:
9408:
9396:
9383:
9368:
9361:
9357:
9356:
9350:
9335:
9328:
9324:
9323:
9311:
9272:
9269:
9268:
9198:
9195:
9194:
9154:
9151:
9150:
9127:
9112:
9105:
9101:
9100:
9088:
9064:
9061:
9060:
9030:
9026:
9006:
9003:
9002:
8967:
8964:
8963:
8933:
8930:
8929:
8909:
8905:
8897:
8894:
8893:
8886:
8880:
8865:
8861:
8853:
8849:
8832:
8828:
8826:
8823:
8822:
8800:
8796:
8794:
8791:
8790:
8755:
8751:
8742:
8738:
8737:
8733:
8719:
8715:
8706:
8702:
8701:
8697:
8683:
8679:
8670:
8666:
8665:
8661:
8647:
8643:
8634:
8630:
8629:
8625:
8611:
8607:
8598:
8594:
8593:
8589:
8575:
8571:
8562:
8558:
8557:
8553:
8551:
8548:
8547:
8525:
8521:
8519:
8516:
8515:
8504:
8503:representation
8495:
8490:
8483:
8482:and every real
8479:
8475:
8450:
8434:
8429:
8428:
8412:
8408:
8392:
8382:
8369:
8365:
8363:
8360:
8359:
8348:
8344:
8328:
8321:
8315:
8308:
8270:
8266:
8257:
8247:
8243:
8239:
8238:
8236:
8233:
8232:
8226:principal value
8190:
8144:
8140:
8116:
8109:
8105:
8076:
8072:
8058:
8047:
8043:
8033:
8029:
8028:
8026:
8023:
8022:
7999:
7995:
7986:
7982:
7976:
7972:
7967:
7964:
7963:
7938:
7934:
7932:
7929:
7928:
7924:
7920:
7912:
7908:
7885:
7881:
7872:
7868:
7862:
7858:
7853:
7850:
7849:
7842:
7838:
7811:
7807:
7801:
7797:
7780:
7769:
7765:
7759:
7755:
7751:
7747:
7746:
7732:
7721:
7717:
7711:
7707:
7703:
7699:
7698:
7689:
7685:
7676:
7672:
7661:
7657:
7652:
7649:
7648:
7623:
7619:
7611:
7608:
7607:
7584:
7580:
7569:
7565:
7548:
7538:
7534:
7530:
7529:
7515:
7511:
7509:
7506:
7505:
7481:
7477:
7473:
7471:
7468:
7467:
7466:integers, then
7463:
7459:
7456:rational number
7434:
7432:
7429:
7428:
7424:
7396:
7392:
7390:
7387:
7386:
7382:
7378:
7372:
7353:
7348:
7346:
7343:
7342:
7321:
7317:
7313:
7311:
7308:
7307:
7303:
7295:
7285:
7279:
7273:
7267:
7261:
7255:
7249:
7242:
7230:
7227:
7224:
7222:
7216:
7213:
7210:
7208:
7202:
7199:
7197:
7191:
7188:
7186:
7180:
7177:
7175:
7169:
7167:
7161:
7159:
7140:
7137:
7134:
7132:
7126:
7123:
7121:
7115:
7112:
7110:
7104:
7101:
7099:
7093:
7091:
7085:
7083:
7061:
7058:
7055:
7053:
7047:
7044:
7042:
7036:
7033:
7031:
7025:
7022:
7020:
7014:
7012:
7006:
7004:
6982:
6979:
6977:
6971:
6968:
6966:
6960:
6957:
6955:
6949:
6947:
6941:
6939:
6933:
6931:
6909:
6906:
6904:
6898:
6895:
6893:
6887:
6884:
6882:
6876:
6874:
6868:
6866:
6861:
6839:
6836:
6834:
6828:
6825:
6823:
6817:
6815:
6809:
6807:
6801:
6799:
6774:
6771:
6769:
6763:
6761:
6755:
6753:
6747:
6745:
6740:
6715:
6713:
6707:
6705:
6700:
6695:
6550:
6523:
6520:
6519:
6464:
6461:
6460:
6458:
6436:
6433:
6432:
6416:
6413:
6412:
6395:
6391:
6380:
6377:
6376:
6360:
6357:
6356:
6340:
6337:
6336:
6315:
6311:
6291:
6288:
6287:
6265:
6262:
6261:
6245:
6242:
6241:
6225:
6222:
6221:
6192:
6189:
6188:
6172:
6169:
6168:
6116:
6113:
6112:
6110:
6091:
6088:
6087:
6070:
6066:
6055:
6052:
6051:
6035:
6032:
6031:
6015:
6012:
6011:
5990:
5986:
5966:
5963:
5962:
5946:
5943:
5942:
5920:
5917:
5916:
5900:
5897:
5896:
5880:
5877:
5876:
5854:
5851:
5850:
5830:
5827:
5826:
5804:
5801:
5800:
5783:
5779:
5759:
5756:
5755:
5745:
5731:
5724:
5718:
5716:Power functions
5684:
5673:
5666:
5657:
5651:
5645:
5638:
5629:
5623:
5619:
5615:
5611:
5601:
5595:
5588:
5574:
5572:
5565:
5558:
5518:
5511:
5504:
5494:
5492:Large exponents
5477:
5469:
5462:
5461:after removing
5458:
5454:
5448:
5441:
5435:
5432:
5423:
5418:
5416:The expression
5395:
5390:
5387:
5386:
5378:
5358:
5354:
5349:
5344:
5341:
5340:
5336:
5332:
5325:
5321:
5314:
5310:
5303:
5299:
5296:
5281:
5274:
5270:
5267:
5258:
5254:
5250:
5246:
5242:
5234:
5230:
5226:
5218:
5200:
5196:
5188:
5181:
5167:
5159:
5140:
5137:
5131:
5122:
5117:
5113:
5112:
5104:
5092:
5090:
5080:
5077:
5074:
5072:
5058:
5055:
5053:
5043:
5035:
5031:
5030:
5025:
5021:
5020:
5016:
5012:
5005:
4999:
4993:
4988:
4925:
4921:
4915:
4907:
4898:
4894:
4890:
4886:
4882:
4874:
4866:
4862:
4848:
4847:, the value of
4844:
4840:
4837:
4831:
4818:
4814:
4799:square matrices
4794:
4790:
4772:
4743:
4739:
4733:
4729:
4701:
4693:
4691:
4685:
4674:
4655:
4651:
4645:
4641:
4634:
4621:
4620:
4619:
4613:
4602:
4589:
4585:
4571:
4568:
4567:
4558:
4556:Powers of a sum
4530:
4526:
4517:
4507:
4503:
4499:
4498:
4496:
4493:
4492:
4460:
4456:
4452:
4451:
4447:
4436:
4432:
4431:
4427:
4425:
4422:
4421:
4398:
4394:
4393:. For example,
4386:
4385:. For example,
4365:
4364:
4358:
4354:
4345:
4341:
4334:
4328:
4324:
4309:
4308:
4296:
4292:
4285:
4279:
4269:
4265:
4261:
4260:
4257:
4256:
4250:
4246:
4237:
4233:
4226:
4214:
4210:
4206:
4204:
4201:
4200:
4191:
4190:
4188:, often called
4182:
4175:
4148:
4144:
4142:
4139:
4138:
4134:
4128:square matrices
4123:
4079:
4076:
4075:
4058:
4054:
4045:
4041:
4026:
4022:
4020:
4017:
4016:
3997:
3994:
3993:
3970:
3966:
3961:
3949:
3945:
3943:
3940:
3939:
3932:
3928:
3925:
3909:
3905:
3901:
3876:
3873:
3872:
3871:also holds for
3852:
3848:
3839:
3835:
3820:
3816:
3814:
3811:
3810:
3769:
3765:
3763:
3760:
3759:
3752:
3748:
3745:
3717:
3713:
3704:
3700:
3694:
3690:
3685:
3682:
3681:
3655:
3651:
3642:
3638:
3623:
3619:
3617:
3614:
3613:
3606:
3602:
3573:
3569:
3554:
3550:
3548:
3545:
3544:
3511:
3507:
3505:
3502:
3501:
3480:
3468:
3425:
3421:
3406:
3402:
3398:
3394:
3390:
3377:
3371:
3365:
3359:
3354:is called "the
3337:
3328:
3322:
3316:
3310:
3304:
3298:
3292:
3286:
3281:is called "the
3268:
3267:The expression
3265:
3244:
3217:
3194:
3186:
3183:
3174:
3168:
3167:by itself; and
3162:
3161:in multiplying
3156:
3150:
3130:
3124:
3113:
3085:
3073:
3067:
3064:
3063:
3062:
3043:
3039:
3036:Nicolas Chuquet
3033:
3028:
3006:as the letters
2974:
2962:
2953:
2949:
2945:
2932:
2926:
2921:
2916:
2857:
2777:
2773:
2769:
2759:
2757:
2754:
2753:
2730:
2722:
2719:
2718:
2690:
2687:
2686:
2669:
2665:
2650:
2646:
2644:
2641:
2640:
2623:
2619:
2617:
2614:
2613:
2597:
2594:
2593:
2564:
2560:
2558:
2555:
2554:
2531:
2527:
2518:
2514:
2512:
2509:
2508:
2492:
2489:
2488:
2464:
2454:
2452:
2449:
2448:
2430:
2421:
2417:
2415:
2412:
2411:
2395:
2392:
2391:
2373:
2371:
2368:
2367:
2347:
2343:
2338:
2323:
2319:
2317:
2314:
2313:
2290:
2286:
2284:
2281:
2280:
2261:
2246:
2242:
2240:
2237:
2236:
2219:
2215:
2210:
2195:
2191:
2189:
2186:
2185:
2168:
2164:
2162:
2159:
2158:
2135:
2131:
2113:
2109:
2100:
2096:
2084:
2080:
2078:
2075:
2074:
2054:
2050:
2048:
2045:
2044:
2018:
2014:
2012:
2009:
2008:
1991:
1987:
1966:
1962:
1953:
1949:
1940:
1936:
1927:
1923:
1914:
1910:
1904:
1900:
1895:
1892:
1891:
1868:
1864:
1862:
1859:
1858:
1832:
1828:
1823:
1817:
1813:
1804:
1800:
1798:
1795:
1794:
1777:
1773:
1771:
1768:
1767:
1750:
1746:
1731:
1727:
1718:
1714:
1705:
1701:
1699:
1696:
1695:
1679:
1676:
1675:
1653:
1650:
1649:
1632:
1628:
1626:
1623:
1622:
1606:
1605:
1599:
1595:
1586:
1582:
1573:
1572:
1565:
1561:
1539:
1537:
1526:
1522:
1500:
1498:
1488:
1487:
1480:
1470:
1448:
1446:
1438:
1426:
1422:
1418:
1416:
1413:
1412:
1394:
1391:
1390:
1389:occurrences of
1374:
1371:
1370:
1353:
1349:
1347:
1344:
1343:
1327:
1324:
1323:
1274:
1247:
1243:
1209:
1207:
1197:
1193:
1191:
1188:
1187:
1183:
1182:of multiplying
1173:
1161:
1157:
1153:
1145:
1137:
1131:
1094:
1046:
1043:
1040:
1039:
1013:
1001:
997:
994:
991:
990:
960:
957:
954:
953:
928:
922:
919:
916:
913:
912:
879:
876:
873:
872:
847:
846:
839:
834:
833:
829:
828:
821:
816:
815:
810:
807:
804:
801:
800:
768:
767:
761:
757:
756:
750:
746:
745:
739:
734:
730:
727:
724:
723:
697:
696:
687:
680:
677:
673:
672:
663:
656:
653:
648:
645:
642:
639:
638:
607:
604:
601:
600:
574:
573:
567:
557:
553:
552:
546:
536:
531:
528:
525:
522:
521:
491:
488:
485:
484:
459:
458:
452:
442:
438:
437:
431:
421:
416:
413:
410:
407:
406:
376:
373:
370:
369:
344:
343:
337:
327:
323:
322:
316:
306:
302:
301:
295:
285:
281:
280:
274:
264:
259:
256:
253:
250:
249:
230:
201:
191:
185:
184:, the value of
178:
174:
170:
166:
155:
152:
149:
148:
146:
145:
141:
140:
139:
132:
126:
125:
124:
115:
109:
108:
107:
100:
94:
93:
92:
86:
76:
53:
42:
35:
28:
23:
22:
15:
12:
11:
5:
29264:
29254:
29253:
29248:
29243:
29226:
29225:
29223:
29222:
29211:
29208:
29207:
29204:
29203:
29201:
29200:
29194:
29191:
29190:
29177:
29176:
29173:
29172:
29170:
29169:
29164:
29158:
29155:
29154:
29141:
29140:
29137:
29136:
29134:
29133:
29131:Sorting number
29128:
29126:Pancake number
29122:
29119:
29118:
29105:
29104:
29101:
29100:
29098:
29097:
29092:
29086:
29083:
29082:
29069:
29068:
29065:
29064:
29062:
29061:
29056:
29051:
29045:
29042:
29041:
29038:Binary numbers
29029:
29028:
29025:
29024:
29021:
29020:
29018:
29017:
29011:
29009:
29005:
29004:
29002:
29001:
28996:
28991:
28986:
28981:
28976:
28971:
28965:
28963:
28959:
28958:
28956:
28955:
28950:
28945:
28940:
28935:
28929:
28927:
28919:
28918:
28916:
28915:
28910:
28905:
28900:
28895:
28890:
28885:
28879:
28877:
28870:
28869:
28867:
28866:
28865:
28864:
28853:
28851:
28848:P-adic numbers
28844:
28843:
28840:
28839:
28837:
28836:
28835:
28834:
28824:
28819:
28814:
28809:
28804:
28799:
28794:
28789:
28783:
28781:
28777:
28776:
28774:
28773:
28767:
28765:
28764:Coding-related
28761:
28760:
28758:
28757:
28752:
28746:
28744:
28740:
28739:
28737:
28736:
28731:
28726:
28721:
28715:
28713:
28704:
28703:
28702:
28701:
28699:Multiplicative
28696:
28685:
28683:
28668:
28667:
28663:Numeral system
28654:
28653:
28650:
28649:
28647:
28646:
28641:
28636:
28631:
28626:
28621:
28616:
28611:
28606:
28601:
28596:
28591:
28586:
28581:
28576:
28571:
28565:
28562:
28561:
28543:
28542:
28539:
28538:
28535:
28534:
28532:
28531:
28526:
28520:
28518:
28512:
28511:
28509:
28508:
28503:
28498:
28493:
28487:
28485:
28479:
28478:
28476:
28475:
28470:
28465:
28460:
28455:
28453:Highly totient
28450:
28444:
28442:
28436:
28435:
28433:
28432:
28427:
28421:
28419:
28413:
28412:
28410:
28409:
28404:
28399:
28394:
28389:
28384:
28379:
28374:
28369:
28364:
28359:
28354:
28349:
28344:
28339:
28334:
28329:
28324:
28319:
28314:
28309:
28307:Almost perfect
28304:
28298:
28296:
28286:
28285:
28269:
28268:
28265:
28264:
28262:
28261:
28256:
28251:
28246:
28241:
28236:
28231:
28226:
28221:
28216:
28211:
28206:
28200:
28197:
28196:
28184:
28183:
28180:
28179:
28177:
28176:
28171:
28166:
28161:
28155:
28152:
28151:
28139:
28138:
28135:
28134:
28132:
28131:
28126:
28121:
28116:
28114:Stirling first
28111:
28106:
28101:
28096:
28091:
28086:
28081:
28076:
28071:
28066:
28061:
28056:
28051:
28046:
28041:
28036:
28030:
28027:
28026:
28016:
28015:
28012:
28011:
28008:
28007:
28004:
28003:
28001:
28000:
27995:
27990:
27984:
27982:
27975:
27969:
27968:
27965:
27964:
27962:
27961:
27955:
27953:
27947:
27946:
27944:
27943:
27938:
27933:
27928:
27923:
27918:
27912:
27910:
27904:
27903:
27901:
27900:
27895:
27890:
27885:
27880:
27874:
27872:
27863:
27857:
27856:
27853:
27852:
27850:
27849:
27844:
27839:
27834:
27829:
27824:
27819:
27814:
27809:
27804:
27798:
27796:
27790:
27789:
27787:
27786:
27781:
27776:
27771:
27766:
27761:
27756:
27751:
27746:
27740:
27738:
27729:
27719:
27718:
27706:
27705:
27702:
27701:
27699:
27698:
27693:
27688:
27683:
27678:
27673:
27667:
27664:
27663:
27653:
27652:
27649:
27648:
27646:
27645:
27640:
27635:
27630:
27625:
27619:
27616:
27615:
27605:
27604:
27601:
27600:
27598:
27597:
27592:
27587:
27582:
27577:
27572:
27567:
27562:
27556:
27553:
27552:
27539:
27538:
27535:
27534:
27532:
27531:
27526:
27521:
27516:
27511:
27505:
27502:
27501:
27491:
27490:
27487:
27486:
27484:
27483:
27478:
27473:
27468:
27463:
27458:
27453:
27447:
27444:
27443:
27429:
27428:
27425:
27424:
27422:
27421:
27416:
27411:
27406:
27401:
27396:
27391:
27386:
27381:
27376:
27371:
27366:
27361:
27356:
27350:
27347:
27346:
27333:
27332:
27324:
27323:
27316:
27309:
27301:
27292:
27291:
27289:
27288:
27283:
27278:
27273:
27268:
27262:
27260:
27256:
27255:
27253:
27252:
27247:
27242:
27237:
27232:
27227:
27222:
27216:
27214:
27210:
27209:
27202:
27199:
27198:
27191:
27190:
27183:
27176:
27168:
27159:
27158:
27156:
27155:
27150:
27145:
27140:
27135:
27129:
27127:
27123:
27122:
27120:
27119:
27114:
27109:
27104:
27099:
27093:
27091:
27087:
27086:
27084:
27083:
27081:Super-root (4)
27078:
27073:
27068:
27063:
27057:
27055:
27048:
27047:
27045:
27044:
27039:
27034:
27029:
27024:
27019:
27014:
27008:
27006:
27002:
27001:
26994:
26993:
26986:
26979:
26971:
26963:
26962:
26956:978-0136073475
26955:
26937:
26899:
26863:
26817:
26802:
26800:(2+51+1 pages)
26784:. p. 15.
26744:
26737:
26719:
26706:
26679:
26635:
26583:
26565:
26537:10.1.1.17.7076
26507:
26494:
26481:
26466:
26457:
26450:
26424:
26409:
26380:
26355:
26347:
26319:
26301:
26279:
26272:
26242:
26235:
26217:
26210:
26181:
26174:
26156:
26148:
26110:Luderer, Bernd
26093:
26076:
26044:
26037:
25998:
25988:
25959:
25952:
25932:
25925:
25905:
25881:
25845:
25757:
25740:
25716:
25702:
25674:
25657:
25629:
25601:
25583:
25555:
25524:
25517:
25489:
25482:
25446:
25432:
25401:
25366:
25365:
25363:
25360:
25357:
25356:
25343:
25330:
25327:
25324:
25300:
25297:
25277:
25274:
25271:
25260:multiplication
25250:
25249:
25247:
25244:
25241:
25240:
25237:
25234:
25231:
25228:
25225:
25222:
25216:
25215:
25212:
25209:
25206:
25203:
25200:
25197:
25191:
25190:
25187:
25184:
25181:
25178:
25175:
25172:
25166:
25165:
25162:
25159:
25156:
25153:
25150:
25147:
25141:
25140:
25137:
25134:
25131:
25128:
25125:
25122:
25116:
25115:
25112:
25109:
25106:
25103:
25100:
25097:
25086:
25085:
25082:
25079:
25076:
25073:
25070:
25067:
25061:
25060:
25057:
25054:
25051:
25048:
25045:
25042:
25036:
25035:
25032:
25029:
25026:
25023:
25020:
25017:
25011:
25010:
25007:
25004:
25001:
24998:
24995:
24992:
24982:
24981:
24978:
24975:
24972:
24969:
24966:
24963:
24957:
24956:
24953:
24950:
24947:
24944:
24941:
24938:
24932:
24931:
24928:
24925:
24922:
24919:
24916:
24913:
24907:
24906:
24903:
24900:
24897:
24894:
24891:
24888:
24882:
24881:
24878:
24875:
24872:
24869:
24866:
24863:
24857:
24856:
24853:
24850:
24847:
24844:
24841:
24838:
24832:
24831:
24828:
24825:
24822:
24819:
24816:
24813:
24807:
24806:
24803:
24800:
24797:
24794:
24791:
24788:
24782:
24781:
24778:
24775:
24772:
24769:
24766:
24763:
24757:
24756:
24753:
24750:
24747:
24744:
24741:
24738:
24732:
24731:
24728:
24725:
24722:
24719:
24716:
24713:
24707:
24706:
24703:
24700:
24697:
24694:
24691:
24688:
24682:
24681:
24678:
24675:
24672:
24669:
24666:
24663:
24657:
24656:
24653:
24650:
24647:
24644:
24641:
24638:
24636:Finite product
24632:
24631:
24628:
24625:
24622:
24619:
24616:
24613:
24607:
24606:
24603:
24600:
24597:
24594:
24591:
24588:
24582:
24581:
24578:
24575:
24572:
24569:
24566:
24563:
24557:
24556:
24551:
24546:
24541:
24536:
24531:
24526:
24524:
24523:
24516:
24509:
24501:
24495:
24494:
24489:
24478:
24473:
24468:
24463:
24458:
24453:
24448:
24446:Hyperoperation
24443:
24438:
24433:
24428:
24422:
24421:
24420:
24404:
24401:
24400:
24399:
24385:
24371:
24341:
24340:
24330:
24323:Math.pow(x, y)
24320:
24313:math:pow(X, Y)
24310:
24303:Math.Pow(x, y)
24300:
24274:
24273:
24263:
24214:
24213:
24203:
24193:
24175:
24049:
23951:
23948:
23936:
23930:
23927:
23924:
23920:
23915:
23912:
23909:
23906:
23903:
23900:
23896:
23892:
23868:
23865:
23862:
23859:
23856:
23853:
23850:
23847:
23844:
23839:
23836:
23832:
23828:
23825:
23822:
23817:
23814:
23810:
23785:
23782:
23759:
23756:
23753:
23750:
23747:
23744:
23741:
23738:
23735:
23732:
23712:
23709:
23706:
23703:
23700:
23697:
23694:
23691:
23688:
23685:
23682:
23662:
23659:
23656:
23653:
23648:
23644:
23623:
23620:
23615:
23611:
23586:
23583:
23580:
23577:
23574:
23571:
23568:
23563:
23559:
23538:
23535:
23532:
23529:
23526:
23523:
23520:
23515:
23512:
23508:
23487:
23484:
23481:
23478:
23475:
23472:
23469:
23466:
23463:
23460:
23457:
23452:
23448:
23444:
23441:
23438:
23418:
23415:
23412:
23409:
23406:
23403:
23400:
23397:
23394:
23391:
23388:
23385:
23380:
23376:
23337:
23334:
23331:
23328:
23325:
23322:
23319:
23316:
23313:
23310:
23307:
23287:
23284:
23281:
23276:
23272:
23258:th iterate of
23241:
23237:
23199:
23198:
23187:
23184:
23181:
23178:
23175:
23172:
23169:
23166:
23163:
23160:
23157:
23154:
23151:
23148:
23145:
23142:
23119:
23116:
23113:
23110:
23093:such that the
23073:
23070:
23054:. Finding the
23027:
23024:
23000:
22997:
22994:
22991:
22988:
22985:
22980:
22976:
22972:
22969:
22966:
22963:
22941:
22940:
22903:
22902:
22880:
22879:
22866:
22865:
22852:
22851:
22844:
22843:
22839:
22838:
22837:2 (2) = 2 = 8
22834:
22833:
22824:
22823:
22811:
22808:
22805:
22802:
22799:
22796:
22791:
22787:
22783:
22780:
22777:
22772:
22768:
22764:
22759:
22755:
22751:
22746:
22742:
22738:
22733:
22729:
22725:
22722:
22686:
22683:
22602:, since pairs
22571:
22570:
22552:
22534:
22510:
22341:
22338:
22335:
22332:
22329:
22324:
22319:
22314:
22311:
22308:
22305:
22302:
22299:
22296:
22293:
22290:
22268:
22264:
22260:
22257:
22254:
22251:
22248:
22245:
22242:
22210:
22207:
22171:hyperoperation
22161:Hyperoperation
22152:
22149:
22132:
22129:
22126:
22123:
22120:
22096:
22093:
22090:
22087:
22084:
22081:
22059:
22055:
22051:
22048:
22008:
22007:
21996:
21993:
21990:
21987:
21984:
21981:
21978:
21975:
21972:
21969:
21966:
21963:
21958:
21954:
21950:
21947:
21944:
21941:
21938:
21913:
21910:
21907:
21903:
21899:
21894:
21890:
21884:
21880:
21876:
21856:
21853:
21850:
21847:
21844:
21841:
21838:
21835:
21813:
21809:
21761:Main article:
21758:
21755:
21687:
21684:
21681:
21678:
21675:
21672:
21638:
21634:
21613:
21610:
21607:
21604:
21601:
21598:
21553:
21549:
21545:
21540:
21511:
21505:
21484:abelian groups
21473:disjoint union
21460:
21440:
21429:
21428:
21417:
21412:
21408:
21404:
21399:
21395:
21391:
21386:
21383:
21380:
21376:
21365:
21354:
21349:
21346:
21343:
21339:
21335:
21330:
21326:
21320:
21316:
21312:
21283:
21279:
21239:
21236:
21233:
21230:
21227:
21224:
21221:
21218:
21190:
21185:
21181:
21177:
21174:
21171:
21166:
21162:
21158:
21136:
21133:
21108:
21104:
21078:
21055:
21050:
21037:direct product
21022:
21018:
20986:
20981:
20977:
20973:
20970:
20967:
20962:
20958:
20954:
20921:
20917:
20890:
20887:
20884:
20881:
20878:
20875:
20872:
20869:
20849:
20846:
20843:
20840:
20837:
20834:
20831:
20828:
20825:
20822:
20803:
20800:
20797:
20794:
20791:
20788:
20785:
20782:
20779:
20776:
20738:
20735:
20732:
20729:
20709:
20706:
20703:
20683:
20680:
20677:
20674:
20671:
20641:
20638:
20619:
20615:
20576:
20572:
20551:
20546:
20541:
20490:
20485:
20441:
20436:
20414:
20409:
20405:
20401:
20398:
20370:
20365:
20360:
20343:
20342:
20325:
20321:
20317:
20314:
20311:
20309:
20304:
20299:
20294:
20289:
20284:
20279:
20275:
20272:
20269:
20268:
20245:
20240:
20235:
20213:
20210:
20205:
20201:
20185:
20184:
20171:
20167:
20163:
20158:
20154:
20150:
20145:
20141:
20137:
20134:
20131:
20128:
20101:
20096:
20091:
20062:
20042:
20037:
20032:
20010:
20007:
20004:
20001:
19998:
19995:
19975:
19970:
19965:
19943:
19940:
19937:
19932:
19928:
19924:
19919:
19916:
19913:
19909:
19905:
19902:
19899:
19894:
19890:
19886:
19883:
19880:
19875:
19871:
19867:
19843:is an element
19830:
19825:
19797:
19792:
19787:
19782:
19779:
19768:
19767:
19756:
19753:
19748:
19744:
19718:
19713:
19708:
19658:
19653:
19649:
19645:
19642:
19576:Main article:
19573:
19570:
19537:
19536:
19525:
19522:
19519:
19516:
19511:
19508:
19505:
19501:
19497:
19494:
19491:
19488:
19485:
19477:
19473:
19469:
19463:
19459:
19453:
19450:
19447:
19444:
19441:
19436:
19431:
19425:
19422:
19418:
19413:
19376:
19373:
19370:
19366:
19363:
19359:
19356:
19353:
19350:
19347:
19344:
19341:
19338:
19334:
19330:
19327:
19307:
19304:
19301:
19298:
19278:
19275:
19271:
19267:
19224:
19220:
19193:
19188:
19184:
19163:
19158:
19154:
19102:
19097:
19092:
19089:
19085:
19081:
19076:
19071:
19068:
19064:
19035:
19031:
18996:
18993:
18969:
18964:
18960:
18956:
18953:
18950:
18945:
18941:
18937:
18934:
18856:
18853:
18848:
18844:
18823:
18820:
18817:
18770:
18767:
18762:
18758:
18741:
18738:
18726:
18721:
18717:
18711:
18707:
18703:
18698:
18694:
18690:
18687:
18684:
18662:
18659:
18655:
18651:
18646:
18642:
18636:
18632:
18628:
18582:), and in the
18580:Sylow theorems
18536:
18533:
18530:
18525:
18521:
18517:
18512:
18508:
18466:
18455:additive group
18411:
18408:
18405:
18383:
18379:
18347:
18344:
18332:
18329:
18326:
18323:
18318:
18314:
18293:
18290:
18287:
18284:
18279:
18276:
18272:
18268:
18248:
18243:
18239:
18235:
18232:
18229:
18209:
18206:
18203:
18200:
18195:
18191:
18187:
18176:
18175:
18164:
18161:
18158:
18155:
18152:
18149:
18146:
18143:
18140:
18137:
18134:
18131:
18128:
18125:
18122:
18119:
18116:
18113:
18110:
18107:
18104:
18099:
18096:
18092:
18088:
18074:
18073:
18062:
18059:
18056:
18053:
18050:
18047:
18044:
18041:
18038:
18035:
18031:
18028:
18025:
18022:
18019:
18014:
18010:
18006:
18003:
18000:
17997:
17994:
17991:
17988:
17985:
17982:
17979:
17974:
17970:
17966:
17937:
17934:
17930:
17907:
17903:
17840:
17839:
17819:
17816:
17813:
17810:
17807:
17804:
17793:
17789:
17783:
17779:
17775:
17772:
17770:
17766:
17762:
17758:
17755:
17752:
17749:
17748:
17743:
17740:
17736:
17732:
17729:
17727:
17723:
17719:
17713:
17709:
17705:
17702:
17701:
17696:
17692:
17686:
17682:
17678:
17675:
17673:
17669:
17666:
17663:
17659:
17655:
17654:
17651:
17648:
17645:
17643:
17639:
17635:
17631:
17630:
17589:
17584:
17581:
17576:
17571:
17568:
17564:
17560:
17549:is defined as
17512:
17508:
17492:
17491:
17473:
17469:
17465:
17462:
17457:
17454:
17451:
17447:
17436:
17425:
17422:
17419:
17414:
17410:
17361:
17358:
17331:
17328:
17325:
17322:
17319:
17316:
17313:
17310:
17280:transcendental
17210:Main article:
17207:
17204:
17202:
17201:
17189:
17182:
17178:
17174:
17171:
17168:
17164:
17160:
17155:
17150:
17145:
17141:
17137:
17100:
17097:
17094:
17091:
17088:
17085:
17082:
17062:
17059:
17056:
17053:
17050:
17047:
17044:
17041:
17038:
17035:
17032:
17029:
17025:
17021:
17018:
17015:
17012:
17009:
17006:
17003:
17000:
16997:
16994:
16991:
16988:
16985:
16982:
16979:
16976:
16973:
16970:
16967:
16964:
16960:
16956:
16953:
16923:
16920:
16917:
16914:
16911:
16908:
16905:
16902:
16899:
16877:
16873:
16852:
16849:
16846:
16843:
16840:
16837:
16815:
16811:
16796:
16795:
16778:
16775:
16768:
16764:
16758:
16754:
16750:
16747:
16743:
16732:
16718:
16714:
16708:
16704:
16700:
16695:
16692:
16689:
16685:
16663:
16660:
16653:
16649:
16643:
16639:
16635:
16632:
16628:
16622:
16619:
16616:
16613:
16609:
16603:
16599:
16588:
16574:
16571:
16567:
16563:
16558:
16553:
16548:
16544:
16540:
16517:
16514:
16507:
16503:
16497:
16493:
16489:
16486:
16483:
16480:
16477:
16474:
16471:
16468:
16464:
16453:
16441:
16438:
16435:
16432:
16429:
16426:
16423:
16420:
16399:
16396:
16391:
16388:
16385:
16382:
16379:
16376:
16371:
16366:
16363:
16360:
16357:
16354:
16351:
16347:
16343:
16331:
16320:
16317:
16314:
16311:
16308:
16305:
16300:
16297:
16294:
16291:
16287:
16281:
16277:
16273:
16268:
16265:
16262:
16259:
16256:
16253:
16249:
16204:
16176:
16173:
16170:
16165:
16162:
16157:
16148:
16145:
16140:
16136:
16133:
16130:
16123:
16120:
16115:
16109:
16106:
16103:
16097:
16094:
16089:
16085:
16082:
16079:
16076:
16070:
16067:
16061:
16055:
16052:
16048:
16043:
16021:
16018:
16015:
16009:
16006:
16001:
15997:
15994:
15991:
15985:
15982:
15977:
15973:
15970:
15967:
15964:
15961:
15958:
15952:
15949:
15944:
15940:
15937:
15934:
15931:
15928:
15925:
15869:
15853:
15848:
15844:
15841:
15838:
15835:
15832:
15829:
15826:
15823:
15820:
15817:
15814:
15811:
15808:
15805:
15801:
15797:
15794:
15792:
15789:
15785:
15782:
15779:
15776:
15772:
15768:
15767:
15763:
15758:
15754:
15751:
15748:
15745:
15742:
15739:
15736:
15733:
15730:
15727:
15724:
15721:
15718:
15715:
15712:
15709:
15706:
15703:
15700:
15697:
15694:
15691:
15687:
15683:
15680:
15678:
15675:
15669:
15665:
15661:
15658:
15654:
15650:
15649:
15579:
15576:
15573:
15570:
15566:
15563:
15558:
15555:
15552:
15549:
15546:
15541:
15537:
15533:
15530:
15508:
15505:
15502:
15499:
15494:
15490:
15487:
15484:
15477:
15474:
15471:
15466:
15462:
15458:
15455:
15452:
15448:
15444:
15441:
15438:
15435:
15432:
15429:
15426:
15423:
15420:
15417:
15414:
15411:
15408:
15405:
15402:
15399:
15396:
15393:
15390:
15387:
15384:
15381:
15378:
15375:
15370:
15366:
15362:
15359:
15356:
15353:
15350:
15347:
15307:
15298:
15295:
15272:
15268:
15256:
15255:
15243:
15240:
15237:
15234:
15231:
15207:
15204:
15201:
15198:
15195:
15192:
15189:
15186:
15183:
15180:
15177:
15174:
15171:
15168:
15165:
15162:
15159:
15156:
15153:
15150:
15147:
15142:
15139:
15136:
15133:
15130:
15127:
15124:
15121:
15118:
15114:
15108:
15104:
15100:
15097:
15094:
15092:
15090:
15087:
15084:
15081:
15078:
15075:
15072:
15069:
15066:
15063:
15060:
15057:
15054:
15051:
15048:
15045:
15042:
15039:
15036:
15033:
15030:
15027:
15024:
15021:
15018:
15015:
15012:
15009:
15006:
15003:
15000:
14997:
14994:
14991:
14988:
14985:
14982:
14979:
14976:
14971:
14968:
14965:
14962:
14959:
14956:
14953:
14950:
14947:
14943:
14937:
14933:
14929:
14926:
14924:
14920:
14917:
14914:
14911:
14907:
14903:
14900:
14897:
14894:
14893:
14873:
14868:
14865:
14861:
14857:
14854:
14851:
14848:
14821:
14818:
14815:
14812:
14808:
14804:
14801:
14798:
14788:
14777:
14774:
14767:
14764:
14759:
14755:
14732:
14728:
14707:
14702:
14699:
14696:
14693:
14689:
14681:
14678:
14673:
14669:
14665:
14660:
14657:
14654:
14651:
14647:
14643:
14638:
14634:
14613:
14609:
14605:
14602:
14599:
14596:
14591:
14588:
14582:
14578:
14575:
14572:
14569:
14566:
14546:
14543:
14540:
14520:
14515:
14511:
14507:
14504:
14500:
14496:
14493:
14466:
14462:
14449:
14446:
14445:
14444:
14432:
14429:
14426:
14406:
14402:
14398:
14395:
14392:
14389:
14386:
14383:
14380:
14377:
14374:
14371:
14368:
14365:
14362:
14359:
14356:
14353:
14350:
14347:
14344:
14341:
14338:
14335:
14332:
14329:
14326:
14323:
14320:
14317:
14314:
14311:
14308:
14305:
14302:
14299:
14295:
14289:
14286:
14283:
14280:
14277:
14274:
14271:
14268:
14265:
14261:
14255:
14251:
14247:
14242:
14238:
14217:
14212:
14208:
14204:
14199:
14196:
14193:
14190:
14186:
14163:
14159:
14153:
14149:
14145:
14140:
14137:
14134:
14130:
14115:
14104:
14101:
14098:
14078:
14075:
14072:
14069:
14066:
14063:
14060:
14057:
14054:
14051:
14048:
14045:
14042:
14039:
14036:
14033:
14030:
14027:
14024:
14021:
14018:
14015:
14012:
14009:
14006:
14003:
14000:
13997:
13994:
13991:
13988:
13985:
13982:
13979:
13976:
13973:
13953:
13950:
13947:
13944:
13916:
13913:
13910:
13907:
13904:
13901:
13880:
13877:
13874:
13871:
13868:
13856:
13840:
13837:
13834:
13831:
13807:
13787:
13784:
13781:
13778:
13775:
13772:
13769:
13766:
13763:
13760:
13757:
13733:
13715:
13712:
13709:
13706:
13703:
13700:
13697:
13694:
13691:
13667:
13663:
13659:
13654:
13650:
13644:
13641:
13621:
13618:
13615:
13612:
13609:
13606:
13603:
13600:
13597:
13594:
13591:
13588:
13585:
13580:
13577:
13573:
13569:
13566:
13563:
13531:
13528:
13525:
13522:
13519:
13516:
13473:
13469:
13448:
13445:
13442:
13439:
13429:canonical form
13424:
13421:
13406:
13402:
13369:
13366:
13363:
13360:
13321:
13318:
13315:
13312:
13286:
13282:
13261:
13258:
13255:
13252:
13219:
13216:
13211:
13208:
13186:
13183:
13180:
13177:
13157:
13154:
13151:
13129:
13125:
13121:
13116:
13112:
13069:
13065:
13042:
13041:
13030:
13025:
13022:
13019:
13016:
13013:
13009:
13003:
12999:
12995:
12990:
12987:
12984:
12981:
12978:
12975:
12972:
12969:
12966:
12963:
12960:
12956:
12930:
12926:
12901:
12898:
12895:
12892:
12889:
12886:
12883:
12880:
12877:
12857:
12854:
12851:
12818:
12814:
12793:
12790:
12787:
12775:
12772:
12755:
12752:
12748:
12744:
12741:
12738:
12716:
12712:
12678:
12674:
12670:
12667:
12664:
12661:
12658:
12655:
12632:
12629:
12626:
12606:
12601:
12598:
12595:
12592:
12588:
12584:
12579:
12575:
12565:is defined as
12552:
12548:
12525:
12522:
12519:
12516:
12513:
12510:
12507:
12504:
12468:
12465:
12462:
12459:
12448:
12447:
12436:
12433:
12430:
12427:
12424:
12421:
12418:
12415:
12412:
12390:
12389:
12378:
12375:
12372:
12367:
12364:
12361:
12357:
12329:
12326:
12306:
12303:
12291:
12290:
12279:
12276:
12271:
12268:
12265:
12261:
12233:
12230:
12227:
12216:
12215:
12204:
12199:
12196:
12193:
12190:
12186:
12182:
12177:
12173:
12138:
12134:
12107:
12103:
12092:
12089:
12087:, which is 1.
12074:
12070:
12066:
12062:
12010:
12006:
12003:
12000:
11997:
11991:
11981:As the number
11914:
11911:
11908:
11888:
11868:
11865:
11862:
11831:
11825:
11821:
11818:
11815:
11812:
11806:
11802:
11797:
11793:
11760:
11755:
11752:
11749:
11745:
11741:
11736:
11732:
11728:
11723:
11719:
11715:
11712:
11709:
11704:
11700:
11696:
11691:
11687:
11683:
11680:
11657:
11653:
11650:
11647:
11641:
11637:
11634:
11567:Main article:
11564:
11563:Roots of unity
11561:
11544:
11540:
11536:
11533:
11530:
11526:
11499:
11496:
11493:
11426:
11423:
11420:
11417:
11414:
11411:
11408:
11393:principal root
11351:
11347:
11343:
11340:
11337:
11317:
11297:
11294:
11283:
11282:
11271:
11265:
11261:
11258:
11252:
11244:
11240:
11235:
11229:
11226:
11220:
11214:
11211:
11207:
11203:
11199:
11159:
11139:
11136:
11133:
11130:
11127:
11107:
11072:
11057:absolute value
11044:
11033:
11032:
11021:
11018:
11015:
11012:
11009:
11006:
11003:
11000:
10997:
10994:
10991:
10988:
10985:
10980:
10977:
10973:
10969:
10966:
10963:
10940:
10934:
10909:
10906:
10902:
10898:
10882:
10879:
10878:
10877:
10866:
10863:
10860:
10857:
10854:
10851:
10848:
10845:
10842:
10839:
10836:
10833:
10830:
10827:
10824:
10821:
10818:
10815:
10812:
10809:
10806:
10801:
10797:
10793:
10788:
10785:
10782:
10779:
10776:
10772:
10766:
10762:
10758:
10753:
10750:
10746:
10740:
10736:
10732:
10727:
10724:
10721:
10718:
10714:
10695:absolute value
10691:
10690:
10679:
10676:
10673:
10670:
10667:
10664:
10661:
10658:
10655:
10652:
10649:
10646:
10643:
10640:
10637:
10634:
10631:
10628:
10625:
10622:
10619:
10614:
10610:
10606:
10601:
10598:
10595:
10592:
10588:
10554:
10550:
10534:
10533:
10522:
10519:
10516:
10513:
10510:
10507:
10504:
10501:
10498:
10495:
10490:
10487:
10483:
10454:
10453:
10442:
10437:
10434:
10430:
10426:
10421:
10416:
10411:
10407:
10403:
10364:
10359:
10354:
10350:
10346:
10333:
10332:
10321:
10316:
10312:
10306:
10302:
10298:
10293:
10290:
10287:
10283:
10248:
10245:
10242:
10231:
10230:
10219:
10214:
10211:
10208:
10205:
10202:
10199:
10195:
10191:
10186:
10182:
10144:
10141:
10116:
10113:
10110:
10107:
10103:
10091:
10090:
10077:
10074:
10071:
10068:
10064:
10060:
10055:
10050:
10045:
10042:
10039:
10035:
10031:
10026:
10021:
10017:
10004:one must have
9993:
9988:
9985:
9981:
9977:
9972:
9968:
9962:
9958:
9954:
9936:
9935:
9922:
9919:
9916:
9912:
9908:
9905:
9902:
9899:
9896:
9893:
9890:
9887:
9884:
9881:
9825:
9822:
9805:
9800:
9796:
9775:
9772:
9769:
9766:
9763:
9726:
9722:
9718:
9715:
9712:
9709:
9706:
9703:
9673:
9669:
9665:
9662:
9659:
9656:
9653:
9650:
9630:
9627:
9624:
9621:
9618:
9615:
9612:
9589:
9586:
9583:
9580:
9577:
9574:
9571:
9568:
9565:
9562:
9559:
9556:
9553:
9550:
9547:
9544:
9541:
9538:
9514:
9510:
9505:
9502:
9488:
9487:
9476:
9471:
9466:
9458:
9454:
9449:
9446:
9440:
9435:
9431:
9428:
9425:
9419:
9416:
9412:
9405:
9402:
9399:
9395:
9391:
9386:
9381:
9375:
9372:
9367:
9364:
9360:
9353:
9348:
9342:
9339:
9334:
9331:
9327:
9320:
9317:
9314:
9310:
9306:
9303:
9300:
9297:
9294:
9291:
9288:
9285:
9282:
9279:
9276:
9253:
9250:
9247:
9244:
9241:
9238:
9235:
9232:
9229:
9226:
9223:
9220:
9217:
9214:
9211:
9208:
9205:
9202:
9179:
9176:
9173:
9170:
9167:
9164:
9161:
9158:
9147:
9146:
9135:
9130:
9125:
9119:
9116:
9111:
9108:
9104:
9097:
9094:
9091:
9087:
9083:
9080:
9077:
9074:
9071:
9068:
9050:
9049:
9038:
9033:
9029:
9025:
9022:
9019:
9016:
9013:
9010:
8986:
8983:
8980:
8977:
8974:
8971:
8956:Euler's number
8943:
8940:
8937:
8917:
8912:
8908:
8904:
8901:
8882:Main article:
8879:
8876:
8835:
8831:
8808:
8803:
8799:
8783:
8782:
8771:
8768:
8764:
8758:
8754:
8750:
8745:
8741:
8736:
8732:
8728:
8722:
8718:
8714:
8709:
8705:
8700:
8696:
8692:
8686:
8682:
8678:
8673:
8669:
8664:
8660:
8656:
8650:
8646:
8642:
8637:
8633:
8628:
8624:
8620:
8614:
8610:
8606:
8601:
8597:
8592:
8588:
8584:
8578:
8574:
8570:
8565:
8561:
8556:
8533:
8528:
8524:
8472:
8471:
8460:
8457:
8453:
8449:
8446:
8442:
8437:
8432:
8427:
8424:
8421:
8415:
8411:
8405:
8402:
8399:
8395:
8391:
8388:
8385:
8381:
8377:
8372:
8368:
8355:with the rule
8307:
8304:
8290:
8289:
8276:
8273:
8269:
8265:
8260:
8255:
8250:
8246:
8242:
8189:
8188:Real exponents
8186:
8170:
8169:
8158:
8155:
8152:
8147:
8143:
8139:
8136:
8133:
8130:
8123:
8120:
8115:
8112:
8108:
8104:
8101:
8098:
8095:
8092:
8089:
8083:
8080:
8075:
8071:
8065:
8062:
8056:
8050:
8046:
8042:
8039:
8036:
8032:
8005:
8002:
7998:
7994:
7989:
7985:
7979:
7975:
7971:
7951:
7945:
7942:
7937:
7891:
7888:
7884:
7880:
7875:
7871:
7865:
7861:
7857:
7824:
7818:
7815:
7810:
7804:
7800:
7796:
7793:
7787:
7784:
7778:
7772:
7768:
7762:
7758:
7754:
7750:
7745:
7739:
7736:
7730:
7724:
7720:
7714:
7710:
7706:
7702:
7697:
7692:
7688:
7684:
7679:
7675:
7668:
7665:
7660:
7656:
7636:
7630:
7627:
7622:
7618:
7615:
7604:
7603:
7592:
7587:
7583:
7576:
7573:
7568:
7564:
7561:
7555:
7552:
7546:
7541:
7537:
7533:
7528:
7522:
7519:
7514:
7501:is defined as
7488:
7484:
7480:
7476:
7441:
7438:
7410:
7407:
7404:
7399:
7395:
7356:
7352:
7328:
7324:
7320:
7316:
7241:
7238:
7235:
7234:
7220:
7206:
7195:
7184:
7173:
7165:
7157:
7154:
7151:
7145:
7144:
7130:
7119:
7108:
7097:
7089:
7081:
7078:
7075:
7072:
7066:
7065:
7051:
7040:
7029:
7018:
7010:
7002:
6999:
6996:
6993:
6987:
6986:
6975:
6964:
6953:
6945:
6937:
6929:
6926:
6923:
6920:
6914:
6913:
6902:
6891:
6880:
6872:
6864:
6859:
6856:
6853:
6850:
6844:
6843:
6832:
6821:
6813:
6805:
6797:
6794:
6791:
6788:
6785:
6779:
6778:
6767:
6759:
6751:
6743:
6738:
6735:
6732:
6729:
6726:
6720:
6719:
6711:
6703:
6698:
6693:
6690:
6687:
6684:
6681:
6678:
6672:
6671:
6668:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6638:
6637:
6634:
6631:
6628:
6625:
6622:
6619:
6616:
6613:
6610:
6604:
6603:
6598:
6593:
6588:
6583:
6578:
6573:
6568:
6563:
6558:
6549:
6546:
6533:
6530:
6527:
6498:
6495:
6492:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6468:
6446:
6443:
6440:
6420:
6398:
6394:
6390:
6387:
6384:
6364:
6344:
6318:
6314:
6310:
6307:
6304:
6301:
6298:
6295:
6275:
6272:
6269:
6249:
6229:
6205:
6202:
6199:
6196:
6176:
6162:even functions
6147:
6144:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
6120:
6095:
6073:
6069:
6065:
6062:
6059:
6039:
6019:
5993:
5989:
5985:
5982:
5979:
5976:
5973:
5970:
5950:
5930:
5927:
5924:
5904:
5895:even, and for
5884:
5864:
5861:
5858:
5834:
5814:
5811:
5808:
5786:
5782:
5778:
5775:
5772:
5769:
5766:
5763:
5720:Main article:
5717:
5714:
5691:
5690:
5608:
5607:
5582:
5581:
5552:absolute value
5525:
5524:
5493:
5490:
5431:
5428:
5402:
5398:
5394:
5364:
5361:
5357:
5352:
5348:
5295:
5294:Powers of zero
5292:
5266:
5263:
5172:members has a
5133:Main article:
5130:
5127:
5067:in vacuum, in
5065:speed of light
5001:Main article:
4992:
4991:Powers of ten
4989:
4987:
4984:
4981:
4980:
4977:
4973:
4972:
4969:
4965:
4964:
4961:
4957:
4956:
4953:
4949:
4948:
4945:
4941:
4940:
4937:
4933:
4932:
4912:
4869:elements (see
4830:
4827:
4769:
4768:
4757:
4752:
4749:
4746:
4742:
4736:
4732:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4699:
4696:
4688:
4683:
4680:
4677:
4673:
4669:
4664:
4661:
4658:
4654:
4648:
4644:
4637:
4632:
4629:
4624:
4616:
4611:
4608:
4605:
4601:
4597:
4592:
4588:
4584:
4581:
4578:
4575:
4557:
4554:
4553:
4552:
4541:
4536:
4533:
4529:
4525:
4520:
4515:
4510:
4506:
4502:
4486:
4485:
4474:
4468:
4463:
4459:
4455:
4450:
4446:
4439:
4435:
4430:
4379:
4378:
4361:
4357:
4353:
4348:
4344:
4340:
4337:
4335:
4331:
4327:
4323:
4320:
4317:
4314:
4311:
4310:
4305:
4302:
4299:
4295:
4291:
4288:
4286:
4282:
4277:
4272:
4268:
4264:
4259:
4258:
4253:
4249:
4245:
4240:
4236:
4232:
4229:
4227:
4223:
4220:
4217:
4213:
4209:
4208:
4192:exponent rules
4184:The following
4174:
4171:
4159:
4154:
4151:
4147:
4114:, that is, an
4092:
4089:
4086:
4083:
4061:
4057:
4053:
4048:
4044:
4040:
4035:
4032:
4029:
4025:
4001:
3990:
3989:
3973:
3969:
3965:
3960:
3955:
3952:
3948:
3924:
3921:
3886:
3883:
3880:
3869:
3868:
3855:
3851:
3847:
3842:
3838:
3834:
3829:
3826:
3823:
3819:
3792:
3791:
3780:
3777:
3772:
3768:
3744:
3741:
3740:
3739:
3728:
3723:
3720:
3716:
3712:
3707:
3703:
3697:
3693:
3689:
3675:
3674:
3663:
3658:
3654:
3650:
3645:
3641:
3637:
3632:
3629:
3626:
3622:
3599:
3598:
3587:
3584:
3581:
3576:
3572:
3568:
3563:
3560:
3557:
3553:
3534:
3533:
3522:
3519:
3514:
3510:
3479:
3476:
3467:
3464:
3411:5th power of 3
3264:
3261:
3248:Leonhard Euler
3243:
3240:
3216:
3213:
3181:
3137:René Descartes
3112:
3109:
3093:Robert Recorde
3084:
3081:
3055:Michael Stifel
3032:
3029:
3027:
3024:
2998:, which later
2973:
2963:
2961:
2958:
2928:Main article:
2925:
2922:
2920:
2917:
2915:
2912:
2902:mathematician
2856:
2853:
2847:behavior, and
2784:
2780:
2776:
2772:
2768:
2763:
2738:
2735:
2729:
2726:
2706:
2703:
2700:
2697:
2694:
2672:
2668:
2664:
2659:
2656:
2653:
2649:
2626:
2622:
2601:
2581:
2578:
2573:
2570:
2567:
2563:
2542:
2539:
2534:
2530:
2526:
2521:
2517:
2496:
2476:
2473:
2468:
2463:
2458:
2434:
2429:
2424:
2420:
2399:
2377:
2350:
2346:
2341:
2337:
2334:
2329:
2326:
2322:
2301:
2298:
2293:
2289:
2268:
2264:
2260:
2257:
2252:
2249:
2245:
2222:
2218:
2213:
2209:
2206:
2201:
2198:
2194:
2171:
2167:
2146:
2143:
2138:
2134:
2130:
2125:
2122:
2119:
2116:
2112:
2108:
2103:
2099:
2095:
2090:
2087:
2083:
2060:
2057:
2053:
2029:
2026:
2021:
2017:
1994:
1990:
1986:
1981:
1978:
1975:
1972:
1969:
1965:
1961:
1956:
1952:
1948:
1943:
1939:
1935:
1930:
1926:
1922:
1917:
1913:
1907:
1903:
1899:
1879:
1876:
1871:
1867:
1857:The fact that
1843:
1840:
1835:
1831:
1826:
1820:
1816:
1812:
1807:
1803:
1780:
1776:
1753:
1749:
1745:
1740:
1737:
1734:
1730:
1726:
1721:
1717:
1713:
1708:
1704:
1683:
1663:
1660:
1657:
1635:
1631:
1602:
1598:
1594:
1589:
1585:
1581:
1578:
1576:
1574:
1564:
1558:
1554:
1551:
1548:
1545:
1542:
1535:
1525:
1519:
1515:
1512:
1509:
1506:
1503:
1496:
1493:
1491:
1489:
1479:
1476:
1473:
1467:
1463:
1460:
1457:
1454:
1451:
1444:
1441:
1439:
1435:
1432:
1429:
1425:
1421:
1420:
1398:
1378:
1356:
1352:
1331:
1283:raised to the
1256:
1246:
1240:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1205:
1200:
1196:
1170:multiplication
1164:is a positive
1106:exponentiation
1096:
1095:
1093:
1092:
1085:
1078:
1070:
1067:
1066:
1063:
1062:
1037:
1024:
1020:
1015:anti-logarithm
1012:
1009:
1000:
988:
985:
984:
977:
976:
951:
938:
910:
907:
906:
896:
895:
870:
857:
852:
831:
830:
813:
812:
809:
798:
795:
794:
791:Exponentiation
787:
786:
772:
759:
758:
748:
747:
737:
736:
733:
720:
707:
702:
675:
674:
651:
650:
647:
636:
633:
632:
625:
624:
597:
584:
579:
565:
555:
554:
544:
534:
533:
530:
519:
516:
515:
512:Multiplication
508:
507:
482:
469:
464:
450:
440:
439:
429:
419:
418:
415:
404:
401:
400:
393:
392:
367:
354:
349:
335:
325:
324:
314:
304:
303:
293:
283:
282:
272:
262:
261:
258:
247:
244:
243:
232:
231:
229:
228:
221:
214:
206:
67:
66:
55:
54:
51:
48:
47:
26:
9:
6:
4:
3:
2:
29263:
29252:
29249:
29247:
29244:
29242:
29239:
29238:
29236:
29221:
29217:
29213:
29212:
29209:
29199:
29196:
29195:
29192:
29187:
29182:
29178:
29168:
29165:
29163:
29160:
29159:
29156:
29151:
29146:
29142:
29132:
29129:
29127:
29124:
29123:
29120:
29115:
29110:
29106:
29096:
29093:
29091:
29088:
29087:
29084:
29080:
29074:
29070:
29060:
29057:
29055:
29052:
29050:
29047:
29046:
29043:
29039:
29034:
29030:
29016:
29013:
29012:
29010:
29006:
29000:
28997:
28995:
28992:
28990:
28989:Polydivisible
28987:
28985:
28982:
28980:
28977:
28975:
28972:
28970:
28967:
28966:
28964:
28960:
28954:
28951:
28949:
28946:
28944:
28941:
28939:
28936:
28934:
28931:
28930:
28928:
28925:
28920:
28914:
28911:
28909:
28906:
28904:
28901:
28899:
28896:
28894:
28891:
28889:
28886:
28884:
28881:
28880:
28878:
28875:
28871:
28863:
28860:
28859:
28858:
28855:
28854:
28852:
28849:
28845:
28833:
28830:
28829:
28828:
28825:
28823:
28820:
28818:
28815:
28813:
28810:
28808:
28805:
28803:
28800:
28798:
28795:
28793:
28790:
28788:
28785:
28784:
28782:
28778:
28772:
28769:
28768:
28766:
28762:
28756:
28753:
28751:
28748:
28747:
28745:
28743:Digit product
28741:
28735:
28732:
28730:
28727:
28725:
28722:
28720:
28717:
28716:
28714:
28712:
28708:
28700:
28697:
28695:
28692:
28691:
28690:
28687:
28686:
28684:
28682:
28677:
28673:
28669:
28664:
28659:
28655:
28645:
28642:
28640:
28637:
28635:
28632:
28630:
28627:
28625:
28622:
28620:
28617:
28615:
28612:
28610:
28607:
28605:
28602:
28600:
28597:
28595:
28592:
28590:
28587:
28585:
28582:
28580:
28579:Erdős–Nicolas
28577:
28575:
28572:
28570:
28567:
28566:
28563:
28558:
28554:
28548:
28544:
28530:
28527:
28525:
28522:
28521:
28519:
28517:
28513:
28507:
28504:
28502:
28499:
28497:
28494:
28492:
28489:
28488:
28486:
28484:
28480:
28474:
28471:
28469:
28466:
28464:
28461:
28459:
28456:
28454:
28451:
28449:
28446:
28445:
28443:
28441:
28437:
28431:
28428:
28426:
28423:
28422:
28420:
28418:
28414:
28408:
28405:
28403:
28400:
28398:
28397:Superabundant
28395:
28393:
28390:
28388:
28385:
28383:
28380:
28378:
28375:
28373:
28370:
28368:
28365:
28363:
28360:
28358:
28355:
28353:
28350:
28348:
28345:
28343:
28340:
28338:
28335:
28333:
28330:
28328:
28325:
28323:
28320:
28318:
28315:
28313:
28310:
28308:
28305:
28303:
28300:
28299:
28297:
28295:
28291:
28287:
28283:
28279:
28274:
28270:
28260:
28257:
28255:
28252:
28250:
28247:
28245:
28242:
28240:
28237:
28235:
28232:
28230:
28227:
28225:
28222:
28220:
28217:
28215:
28212:
28210:
28207:
28205:
28202:
28201:
28198:
28194:
28189:
28185:
28175:
28172:
28170:
28167:
28165:
28162:
28160:
28157:
28156:
28153:
28149:
28144:
28140:
28130:
28127:
28125:
28122:
28120:
28117:
28115:
28112:
28110:
28107:
28105:
28102:
28100:
28097:
28095:
28092:
28090:
28087:
28085:
28082:
28080:
28077:
28075:
28072:
28070:
28067:
28065:
28062:
28060:
28057:
28055:
28052:
28050:
28047:
28045:
28042:
28040:
28037:
28035:
28032:
28031:
28028:
28021:
28017:
27999:
27996:
27994:
27991:
27989:
27986:
27985:
27983:
27979:
27976:
27974:
27973:4-dimensional
27970:
27960:
27957:
27956:
27954:
27952:
27948:
27942:
27939:
27937:
27934:
27932:
27929:
27927:
27924:
27922:
27919:
27917:
27914:
27913:
27911:
27909:
27905:
27899:
27896:
27894:
27891:
27889:
27886:
27884:
27883:Centered cube
27881:
27879:
27876:
27875:
27873:
27871:
27867:
27864:
27862:
27861:3-dimensional
27858:
27848:
27845:
27843:
27840:
27838:
27835:
27833:
27830:
27828:
27825:
27823:
27820:
27818:
27815:
27813:
27810:
27808:
27805:
27803:
27800:
27799:
27797:
27795:
27791:
27785:
27782:
27780:
27777:
27775:
27772:
27770:
27767:
27765:
27762:
27760:
27757:
27755:
27752:
27750:
27747:
27745:
27742:
27741:
27739:
27737:
27733:
27730:
27728:
27727:2-dimensional
27724:
27720:
27716:
27711:
27707:
27697:
27694:
27692:
27689:
27687:
27684:
27682:
27679:
27677:
27674:
27672:
27671:Nonhypotenuse
27669:
27668:
27665:
27658:
27654:
27644:
27641:
27639:
27636:
27634:
27631:
27629:
27626:
27624:
27621:
27620:
27617:
27610:
27606:
27596:
27593:
27591:
27588:
27586:
27583:
27581:
27578:
27576:
27573:
27571:
27568:
27566:
27563:
27561:
27558:
27557:
27554:
27549:
27544:
27540:
27530:
27527:
27525:
27522:
27520:
27517:
27515:
27512:
27510:
27507:
27506:
27503:
27496:
27492:
27482:
27479:
27477:
27474:
27472:
27469:
27467:
27464:
27462:
27459:
27457:
27454:
27452:
27449:
27448:
27445:
27440:
27434:
27430:
27420:
27417:
27415:
27412:
27410:
27409:Perfect power
27407:
27405:
27402:
27400:
27399:Seventh power
27397:
27395:
27392:
27390:
27387:
27385:
27382:
27380:
27377:
27375:
27372:
27370:
27367:
27365:
27362:
27360:
27357:
27355:
27352:
27351:
27348:
27343:
27338:
27334:
27330:
27322:
27317:
27315:
27310:
27308:
27303:
27302:
27299:
27287:
27284:
27282:
27279:
27277:
27274:
27272:
27269:
27267:
27264:
27263:
27261:
27257:
27251:
27248:
27246:
27243:
27241:
27238:
27236:
27233:
27231:
27228:
27226:
27223:
27221:
27218:
27217:
27215:
27211:
27206:
27200:
27196:
27189:
27184:
27182:
27177:
27175:
27170:
27169:
27166:
27154:
27151:
27149:
27146:
27144:
27141:
27139:
27136:
27134:
27131:
27130:
27128:
27124:
27118:
27115:
27113:
27112:Logarithm (3)
27110:
27108:
27105:
27103:
27100:
27098:
27095:
27094:
27092:
27088:
27082:
27079:
27077:
27074:
27072:
27069:
27067:
27064:
27062:
27059:
27058:
27056:
27053:
27049:
27043:
27040:
27038:
27037:Pentation (5)
27035:
27033:
27032:Tetration (4)
27030:
27028:
27025:
27023:
27020:
27018:
27015:
27013:
27012:Successor (0)
27010:
27009:
27007:
27003:
26999:
26992:
26987:
26985:
26980:
26978:
26973:
26972:
26969:
26958:
26952:
26948:
26941:
26927:
26923:
26919:
26918:1001001, Inc.
26915:
26914:
26909:
26908:"80 Contents"
26903:
26888:
26884:
26880:
26879:
26874:
26867:
26848:
26844:
26837:
26836:
26831:
26827:
26821:
26813:
26806:
26787:
26783:
26776:
26775:
26770:
26766:
26762:
26758:
26754:
26748:
26740:
26734:
26730:
26723:
26709:
26703:
26699:
26695:
26694:
26689:
26683:
26676:
26673:and mentions
26672:
26668:
26657:
26653:
26649:
26645:
26639:
26631:
26627:
26623:
26619:
26615:
26611:
26607:
26603:
26599:
26598:
26593:
26587:
26579:
26575:
26569:
26555:on 2018-07-23
26551:
26547:
26543:
26538:
26533:
26529:
26525:
26518:
26511:
26504:
26498:
26491:
26485:
26477:
26470:
26461:
26453:
26447:
26443:
26438:
26437:
26428:
26420:
26413:
26405:
26401:
26400:
26395:
26391:
26384:
26375:
26374:
26369:
26366:
26359:
26350:
26344:
26340:
26336:
26335:
26330:
26329:Flahive, Mary
26323:
26316:
26312:
26309:
26304:
26298:
26294:
26290:
26283:
26275:
26269:
26265:
26261:
26257:
26253:
26246:
26238:
26232:
26228:
26221:
26213:
26211:9780470647691
26207:
26203:
26198:
26197:
26188:
26186:
26177:
26175:9780134439020
26171:
26167:
26160:
26151:
26145:
26141:
26137:
26133:
26132:
26127:
26123:
26119:
26115:
26111:
26107:
26103:
26097:
26091:
26087:
26083:
26079:
26073:
26069:
26065:
26061:
26057:
26056:
26048:
26040:
26038:3-87144-492-8
26034:
26030:
26026:
26022:
26018:
26017:
26012:
26008:
26002:
25995:
25991:
25989:9789401735964
25985:
25981:
25977:
25973:
25969:
25963:
25955:
25949:
25945:
25944:
25936:
25928:
25922:
25918:
25917:
25909:
25902:
25897:
25896:
25891:
25885:
25876:
25868:
25864:
25863:
25857:
25849:
25841:
25835:
25829:
25823:
25817:
25811:
25808:
25802:
25796:
25790:
25785:
25779:
25778:
25773:
25772:
25767:
25761:
25753:
25752:
25744:
25736:
25733:. Nuremberg:
25732:
25731:
25726:
25720:
25713:. 2017-06-23.
25712:
25706:
25698:
25694:
25690:
25689:
25684:
25678:
25670:
25669:
25661:
25653:
25649:
25648:
25643:
25640:
25633:
25618:
25614:
25608:
25606:
25598:
25594:
25587:
25579:
25575:
25571:
25570:
25565:
25559:
25551:
25547:
25546:
25541:
25538:
25531:
25529:
25520:
25518:90-277-0819-3
25514:
25510:
25506:
25502:
25501:
25493:
25485:
25479:
25475:
25471:
25466:
25465:
25459:
25453:
25451:
25442:
25436:
25421:
25420:
25415:
25412:
25405:
25391:
25387:
25380:
25378:
25376:
25374:
25372:
25367:
25353:
25347:
25328:
25325:
25322:
25314:
25298:
25295:
25275:
25272:
25269:
25261:
25255:
25251:
25238:
25235:
25232:
25229:
25226:
25223:
25221:
25218:
25217:
25213:
25210:
25207:
25204:
25201:
25198:
25196:
25193:
25192:
25188:
25185:
25182:
25179:
25176:
25173:
25171:
25168:
25167:
25163:
25160:
25157:
25154:
25151:
25148:
25146:
25143:
25142:
25138:
25135:
25132:
25129:
25126:
25123:
25121:
25118:
25117:
25113:
25110:
25107:
25104:
25101:
25098:
25095:
25091:
25088:
25087:
25083:
25080:
25077:
25074:
25071:
25068:
25066:
25063:
25062:
25058:
25055:
25052:
25049:
25046:
25043:
25041:
25038:
25037:
25033:
25030:
25027:
25024:
25021:
25018:
25016:
25013:
25012:
25008:
25005:
25002:
24999:
24996:
24993:
24991:
24987:
24984:
24983:
24979:
24976:
24973:
24970:
24967:
24964:
24962:
24959:
24958:
24954:
24951:
24948:
24945:
24942:
24939:
24937:
24934:
24933:
24929:
24926:
24923:
24920:
24917:
24914:
24912:
24909:
24908:
24904:
24901:
24898:
24895:
24892:
24889:
24887:
24884:
24883:
24879:
24876:
24873:
24870:
24867:
24864:
24862:
24859:
24858:
24854:
24851:
24848:
24845:
24842:
24839:
24837:
24834:
24833:
24829:
24826:
24823:
24820:
24817:
24814:
24812:
24809:
24808:
24804:
24801:
24798:
24795:
24792:
24789:
24787:
24784:
24783:
24779:
24776:
24773:
24770:
24767:
24764:
24762:
24759:
24758:
24754:
24751:
24748:
24745:
24742:
24739:
24737:
24734:
24733:
24729:
24726:
24723:
24720:
24717:
24714:
24712:
24709:
24708:
24704:
24701:
24698:
24695:
24692:
24689:
24687:
24684:
24683:
24679:
24676:
24673:
24670:
24667:
24664:
24662:
24659:
24658:
24654:
24651:
24648:
24645:
24642:
24639:
24637:
24634:
24633:
24629:
24626:
24623:
24620:
24617:
24614:
24612:
24609:
24608:
24604:
24601:
24598:
24595:
24592:
24589:
24587:
24584:
24583:
24579:
24576:
24573:
24570:
24567:
24564:
24562:
24559:
24558:
24555:
24552:
24550:
24547:
24545:
24542:
24540:
24537:
24535:
24532:
24530:
24527:
24522:
24517:
24515:
24510:
24508:
24503:
24502:
24500:
24499:
24493:
24490:
24488:
24487:
24483:
24479:
24477:
24474:
24472:
24469:
24467:
24464:
24462:
24459:
24457:
24454:
24452:
24449:
24447:
24444:
24442:
24439:
24437:
24434:
24432:
24429:
24427:
24424:
24423:
24418:
24412:
24407:
24398:as an integer
24386:
24372:
24358:
24357:
24356:
24354:
24350:
24346:
24338:
24331:
24328:
24321:
24318:
24311:
24308:
24301:
24294:
24290:
24283:
24282:
24281:
24279:
24271:
24264:
24261:
24254:
24253:
24252:
24249:
24247:
24243:
24239:
24219:
24211:
24204:
24201:
24194:
24191:
24187:
24183:
24176:
24173:
24169:
24165:
24161:
24157:
24153:
24149:
24145:
24141:
24137:
24133:
24129:
24125:
24121:
24117:
24113:
24109:
24105:
24101:
24097:
24093:
24089:
24085:
24081:
24077:
24073:
24057:
24050:
24047:
24043:
24039:
24035:
24031:
24027:
24023:
24019:
24015:
24011:
24007:
24003:
23999:
23995:
23991:
23984:
23983:
23982:
23980:
23972:
23964:
23960:
23956:
23947:
23934:
23928:
23925:
23922:
23918:
23913:
23907:
23901:
23898:
23894:
23890:
23882:
23866:
23863:
23860:
23857:
23854:
23848:
23842:
23837:
23834:
23830:
23826:
23823:
23820:
23815:
23812:
23808:
23799:
23783:
23780:
23771:
23757:
23748:
23742:
23739:
23733:
23730:
23707:
23701:
23698:
23695:
23689:
23683:
23680:
23657:
23651:
23646:
23642:
23621:
23618:
23613:
23609:
23600:
23584:
23581:
23578:
23575:
23572:
23569:
23566:
23561:
23557:
23536:
23533:
23530:
23527:
23524:
23521:
23518:
23513:
23510:
23506:
23485:
23479:
23473:
23470:
23464:
23458:
23455:
23450:
23442:
23436:
23416:
23407:
23401:
23395:
23392:
23386:
23378:
23374:
23365:
23361:
23357:
23353:
23348:
23335:
23323:
23317:
23311:
23305:
23282:
23274:
23270:
23239:
23235:
23226:
23210:
23179:
23173:
23167:
23164:
23158:
23149:
23146:
23143:
23133:
23132:
23131:
23117:
23114:
23111:
23108:
23100:
23096:
23092:
23088:
23084:
23079:
23069:
23067:
23062:
23057:
23053:
23045:
23025:
23014:
22998:
22995:
22992:
22986:
22983:
22978:
22974:
22967:
22964:
22952:
22946:
22905:
22904:
22882:
22881:
22868:
22867:
22854:
22853:
22846:
22845:
22842:(2) = 2 = 64
22841:
22840:
22836:
22835:
22831:
22830:
22827:
22803:
22800:
22797:
22789:
22785:
22781:
22778:
22770:
22766:
22762:
22757:
22753:
22749:
22744:
22740:
22736:
22731:
22727:
22723:
22720:
22713:
22712:
22711:
22709:
22708:Horner's rule
22699:
22693:
22682:
22675:
22668:
22662:
22656:
22646:
22640:
22634:
22628:
22625:
22618:
22611:
22607:
22599:
22593:
22587:
22582:
22577:
22566:
22553:
22548:
22535:
22530:
22522:
22515:
22511:
22506:
22498:
22491:
22487:
22486:
22485:
22482:
22465:, except for
22458:
22451:
22431:, except for
22429:
22424:
22419:
22413:
22412:has a limit.
22410:
22405:
22398:
22393:
22388:
22382:
22375:
22371:
22363:
22356:
22336:
22333:
22330:
22327:
22322:
22312:
22306:
22303:
22300:
22291:
22288:
22266:
22262:
22258:
22252:
22249:
22246:
22240:
22231:
22224:
22219:
22215:
22206:
22180:
22176:
22172:
22168:
22162:
22158:
22148:
22130:
22127:
22124:
22118:
22110:
22094:
22091:
22088:
22085:
22079:
22057:
22053:
22046:
22038:
22034:
22029:
22020:
22019:right adjoint
21994:
21988:
21985:
21982:
21979:
21976:
21970:
21967:
21964:
21956:
21952:
21948:
21945:
21939:
21936:
21929:
21928:
21927:
21911:
21908:
21905:
21901:
21897:
21892:
21882:
21878:
21854:
21848:
21845:
21842:
21836:
21833:
21811:
21807:
21775:between sets
21774:
21770:
21764:
21754:
21751:
21743:
21733:
21727:
21720:
21715:
21709:
21708:inverse image
21701:
21685:
21679:
21676:
21673:
21654:
21636:
21632:
21611:
21605:
21602:
21599:
21583:
21581:
21577:
21569:
21528:
21493:
21489:
21488:vector spaces
21485:
21481:
21476:
21474:
21458:
21438:
21415:
21410:
21406:
21402:
21397:
21393:
21389:
21384:
21381:
21378:
21374:
21366:
21352:
21347:
21344:
21341:
21337:
21333:
21328:
21318:
21314:
21303:
21302:
21301:
21299:
21281:
21277:
21251:
21237:
21231:
21228:
21225:
21222:
21219:
21208:
21183:
21179:
21175:
21172:
21169:
21164:
21160:
21142:
21132:
21130:
21126:
21122:
21106:
21053:
21038:
21020:
21016:
21002:
20979:
20975:
20971:
20968:
20965:
20960:
20956:
20945:
20919:
20915:
20901:
20888:
20882:
20879:
20876:
20873:
20870:
20847:
20841:
20838:
20832:
20829:
20826:
20801:
20792:
20789:
20786:
20780:
20777:
20766:
20763:
20760:
20756:
20752:
20736:
20733:
20730:
20727:
20707:
20704:
20701:
20678:
20675:
20672:
20662:
20661:ordered pairs
20651:
20647:
20637:
20617:
20613:
20592:
20574:
20570:
20549:
20544:
20525:
20521:
20517:
20512:
20506:
20488:
20473:
20465:
20439:
20412:
20407:
20403:
20399:
20396:
20388:
20385:, called the
20384:
20368:
20363:
20348:
20323:
20319:
20312:
20302:
20287:
20273:
20270:
20259:
20258:
20257:
20243:
20238:
20211:
20208:
20203:
20199:
20169:
20165:
20161:
20156:
20152:
20148:
20143:
20135:
20132:
20129:
20119:
20118:
20117:
20115:
20099:
20094:
20078:
20076:
20060:
20040:
20035:
20005:
20002:
19999:
19993:
19973:
19968:
19938:
19935:
19930:
19926:
19922:
19917:
19914:
19911:
19907:
19903:
19900:
19897:
19892:
19888:
19884:
19881:
19878:
19873:
19869:
19851:
19828:
19813:
19808:
19795:
19790:
19780:
19777:
19754:
19751:
19746:
19742:
19734:
19733:
19732:
19729:
19716:
19711:
19692:
19656:
19651:
19647:
19643:
19640:
19632:
19628:
19624:
19623:finite number
19620:
19615:
19613:
19609:
19605:
19601:
19593:
19589:
19585:
19579:
19572:Finite fields
19569:
19567:
19563:
19559:
19555:
19554:wave equation
19551:
19547:
19546:heat equation
19543:
19523:
19517:
19506:
19499:
19495:
19489:
19483:
19475:
19471:
19467:
19461:
19457:
19451:
19445:
19439:
19434:
19429:
19423:
19420:
19416:
19411:
19402:
19401:
19400:
19397:
19391:
19371:
19364:
19361:
19357:
19351:
19345:
19339:
19336:
19332:
19328:
19302:
19296:
19276:
19273:
19269:
19265:
19257:
19253:
19248:
19246:
19241:
19222:
19218:
19208:
19191:
19186:
19182:
19161:
19156:
19152:
19143:
19138:
19132:
19126:
19121:
19116:
19100:
19095:
19090:
19087:
19083:
19079:
19074:
19069:
19066:
19062:
19052:
19033:
19029:
19020:
19015:
19009:
19003:
18992:
18990:
18983:
18962:
18958:
18954:
18951:
18948:
18943:
18939:
18932:
18925:
18921:
18920:radical ideal
18917:
18909:
18888:
18886:
18882:
18878:
18874:
18854:
18851:
18846:
18842:
18821:
18818:
18815:
18808:(that is, if
18807:
18802:
18801:of the ring.
18800:
18797:, called the
18796:
18792:
18788:
18768:
18765:
18760:
18756:
18747:
18737:
18724:
18719:
18715:
18709:
18705:
18701:
18696:
18688:
18685:
18660:
18657:
18653:
18649:
18644:
18634:
18630:
18616:
18610:
18604:
18601:
18597:
18592:
18587:
18585:
18581:
18577:
18573:
18568:
18534:
18531:
18528:
18523:
18519:
18515:
18510:
18506:
18485:
18481:
18456:
18452:
18441:generated by
18440:
18432:
18427:
18409:
18406:
18403:
18381:
18377:
18363:
18361:
18357:
18353:
18343:
18330:
18324:
18316:
18312:
18288:
18277:
18274:
18270:
18246:
18241:
18233:
18227:
18204:
18193:
18189:
18162:
18153:
18144:
18138:
18132:
18129:
18123:
18117:
18114:
18108:
18097:
18094:
18090:
18079:
18078:
18077:
18060:
18054:
18048:
18045:
18039:
18033:
18026:
18020:
18017:
18012:
18001:
17995:
17989:
17983:
17972:
17968:
17957:
17956:
17955:
17953:
17935:
17932:
17928:
17905:
17901:
17892:
17891:real function
17883:
17881:
17877:
17876:endomorphisms
17873:
17869:
17865:
17861:
17857:
17853:
17849:
17845:
17817:
17814:
17811:
17808:
17805:
17802:
17791:
17787:
17781:
17777:
17773:
17771:
17764:
17756:
17753:
17741:
17738:
17734:
17730:
17728:
17721:
17711:
17707:
17694:
17690:
17684:
17680:
17676:
17674:
17667:
17664:
17661:
17657:
17649:
17646:
17644:
17637:
17633:
17621:
17620:
17619:
17600:
17587:
17582:
17579:
17574:
17569:
17566:
17562:
17558:
17547:
17541:
17532:
17510:
17506:
17471:
17467:
17463:
17460:
17455:
17452:
17449:
17445:
17437:
17423:
17420:
17417:
17412:
17408:
17400:
17399:
17398:
17392:
17384:
17379:
17377:
17372:
17367:
17357:
17354:
17329:
17323:
17320:
17317:
17311:
17308:
17295:
17281:
17276:
17263:
17259:
17251:
17246:
17237:
17232:
17223:
17213:
17187:
17180:
17176:
17172:
17169:
17166:
17162:
17158:
17153:
17148:
17143:
17139:
17135:
17124:
17120:
17098:
17095:
17092:
17089:
17086:
17083:
17080:
17060:
17054:
17051:
17048:
17045:
17042:
17039:
17033:
17030:
17027:
17023:
17016:
17013:
17010:
17007:
17004:
17001:
16995:
16992:
16989:
16986:
16980:
16977:
16974:
16971:
16968:
16965:
16958:
16954:
16951:
16942:
16938:
16921:
16915:
16912:
16909:
16906:
16900:
16897:
16875:
16871:
16850:
16844:
16838:
16835:
16813:
16809:
16790:(dividing by
16776:
16773:
16766:
16762:
16756:
16752:
16748:
16745:
16741:
16733:
16716:
16712:
16706:
16702:
16698:
16693:
16690:
16687:
16683:
16661:
16658:
16651:
16647:
16641:
16637:
16633:
16630:
16626:
16620:
16617:
16614:
16611:
16607:
16601:
16597:
16589:
16572:
16569:
16565:
16561:
16556:
16551:
16546:
16542:
16538:
16515:
16512:
16505:
16501:
16495:
16491:
16487:
16484:
16481:
16478:
16475:
16472:
16469:
16466:
16462:
16454:
16436:
16433:
16430:
16427:
16424:
16421:
16397:
16394:
16389:
16386:
16383:
16380:
16377:
16374:
16369:
16364:
16361:
16358:
16355:
16352:
16349:
16345:
16341:
16332:
16318:
16315:
16312:
16309:
16306:
16303:
16298:
16295:
16292:
16289:
16285:
16279:
16275:
16271:
16266:
16263:
16260:
16257:
16254:
16251:
16247:
16239:
16238:
16232:
16228:
16215:
16211:
16206:The identity
16205:
16203:is incorrect.
16174:
16171:
16168:
16163:
16160:
16155:
16146:
16143:
16134:
16131:
16121:
16118:
16113:
16107:
16104:
16101:
16095:
16092:
16083:
16080:
16074:
16068:
16065:
16059:
16053:
16050:
16046:
16041:
16019:
16016:
16013:
16007:
16004:
15995:
15992:
15983:
15980:
15971:
15968:
15962:
15959:
15956:
15950:
15947:
15938:
15935:
15932:
15929:
15926:
15902:
15898:
15894:
15890:
15883:
15880:
15876:
15870:
15851:
15842:
15839:
15836:
15833:
15830:
15827:
15824:
15821:
15818:
15815:
15812:
15809:
15806:
15803:
15799:
15795:
15793:
15787:
15783:
15780:
15777:
15774:
15770:
15761:
15752:
15749:
15746:
15743:
15740:
15737:
15734:
15731:
15728:
15725:
15722:
15719:
15716:
15713:
15710:
15707:
15704:
15701:
15698:
15695:
15692:
15689:
15685:
15681:
15679:
15673:
15667:
15663:
15659:
15656:
15652:
15640:
15628:
15620:
15614:
15613:proper subset
15609:
15605:
15598:
15591:
15574:
15571:
15568:
15564:
15556:
15553:
15550:
15547:
15544:
15539:
15535:
15531:
15528:
15519:
15506:
15503:
15500:
15497:
15492:
15488:
15485:
15482:
15475:
15472:
15464:
15460:
15456:
15453:
15450:
15446:
15439:
15436:
15433:
15430:
15424:
15421:
15415:
15412:
15409:
15406:
15403:
15400:
15397:
15391:
15388:
15382:
15379:
15376:
15368:
15360:
15357:
15348:
15345:
15336:
15323:
15319:
15315:
15310:The identity
15309:
15308:
15306:
15304:
15294:
15288:
15270:
15266:
15241:
15238:
15235:
15232:
15229:
15205:
15196:
15193:
15190:
15187:
15181:
15178:
15175:
15172:
15166:
15163:
15160:
15157:
15151:
15148:
15137:
15134:
15131:
15128:
15125:
15119:
15116:
15112:
15106:
15102:
15098:
15095:
15093:
15076:
15073:
15070:
15067:
15064:
15058:
15055:
15052:
15049:
15046:
15043:
15037:
15034:
15031:
15028:
15019:
15016:
15013:
15010:
15007:
15001:
14998:
14995:
14992:
14989:
14986:
14980:
14977:
14966:
14963:
14960:
14957:
14954:
14948:
14945:
14941:
14935:
14931:
14927:
14925:
14918:
14915:
14912:
14909:
14901:
14898:
14871:
14866:
14863:
14859:
14855:
14852:
14849:
14846:
14819:
14816:
14813:
14810:
14802:
14799:
14789:
14775:
14772:
14765:
14762:
14757:
14753:
14730:
14726:
14705:
14700:
14697:
14694:
14691:
14687:
14679:
14676:
14671:
14667:
14663:
14658:
14655:
14652:
14649:
14645:
14641:
14636:
14632:
14611:
14607:
14603:
14600:
14597:
14594:
14589:
14586:
14580:
14576:
14573:
14570:
14567:
14564:
14544:
14541:
14538:
14518:
14513:
14509:
14505:
14502:
14498:
14494:
14491:
14464:
14460:
14452:
14451:
14430:
14427:
14424:
14404:
14400:
14393:
14390:
14387:
14384:
14381:
14378:
14375:
14372:
14369:
14366:
14363:
14360:
14354:
14351:
14348:
14345:
14339:
14336:
14333:
14330:
14327:
14324:
14321:
14318:
14315:
14312:
14309:
14306:
14300:
14297:
14293:
14284:
14281:
14278:
14275:
14272:
14266:
14263:
14259:
14253:
14249:
14245:
14240:
14236:
14215:
14210:
14206:
14202:
14197:
14194:
14191:
14188:
14184:
14161:
14157:
14151:
14147:
14143:
14138:
14135:
14132:
14128:
14119:
14116:
14102:
14099:
14096:
14076:
14070:
14067:
14064:
14061:
14058:
14055:
14052:
14049:
14046:
14043:
14040:
14037:
14031:
14028:
14022:
14019:
14016:
14013:
14010:
14007:
14004:
14001:
13998:
13995:
13992:
13989:
13983:
13980:
13977:
13974:
13971:
13951:
13948:
13945:
13942:
13914:
13911:
13908:
13905:
13902:
13899:
13891:
13878:
13875:
13872:
13869:
13866:
13857:
13838:
13835:
13832:
13829:
13821:
13805:
13785:
13782:
13779:
13776:
13773:
13770:
13767:
13764:
13761:
13758:
13755:
13747:
13743:
13738:
13734:
13731:
13730:
13710:
13707:
13704:
13698:
13695:
13692:
13689:
13665:
13661:
13657:
13652:
13648:
13642:
13639:
13619:
13613:
13610:
13607:
13604:
13601:
13598:
13595:
13592:
13586:
13583:
13578:
13575:
13571:
13567:
13564:
13561:
13529:
13526:
13523:
13520:
13517:
13514:
13506:
13501:
13497:
13496:
13495:
13471:
13467:
13446:
13443:
13440:
13437:
13430:
13420:
13404:
13400:
13383:
13367:
13364:
13361:
13358:
13349:
13347:
13339:
13319:
13316:
13313:
13310:
13284:
13280:
13259:
13256:
13253:
13250:
13242:
13217:
13214:
13209:
13206:
13197:
13184:
13181:
13178:
13175:
13155:
13152:
13149:
13127:
13123:
13119:
13114:
13110:
13101:
13089:
13067:
13063:
13049:
13028:
13023:
13020:
13017:
13014:
13011:
13007:
13001:
12997:
12993:
12985:
12982:
12979:
12976:
12973:
12970:
12967:
12964:
12958:
12954:
12946:
12945:
12944:
12928:
12924:
12899:
12896:
12893:
12890:
12887:
12884:
12881:
12878:
12875:
12855:
12852:
12849:
12840:
12838:
12816:
12812:
12791:
12788:
12785:
12771:
12753:
12750:
12746:
12742:
12739:
12736:
12714:
12710:
12696:
12676:
12672:
12662:
12659:
12656:
12646:The function
12644:
12630:
12627:
12624:
12604:
12599:
12596:
12593:
12590:
12586:
12582:
12577:
12573:
12550:
12546:
12536:
12523:
12520:
12517:
12514:
12511:
12508:
12505:
12502:
12490:
12482:
12481:discontinuous
12466:
12463:
12460:
12457:
12434:
12431:
12428:
12425:
12422:
12419:
12416:
12413:
12410:
12403:
12402:
12401:
12395:
12376:
12373:
12370:
12365:
12362:
12359:
12355:
12347:
12346:
12345:
12327:
12324:
12316:
12312:
12302:
12300:
12277:
12274:
12269:
12266:
12263:
12259:
12251:
12250:
12249:
12247:
12231:
12228:
12225:
12202:
12197:
12194:
12191:
12188:
12184:
12180:
12175:
12171:
12163:
12162:
12161:
12159:
12154:
12152:
12136:
12132:
12121:
12105:
12101:
12088:
12072:
12068:
12064:
12060:
12051:
12047:
12039:
12031:
12008:
12004:
12001:
11998:
11995:
11989:
11979:
11977:
11969:
11968:complex plane
11965:
11956:
11925:
11912:
11909:
11906:
11886:
11866:
11863:
11860:
11848:
11829:
11823:
11819:
11816:
11813:
11810:
11804:
11800:
11795:
11791:
11782:
11758:
11753:
11750:
11747:
11743:
11739:
11734:
11730:
11726:
11721:
11717:
11713:
11710:
11707:
11702:
11698:
11694:
11689:
11685:
11681:
11678:
11655:
11651:
11648:
11645:
11639:
11635:
11632:
11612:
11610:
11606:
11596:
11575:
11570:
11569:Root of unity
11560:
11542:
11538:
11534:
11531:
11528:
11524:
11517:
11514:th roots are
11497:
11494:
11491:
11482:
11476:
11468:
11452:
11448:
11437:that is, the
11424:
11421:
11418:
11415:
11412:
11409:
11406:
11394:
11382:
11349:
11345:
11341:
11338:
11335:
11315:
11295:
11292:
11269:
11263:
11259:
11256:
11250:
11242:
11238:
11233:
11227:
11224:
11218:
11212:
11209:
11205:
11201:
11197:
11188:
11187:
11186:
11171:
11157:
11137:
11134:
11131:
11128:
11125:
11105:
11090:
11086:
11070:
11058:
11042:
11019:
11013:
11010:
11007:
11004:
11001:
10998:
10995:
10992:
10986:
10983:
10978:
10975:
10971:
10967:
10964:
10961:
10954:
10953:
10952:
10950:
10933:
10931:
10907:
10904:
10900:
10896:
10864:
10855:
10852:
10849:
10846:
10840:
10837:
10834:
10831:
10825:
10822:
10819:
10816:
10810:
10807:
10799:
10795:
10791:
10786:
10783:
10780:
10777:
10774:
10770:
10764:
10760:
10756:
10751:
10748:
10744:
10738:
10734:
10730:
10725:
10722:
10719:
10716:
10712:
10704:
10703:
10702:
10700:
10699:trigonometric
10696:
10677:
10668:
10665:
10662:
10659:
10653:
10650:
10647:
10644:
10638:
10635:
10632:
10629:
10623:
10620:
10612:
10608:
10604:
10599:
10596:
10593:
10590:
10586:
10578:
10577:
10576:
10570:
10552:
10548:
10539:
10520:
10517:
10514:
10511:
10508:
10505:
10502:
10499:
10496:
10493:
10488:
10485:
10481:
10473:
10472:
10471:
10469:
10465:
10440:
10435:
10432:
10428:
10424:
10419:
10414:
10409:
10405:
10401:
10392:
10391:
10390:
10388:
10387:
10381:
10362:
10357:
10352:
10348:
10344:
10319:
10314:
10310:
10304:
10300:
10296:
10291:
10288:
10285:
10281:
10273:
10272:
10271:
10268:
10262:
10246:
10243:
10240:
10217:
10209:
10206:
10203:
10200:
10193:
10189:
10184:
10180:
10172:
10171:
10170:
10168:
10167:
10158:
10140:
10133:
10114:
10111:
10108:
10105:
10101:
10075:
10072:
10069:
10066:
10062:
10058:
10053:
10048:
10043:
10040:
10037:
10033:
10029:
10024:
10019:
10015:
10007:
10006:
10005:
9991:
9986:
9983:
9979:
9975:
9970:
9960:
9956:
9942:
9920:
9917:
9914:
9910:
9906:
9900:
9897:
9894:
9888:
9885:
9882:
9879:
9872:
9871:
9870:
9867:
9862:
9856:
9851:
9847:
9838:
9832:
9821:
9803:
9798:
9794:
9770:
9764:
9761:
9749:
9744:
9724:
9720:
9716:
9710:
9704:
9701:
9671:
9667:
9663:
9657:
9651:
9648:
9625:
9619:
9616:
9613:
9610:
9601:
9584:
9581:
9578:
9572:
9569:
9566:
9560:
9554:
9551:
9545:
9539:
9536:
9512:
9508:
9503:
9500:
9474:
9469:
9464:
9456:
9452:
9447:
9444:
9438:
9433:
9429:
9426:
9423:
9417:
9414:
9410:
9397:
9389:
9384:
9379:
9373:
9370:
9365:
9362:
9358:
9351:
9346:
9340:
9337:
9332:
9329:
9325:
9312:
9304:
9298:
9292:
9289:
9283:
9277:
9274:
9267:
9266:
9265:
9248:
9242:
9239:
9233:
9227:
9224:
9221:
9215:
9212:
9209:
9203:
9200:
9193:
9177:
9174:
9171:
9165:
9159:
9156:
9133:
9128:
9123:
9117:
9114:
9109:
9106:
9102:
9089:
9081:
9075:
9069:
9066:
9059:
9058:
9057:
9055:
9036:
9031:
9027:
9023:
9017:
9011:
9008:
9001:
9000:
8999:
8984:
8978:
8972:
8969:
8961:
8957:
8941:
8938:
8935:
8915:
8910:
8906:
8899:
8891:
8885:
8875:
8873:
8872:
8859:
8833:
8829:
8821:This defines
8819:
8806:
8801:
8797:
8788:
8769:
8766:
8762:
8756:
8752:
8748:
8743:
8739:
8734:
8730:
8726:
8720:
8716:
8712:
8707:
8703:
8698:
8694:
8690:
8684:
8680:
8676:
8671:
8667:
8662:
8658:
8654:
8648:
8644:
8640:
8635:
8631:
8626:
8622:
8618:
8612:
8608:
8604:
8599:
8595:
8590:
8586:
8582:
8576:
8572:
8568:
8563:
8559:
8554:
8546:
8545:
8544:
8531:
8526:
8522:
8513:
8507:
8502:
8493:
8487:
8458:
8447:
8444:
8440:
8435:
8425:
8422:
8413:
8409:
8403:
8389:
8383:
8375:
8370:
8366:
8358:
8357:
8356:
8354:
8342:
8338:
8324:
8318:
8314:The limit of
8312:
8303:
8301:
8297:
8296:
8292:is true; see
8274:
8271:
8267:
8263:
8258:
8253:
8248:
8244:
8240:
8231:
8230:
8229:
8227:
8221:
8219:
8215:
8211:
8210:
8205:
8201:
8197:
8196:
8185:
8183:
8182:
8177:
8176:
8156:
8153:
8150:
8145:
8137:
8134:
8128:
8121:
8118:
8113:
8110:
8102:
8099:
8093:
8090:
8087:
8081:
8078:
8073:
8069:
8063:
8060:
8054:
8048:
8040:
8037:
8030:
8021:
8020:
8019:
8003:
8000:
7996:
7992:
7987:
7977:
7973:
7962:the identity
7949:
7943:
7940:
7935:
7918:
7905:
7889:
7886:
7882:
7878:
7873:
7863:
7859:
7846:
7835:
7822:
7816:
7813:
7802:
7798:
7791:
7785:
7782:
7776:
7770:
7760:
7756:
7748:
7743:
7737:
7734:
7728:
7722:
7712:
7708:
7700:
7695:
7690:
7686:
7682:
7677:
7666:
7663:
7658:
7634:
7628:
7625:
7620:
7616:
7613:
7590:
7585:
7574:
7571:
7566:
7559:
7553:
7550:
7544:
7539:
7535:
7531:
7526:
7520:
7517:
7512:
7504:
7503:
7502:
7486:
7482:
7478:
7474:
7457:
7439:
7436:
7421:
7408:
7405:
7402:
7397:
7393:
7376:
7354:
7350:
7326:
7322:
7318:
7314:
7301:
7288:
7282:
7276:
7270:
7264:
7258:
7252:
7246:
7221:
7207:
7196:
7185:
7174:
7166:
7158:
7155:
7152:
7150:
7147:
7146:
7131:
7120:
7109:
7098:
7090:
7082:
7079:
7076:
7073:
7071:
7068:
7067:
7052:
7041:
7030:
7019:
7011:
7003:
7000:
6997:
6994:
6992:
6989:
6988:
6976:
6965:
6954:
6946:
6938:
6930:
6927:
6924:
6921:
6919:
6916:
6915:
6903:
6892:
6881:
6873:
6865:
6860:
6857:
6854:
6851:
6849:
6846:
6845:
6833:
6822:
6814:
6806:
6798:
6795:
6792:
6789:
6786:
6784:
6781:
6780:
6768:
6760:
6752:
6744:
6739:
6736:
6733:
6730:
6727:
6725:
6722:
6721:
6712:
6704:
6699:
6694:
6691:
6688:
6685:
6682:
6679:
6677:
6674:
6673:
6669:
6666:
6663:
6660:
6657:
6654:
6651:
6648:
6645:
6643:
6640:
6639:
6635:
6632:
6629:
6626:
6623:
6620:
6617:
6614:
6611:
6609:
6606:
6605:
6602:
6599:
6597:
6594:
6592:
6589:
6587:
6584:
6582:
6579:
6577:
6574:
6572:
6569:
6567:
6564:
6562:
6559:
6557:
6554:
6553:
6545:
6531:
6528:
6525:
6516:
6514:
6513:odd functions
6493:
6487:
6484:
6481:
6475:
6472:
6466:
6444:
6441:
6438:
6418:
6396:
6392:
6388:
6385:
6382:
6362:
6342:
6334:
6316:
6312:
6308:
6305:
6299:
6293:
6273:
6270:
6267:
6247:
6227:
6219:
6200:
6194:
6174:
6165:
6163:
6142:
6136:
6133:
6127:
6124:
6118:
6109:
6093:
6071:
6067:
6063:
6060:
6057:
6037:
6017:
6009:
5991:
5987:
5983:
5980:
5974:
5968:
5948:
5928:
5925:
5922:
5902:
5882:
5862:
5859:
5856:
5848:
5832:
5812:
5809:
5806:
5784:
5780:
5776:
5773:
5767:
5761:
5748:
5742:
5734:
5728:
5723:
5713:
5711:
5710:
5705:
5700:
5698:
5697:
5687:
5681:
5677:
5672:
5671:
5670:
5663:
5660:
5654:
5648:
5641:
5635:
5632:
5626:
5604:
5598:
5591:
5587:
5586:
5585:
5579:| < 1
5577:
5568:
5561:
5557:
5556:
5555:
5553:
5548:
5546:
5542:
5538:
5534:
5530:
5521:
5514:
5507:
5503:
5502:
5501:
5499:
5489:
5487:
5486:
5480:
5475:
5466:
5451:
5445:
5438:
5427:
5421:
5414:
5400:
5396:
5392:
5382:
5362:
5359:
5355:
5350:
5346:
5328:
5324:is negative (
5318:
5306:
5302:is positive (
5291:
5288:
5284:
5278:
5277:is negative.
5265:Powers of one
5262:
5240:
5224:
5216:
5212:
5211:binary number
5208:
5203:
5194:
5185:
5179:
5175:
5170:
5165:
5156:
5154:
5153:
5148:
5147:
5136:
5129:Powers of two
5126:
5110:
5102:
5098:
5088:
5070:
5066:
5051:
5046:
5040:
5010:
5004:
4998:
4978:
4975:
4974:
4970:
4967:
4966:
4962:
4959:
4958:
4954:
4951:
4950:
4947:(1, 1, 1, 1)
4946:
4943:
4942:
4938:
4935:
4934:
4929:
4918:
4913:
4910:
4906:
4905:
4902:
4880:
4872:
4860:
4856:
4851:
4836:
4826:
4824:
4812:
4808:
4804:
4800:
4788:
4784:
4779:
4775:
4755:
4750:
4747:
4744:
4740:
4734:
4730:
4723:
4717:
4714:
4711:
4705:
4702:
4697:
4694:
4686:
4681:
4678:
4675:
4671:
4667:
4662:
4659:
4656:
4652:
4646:
4642:
4630:
4627:
4614:
4609:
4606:
4603:
4599:
4595:
4590:
4582:
4579:
4576:
4566:
4565:
4564:
4563:
4539:
4534:
4531:
4527:
4523:
4518:
4513:
4508:
4504:
4500:
4491:
4490:
4489:
4472:
4466:
4461:
4457:
4453:
4448:
4444:
4437:
4433:
4428:
4420:
4419:
4418:
4416:
4412:
4408:
4404:
4392:
4387:2 = 8 ≠ 3 = 9
4384:
4359:
4355:
4351:
4346:
4342:
4338:
4336:
4329:
4321:
4318:
4315:
4303:
4300:
4297:
4293:
4289:
4287:
4280:
4275:
4270:
4266:
4262:
4251:
4247:
4243:
4238:
4234:
4230:
4228:
4221:
4218:
4215:
4211:
4199:
4198:
4197:
4195:
4187:
4180:
4170:
4157:
4152:
4149:
4145:
4133:
4129:
4121:
4117:
4113:
4109:
4104:
4090:
4087:
4084:
4081:
4059:
4055:
4051:
4046:
4042:
4038:
4033:
4030:
4027:
4023:
4013:
3971:
3967:
3963:
3958:
3953:
3950:
3946:
3938:
3937:
3936:
3920:
3917:
3898:
3884:
3881:
3878:
3853:
3849:
3845:
3840:
3836:
3832:
3827:
3824:
3821:
3817:
3809:
3808:
3807:
3805:
3801:
3797:
3796:empty product
3778:
3775:
3770:
3766:
3758:
3757:
3756:
3743:Zero exponent
3726:
3721:
3718:
3714:
3710:
3705:
3695:
3691:
3680:
3679:
3678:
3661:
3656:
3652:
3648:
3643:
3639:
3635:
3630:
3627:
3624:
3620:
3612:
3611:
3610:
3585:
3582:
3579:
3574:
3570:
3566:
3561:
3558:
3555:
3551:
3543:
3542:
3541:
3539:
3520:
3517:
3512:
3508:
3500:
3499:
3498:
3495:
3493:
3489:
3485:
3475:
3473:
3463:
3461:
3457:
3453:
3449:
3445:
3441:
3437:
3433:
3428:
3418:
3416:
3412:
3388:
3383:
3380:
3374:
3368:
3362:
3357:
3352:
3348:
3344:
3340:
3334:
3331:
3325:
3319:
3313:
3307:
3301:
3295:
3289:
3284:
3279:
3275:
3271:
3259:
3257:
3251:
3249:
3239:
3237:
3233:
3229:
3225:
3221:
3212:
3209:
3205:
3201:
3197:
3192:
3180:
3177:
3171:
3165:
3159:
3153:
3146:
3144:
3143:
3138:
3133:
3127:
3122:
3118:
3108:
3106:
3102:
3098:
3094:
3090:
3080:
3077:
3060:
3056:
3052:
3047:
3042:to represent
3037:
3023:
3021:
3017:
3016:
3011:
3010:
3005:
3001:
2997:
2993:
2992:
2987:
2983:
2979:
2971:
2967:
2957:
2943:
2939:
2938:
2931:
2911:
2909:
2905:
2901:
2897:
2893:
2892:ancient Greek
2889:
2885:
2881:
2877:
2873:
2869:
2866:
2862:
2852:
2850:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2818:
2814:
2809:
2807:
2803:
2798:
2782:
2778:
2774:
2770:
2766:
2761:
2736:
2733:
2727:
2724:
2717:. Therefore,
2704:
2701:
2698:
2695:
2692:
2670:
2666:
2662:
2657:
2654:
2651:
2647:
2624:
2620:
2599:
2579:
2576:
2571:
2568:
2565:
2561:
2540:
2537:
2532:
2528:
2524:
2519:
2515:
2494:
2474:
2471:
2466:
2461:
2456:
2432:
2427:
2422:
2418:
2397:
2375:
2364:
2348:
2344:
2339:
2335:
2332:
2327:
2324:
2320:
2299:
2296:
2291:
2287:
2266:
2262:
2258:
2255:
2250:
2247:
2243:
2220:
2216:
2211:
2207:
2204:
2199:
2196:
2192:
2169:
2165:
2144:
2141:
2136:
2132:
2128:
2123:
2120:
2117:
2114:
2110:
2106:
2101:
2097:
2093:
2088:
2085:
2081:
2058:
2055:
2051:
2041:
2027:
2024:
2019:
2015:
1992:
1988:
1984:
1979:
1976:
1973:
1970:
1967:
1963:
1959:
1954:
1950:
1946:
1941:
1937:
1933:
1928:
1924:
1920:
1915:
1905:
1901:
1877:
1874:
1869:
1865:
1855:
1841:
1838:
1833:
1829:
1824:
1818:
1814:
1810:
1805:
1801:
1778:
1774:
1751:
1747:
1743:
1738:
1735:
1732:
1728:
1724:
1719:
1715:
1711:
1706:
1702:
1681:
1661:
1658:
1655:
1633:
1629:
1619:
1600:
1596:
1592:
1587:
1583:
1579:
1577:
1562:
1556:
1552:
1549:
1546:
1543:
1540:
1533:
1523:
1517:
1513:
1510:
1507:
1504:
1501:
1494:
1492:
1477:
1474:
1471:
1465:
1461:
1458:
1455:
1452:
1449:
1442:
1440:
1433:
1430:
1427:
1423:
1410:
1396:
1376:
1354:
1350:
1329:
1320:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1277:
1272:
1267:
1254:
1244:
1238:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1203:
1198:
1194:
1181:
1176:
1171:
1167:
1151:
1143:
1134:
1129:
1125:
1121:
1120:
1115:
1111:
1107:
1103:
1091:
1086:
1084:
1079:
1077:
1072:
1071:
1069:
1068:
1038:
1022:
1007:
998:
989:
986:
982:
978:
952:
936:
911:
908:
904:
902:
897:
871:
855:
850:
799:
796:
792:
788:
785:
731:
721:
705:
700:
637:
634:
630:
626:
623:
598:
582:
577:
563:
542:
520:
517:
513:
509:
483:
467:
462:
448:
427:
405:
402:
398:
394:
368:
352:
347:
333:
312:
291:
270:
248:
245:
241:
237:
234:
233:
227:
222:
220:
215:
213:
208:
207:
204:
200:
199:
188:
181:
135:
120:
118:
103:
89:
83:
79:
73:
65:
62:and exponent
61:
56:
49:
45:
40:
37:
33:
19:
29241:Exponentials
28953:Transposable
28817:Narcissistic
28724:Digital root
28644:Super-Poulet
28604:Jordan–Pólya
28553:prime factor
28458:Noncototient
28425:Almost prime
28407:Superperfect
28382:Refactorable
28377:Quasiperfect
28352:Hyperperfect
28193:Pseudoprimes
28164:Wall–Sun–Sun
28099:Ordered Bell
28069:Fuss–Catalan
27981:non-centered
27931:Dodecahedral
27908:non-centered
27794:non-centered
27696:Wolstenholme
27441:× 2 ± 1
27438:
27437:Of the form
27404:Eighth power
27384:Fourth power
27341:
27204:
27107:Division (2)
27071:Division (2)
27042:Hexation (6)
27026:
27017:Addition (1)
26946:
26940:
26929:. Retrieved
26911:
26902:
26891:. Retrieved
26876:
26866:
26854:. Retrieved
26834:
26820:
26811:
26805:
26793:. Retrieved
26773:
26769:Sayre, David
26765:Sayre, David
26747:
26728:
26722:
26711:. Retrieved
26692:
26682:
26660:. Retrieved
26651:
26638:
26601:
26595:
26586:
26577:
26568:
26557:. Retrieved
26550:the original
26527:
26523:
26510:
26502:
26497:
26489:
26484:
26475:
26469:
26460:
26435:
26427:
26418:
26412:
26403:
26397:
26383:
26371:
26358:
26333:
26327:Cull, Paul;
26322:
26288:
26282:
26255:
26245:
26226:
26220:
26195:
26165:
26159:
26130:
26096:
26054:
26047:
26020:
26015:
26001:
25993:
25971:
25962:
25942:
25935:
25915:
25908:
25899:
25894:
25884:
25860:
25856:"involution"
25848:
25839:
25833:
25827:
25821:
25815:
25806:
25800:
25794:
25788:
25783:
25781:
25776:
25771:La Géométrie
25769:
25760:
25750:
25743:
25729:
25719:
25705:
25687:
25677:
25667:
25660:
25645:
25632:
25621:. Retrieved
25586:
25568:
25558:
25543:
25499:
25492:
25463:
25435:
25423:. Retrieved
25417:
25404:
25393:. Retrieved
25390:Math Insight
25389:
25346:
25254:
25094:power series
24810:
24760:
24710:
24485:
24481:
24342:
24275:
24250:
24232:is equal to
24215:
24092:CoffeeScript
24062:and so used
23953:
23772:
23363:
23359:
23349:
23221:
23211:
23200:
23081:
23060:
23055:
22950:
22947:
22944:
22869:2 (2) = 2 =
22825:
22697:
22691:
22688:
22673:
22666:
22660:
22654:
22644:
22638:
22632:
22629:
22623:
22616:
22609:
22605:
22597:
22591:
22585:
22580:
22575:
22572:
22564:
22546:
22528:
22520:
22513:
22504:
22496:
22489:
22483:
22456:
22449:
22427:
22417:
22414:
22408:
22396:
22386:
22380:
22373:
22369:
22361:
22354:
22232:
22222:
22212:
22164:
22108:
22030:
22009:
21766:
21749:
21741:
21731:
21725:
21716:
21651:denotes the
21584:
21580:Zorn's lemma
21574:, while the
21477:
21430:
21252:
21144:
21121:vector space
21003:
20902:
20765:isomorphisms
20643:
20513:
20472:Galois group
20344:
20186:
20079:
19849:
19809:
19769:
19730:
19627:prime number
19619:finite field
19618:
19616:
19604:real numbers
19581:
19578:Finite field
19538:
19395:
19389:
19249:
19239:
19206:
19142:Markov chain
19136:
19130:
19124:
19117:
19050:
19019:matrix power
19013:
19011:with itself
19007:
19001:
18998:
18889:
18879:, since the
18803:
18743:
18614:
18608:
18602:
18599:
18595:
18588:
18575:
18572:group theory
18569:
18439:cyclic group
18428:
18369:is a group,
18364:
18349:
18177:
18075:
17884:
17841:
17601:
17545:
17539:
17493:
17380:
17370:
17363:
17352:
17296:
17274:
17264:), and both
17262:not rational
17261:
17244:
17230:
17215:
17122:
17118:
16940:
16936:
16411:(taking the
16213:
16209:
15900:
15896:
15892:
15888:
15881:
15878:
15874:
15626:
15618:
15607:
15603:
15596:
15592:
15520:
15338:
15321:
15317:
15313:
15302:
15300:
15257:
14118:Final result
14117:
13858:
13818:denotes the
13735:
13727:
13498:
13428:
13426:
13350:
13299:has exactly
13198:
13050:
13043:
12841:
12777:
12697:
12645:
12537:
12487:, and it is
12449:
12391:
12308:
12292:
12217:
12155:
12094:
12041:
12033:
11980:
11957:
11947:th roots of
11926:
11776:
11613:
11594:
11583:
11548:e^{2i\pi /n}
11483:
11383:
11308:is added to
11284:
11172:
11034:
10942:
10884:
10692:
10535:
10466:
10455:
10384:
10379:
10335:In general,
10334:
10269:
10259:denotes the
10232:
10169:, above) as
10164:
10146:
10131:
10092:
9940:
9937:
9865:
9854:
9836:
9830:
9827:
9745:
9602:
9489:
9191:
9148:
9051:
8889:
8887:
8869:
8820:
8784:
8512:monotonicity
8508:= 3.14159...
8505:
8491:
8488:
8473:
8334:
8322:
8316:
8293:
8291:
8222:
8207:
8193:
8191:
8179:
8173:
8171:
7906:
7847:
7836:
7647:and writing
7605:
7422:
7293:
7286:
7280:
7274:
7268:
7262:
7256:
7250:
7148:
7069:
6990:
6917:
6847:
6782:
6723:
6675:
6641:
6607:
6600:
6595:
6590:
6585:
6580:
6575:
6570:
6565:
6560:
6555:
6517:
6240:to negative
6166:
5753:
5746:
5732:
5707:
5701:
5694:
5692:
5685:
5679:
5675:
5664:
5658:
5652:
5646:
5639:
5636:
5630:
5624:
5609:
5602:
5596:
5589:
5583:
5575:
5566:
5559:
5549:
5544:
5540:
5532:
5528:
5526:
5519:
5512:
5505:
5495:
5483:
5478:
5467:
5449:
5446:
5436:
5433:
5415:
5380:
5326:
5319:
5304:
5297:
5286:
5282:
5279:
5268:
5239:binary point
5201:
5186:
5180:, which has
5168:
5157:
5150:
5144:
5138:
5135:Power of two
5099:
5087:approximated
5041:
5006:
4927:
4916:
4908:
4849:
4838:
4822:
4777:
4773:
4770:
4559:
4487:
4414:
4410:
4395:(2) = 8 = 64
4380:
4189:
4183:
4105:
4014:
3991:
3931:and nonzero
3926:
3900:The case of
3899:
3870:
3793:
3746:
3676:
3600:
3535:
3496:
3481:
3469:
3459:
3455:
3451:
3447:
3446:th power", "
3443:
3439:
3435:
3431:
3426:
3419:
3414:
3410:
3384:
3378:
3372:
3366:
3360:
3350:
3346:
3342:
3338:
3335:
3329:
3323:
3317:
3311:
3305:
3299:
3293:
3287:
3277:
3273:
3269:
3266:
3253:
3245:
3231:
3227:
3223:
3220:Samuel Jeake
3218:
3207:
3203:
3199:
3195:
3188:
3175:
3169:
3163:
3157:
3151:
3148:
3142:La Géométrie
3140:
3131:
3125:
3120:
3114:
3104:
3097:fourth power
3088:
3086:
3075:
3045:
3034:
3013:
3007:
2989:
2981:
2978:Al-Khwarizmi
2975:
2969:
2965:
2946:10 · 10 = 10
2935:
2933:
2895:
2879:
2875:
2867:
2860:
2858:
2810:
2799:
2410:, such that
2365:
2042:
1856:
1620:
1411:
1321:
1316:
1312:
1308:
1304:
1300:
1299:th power of
1296:
1292:
1288:
1287:th power", "
1284:
1280:
1275:
1268:
1174:
1149:
1141:
1132:
1127:
1123:
1117:
1105:
1099:
900:
790:
569:multiplicand
186:
179:
116:
87:
81:
77:
63:
59:
43:
36:
28974:Extravagant
28969:Equidigital
28924:permutation
28883:Palindromic
28857:Automorphic
28755:Sum-product
28734:Sum-product
28689:Persistence
28584:Erdős–Woods
28506:Untouchable
28387:Semiperfect
28337:Hemiperfect
27998:Tesseractic
27936:Icosahedral
27916:Tetrahedral
27847:Dodecagonal
27548:Recursively
27419:Prime power
27394:Sixth power
27389:Fifth power
27369:Power of 10
27327:Classes of
27250:millisecond
27245:microsecond
27230:femtosecond
26530:: 129–146.
26114:Wanka, Gert
25092:(including
24370:as integers
24349:type safety
24333:::Pow(x, y)
24260:Common Lisp
24220:, that is,
24044:, and most
24010:Mathematica
22463:−∞ ≤ y ≤ +∞
22281:defined on
21576:Hamel bases
21480:direct sums
21269:is denoted
20755:associative
20751:commutative
20464:composition
19631:prime power
19588:associative
18593:; that is,
18591:conjugation
18304:is denoted
18220:is denoted
17954:. That is,
17537:is denoted
16237:, we have:
13423:Computation
13100:periodicity
12489:holomorphic
11964:unit circle
10460:is real or
9786:, and thus
8958:. To avoid
8868:. See also
8220:exponents.
7300:real number
6511:are called
6160:are called
5123:1000 m
5101:SI prefixes
5095:10 m/s
5083:10 m/s
5048:is used in
5003:Power of 10
4817:instead of
4787:commutative
4399:2 = 2 = 512
4391:associative
4383:commutative
3492:associative
3393:. The base
3263:Terminology
3191:polynomials
1567: times
1528: times
1482: times
1279:is called "
1271:superscript
1249: times
1114:two numbers
1102:mathematics
689:denominator
397:Subtraction
29235:Categories
29186:Graphemics
29059:Pernicious
28913:Undulating
28888:Pandigital
28862:Trimorphic
28463:Nontotient
28312:Arithmetic
27926:Octahedral
27827:Heptagonal
27817:Pentagonal
27802:Triangular
27643:Sierpiński
27565:Jacobsthal
27364:Power of 3
27359:Power of 2
27281:gigasecond
27276:megasecond
27271:kilosecond
27240:nanosecond
27235:picosecond
27225:attosecond
26931:2020-02-06
26893:2020-02-06
26861:(29 pages)
26856:2022-07-04
26795:2022-07-04
26738:0201700522
26713:2016-01-18
26662:2020-08-04
26559:2024-01-11
26406:: 286–287.
26256:Analysis I
25877:required.)
25695:. p.
25623:2020-04-16
25576:. p.
25507:. p.
25425:2020-08-27
25395:2020-08-27
25362:References
25315:are used;
25195:Derivative
24611:Finite sum
24337:PowerShell
24256:(expt x y)
24244:, and the
24112:JavaScript
23673:both mean
23225:th iterate
23201:for every
23076:See also:
22906:(2) = 2 =
22883:(2) = 2 =
22855:(2) = 2 =
22847:(2) = 2 =
22689:Computing
22583:values of
22203:=3 = 3 = 3
21139:See also:
21094:copies of
20694:such that
20597:, even if
20425:the field
19986:There are
19858:(that is,
19770:for every
19691:isomorphic
19542:semigroups
19256:derivative
18916:zero ideal
18806:zero ideal
18799:nilradical
18451:isomorphic
18346:In a group
18178:Commonly,
17618:integers:
17260:(that is,
17258:irrational
13500:Polar form
13094:such that
12400:satisfies
12293:for every
12248:such that
12042:principal
11777:primitive
11671:, that is
11445:th root a
10949:polar form
10693:where the
10538:polar form
9938:for every
9052:There are
8353:continuity
8335:Since any
7385:such that
6218:asymptotic
5610:Powers of
5164:set theory
5162:appear in
5158:Powers of
4995:See also:
4833:See also:
4807:algorithms
4797:are, say,
4397:, whereas
4186:identities
3538:recurrence
3484:formalized
3228:involution
3117:James Hume
3105:Biquadrate
3103:(eighth).
3059:Jost Bürgi
2942:Archimedes
2868:exponentem
1112:involving
559:multiplier
493:difference
454:subtrahend
131:base
114:base
99:base
75:Graphs of
28943:Parasitic
28792:Factorion
28719:Digit sum
28711:Digit sum
28529:Fortunate
28516:Primorial
28430:Semiprime
28367:Practical
28332:Descartes
28327:Deficient
28317:Betrothed
28159:Wieferich
27988:Pentatope
27951:pyramidal
27842:Decagonal
27837:Nonagonal
27832:Octagonal
27822:Hexagonal
27681:Practical
27628:Congruent
27560:Fibonacci
27524:Loeschian
26926:0744-7868
26878:InfoWorld
26761:Nutt, Roy
26690:(1952) .
26671:1813 work
26630:118124706
26532:CiteSeerX
26373:MathWorld
26293:MIT Press
26124:(2013) .
25768:(1637). "
25574:Macmillan
25505:D. Reidel
25419:MathWorld
25326:⋅
25313:variables
25273:×
24861:Logarithm
24456:Pentation
24451:Tetration
24388:x.powi(y)
24374:x.powf(y)
24299:library).
24285:pow(x, y)
24278:libraries
24080:KornShell
24022:Analytica
23926:
23902:
23861:
23843:
23835:−
23821:
23813:−
23781:−
23743:
23734:
23702:
23696:⋅
23684:
23652:
23619:
23579:⋅
23573:⋅
23531:∘
23525:∘
23511:∘
23471:⋅
23147:∘
23112:∘
23091:functions
23042:s in the
23023:♯
23011:by using
22993:−
22990:⌋
22984:
22971:⌊
22962:♯
22677:tends to
22541:(+∞) = +∞
22415:In fact,
22313:∈
22167:tetration
22157:Tetration
22128:×
22122:→
22089:×
22083:→
22050:→
21980:×
21971:
21965:≅
21940:
21909:×
21898:≅
21837:
21773:morphisms
21653:power set
21459:⊔
21439:×
21403:×
21390:≅
21382:⊔
21345:×
21334:≅
21226:…
21173:…
20969:…
20934:of a set
20907:th power
20762:canonical
20731:∈
20705:∈
20507:of order
20381:and is a
20316:↦
20293:→
20274::
20116:identity
20061:φ
20003:−
19994:φ
19915:−
19901:…
19781:∈
19088:−
19067:−
18952:…
18852:≠
18819:≠
18787:nilpotent
18740:In a ring
18407:∈
18275:∘
18154:⋯
18130:⋯
18095:∘
18046:⋯
17933:∘
17860:functions
17580:−
17567:−
17173:
17090:
17084:
17049:π
17034:
17011:π
16996:
16990:
16975:π
16955:
16913:
16901:
16839:
16753:π
16746:−
16638:π
16631:−
16615:π
16492:π
16485:−
16476:π
16431:π
16384:π
16359:π
16310:⋅
16293:π
16261:π
16193:(−1 ⋅ −1)
16172:−
16132:−
16108:≠
16081:−
16051:−
16017:−
15993:−
15969:−
15963:≠
15936:−
15933:⋅
15927:−
15843:∈
15837:∣
15828:π
15822:⋅
15810:
15781:
15753:∈
15741:∣
15732:π
15717:π
15711:⋅
15699:
15693:⋅
15660:
15572:π
15554:
15545:≡
15532:
15507:π
15501:−
15489:π
15483:−
15457:π
15451:−
15440:
15422:−
15416:
15407:≠
15404:π
15389:−
15383:
15358:−
15349:
15287:real part
15236:
15194:
15182:
15164:
15152:
15138:π
15126:π
15117:−
15099:−
15077:π
15065:π
15050:
15038:
15020:π
15008:π
14993:
14981:
14967:π
14955:π
14946:−
14899:−
14867:π
14847:−
14800:−
14773:≈
14763:π
14758:−
14701:π
14692:−
14677:π
14672:−
14656:
14604:π
14587:π
14568:
14557:are thus
14542:
14506:π
14394:π
14379:θ
14370:ρ
14367:
14355:
14340:π
14325:θ
14316:ρ
14313:
14301:
14285:π
14273:θ
14264:−
14250:ρ
14228:one gets
14195:
14071:π
14056:θ
14047:ρ
14044:
14023:π
14011:−
14008:θ
14002:−
13999:ρ
13996:
13978:
13949:
13873:
13839:π
13783:θ
13774:ρ
13771:
13759:
13737:Logarithm
13699:
13690:θ
13640:ρ
13614:θ
13611:
13599:θ
13596:
13587:ρ
13579:θ
13568:ρ
13362:≠
13179:π
13153:−
13021:π
12983:
12974:π
12894:
12885:π
12853:
12789:
12669:→
12628:
12597:
12518:
12506:
12432:π
12429:≤
12423:
12414:π
12411:−
12363:
12267:
12229:
12195:
12002:π
11907:−
11861:−
11817:π
11792:ω
11751:−
11744:ω
11731:ω
11718:ω
11711:ω
11699:ω
11686:ω
11649:π
11633:ω
11535:π
11495:π
11422:π
11419:≤
11416:θ
11410:π
11407:−
11342:π
11316:θ
11296:π
11260:θ
11239:ρ
11213:θ
11202:ρ
11138:π
11126:θ
11106:θ
11071:θ
11043:ρ
11014:θ
11011:
10999:θ
10996:
10987:ρ
10979:θ
10968:ρ
10853:
10841:
10823:
10811:
10784:
10666:
10654:
10636:
10624:
10575:, namely
10515:
10500:
10425:≠
10244:
10207:
10159:exponent
10112:
10073:
10041:
9918:
9898:
9889:
9846:logarithm
9765:
9750:value of
9705:
9694:is real,
9652:
9620:
9573:
9555:
9540:
9404:∞
9401:→
9319:∞
9316:→
9293:
9278:
9243:
9228:
9204:
9160:
9096:∞
9093:→
9070:
9012:
8973:
8939:≈
8903:↦
8852:and real
8802:π
8787:sequences
8770:…
8527:π
8448:∈
8426:∈
8401:→
8390:∈
8200:logarithm
8154:−
8135:−
8114:⋅
8100:−
8094:≠
8038:−
6485:−
6473:−
6125:−
5860:≥
5810:≠
5749:= 2, 4, 6
5735:= 1, 3, 5
5722:Power law
5535:tends to
5474:sequences
5455:(−1) = −1
5360:−
5217:may take
5184:members.
5174:power set
5085:and then
5061: m/s
4926:{1, ...,
4920:possible
4855:functions
4783:structure
4748:−
4715:−
4672:∑
4660:−
4600:∑
4352:⋅
4319:⋅
4301:⋅
4244:⋅
4150:−
4088:−
4052:⋅
4000:∞
3951:−
3846:⋅
3751:power is
3649:⋅
3580:⋅
3488:induction
3486:by using
3246:In 1748,
3215:"Indices"
3123:he wrote
3115:In 1636,
3087:The word
2919:Antiquity
2894:δύναμις (
2859:The term
2855:Etymology
2821:chemistry
2813:economics
2639:, giving
2525:×
2462:×
2325:−
2248:−
2197:−
2115:−
2094:×
2086:−
2056:−
1947:×
1934:×
1712:×
1659:≠
1593:×
1557:⏟
1550:×
1547:⋯
1544:×
1534:×
1518:⏟
1511:×
1508:⋯
1505:×
1466:⏟
1459:×
1456:⋯
1453:×
1239:⏟
1232:×
1226:×
1223:⋯
1220:×
1214:×
1110:operation
1048:logarithm
1008:
981:Logarithm
682:numerator
564:×
543:×
449:−
428:−
29015:Friedman
28948:Primeval
28893:Repdigit
28850:-related
28797:Kaprekar
28771:Meertens
28694:Additive
28681:dynamics
28589:Friendly
28501:Sociable
28491:Amicable
28302:Abundant
28282:dynamics
28104:Schröder
28094:Narayana
28064:Eulerian
28054:Delannoy
28049:Dedekind
27870:centered
27736:centered
27623:Amenable
27580:Narayana
27570:Leonardo
27466:Mersenne
27414:Powerful
27354:Achilles
26913:80 Micro
26887:Archived
26847:Archived
26786:Archived
26656:Archived
26646:(1820).
26576:(1903).
26505:, V.4.2.
26392:(1827).
26311:Archived
26062:(NIST),
25901:habeant.
25892:(1748).
25727:(1544).
25685:(1928).
25566:(1915).
25460:(2015).
25220:Integral
24686:Variable
24561:Constant
24403:See also
24360:x.pow(y)
24351:such as
24343:In some
24266:pown x y
24030:TI-BASIC
23959:operator
23879:For the
23723:and not
23095:codomain
23015:, where
22706:, apply
22581:positive
22559:(+∞) = 0
21737:, where
21298:Currying
21207:function
20593:for any
19731:One has
19612:infinite
19606:and the
19365:′
18834:implies
18606:, where
18431:subgroup
17799:if
17312:∉
15615:. Using
14448:Examples
12394:argument
12392:and the
12030:argument
11972:regular
11599:, where
11451:radicand
11085:argument
9190:and the
9149:One has
8510:and the
7464:q > 0
6333:infinity
6187:is odd,
6108:symmetry
6008:infinity
5961:is even
5799:, where
5594:for all
5442:(−1) = 1
5241:, where
5209:integer
4881:from an
4785:that is
4122:denoted
3804:identity
3536:and the
3405:. Here,
3397:appears
3182:—
3089:exponent
3012:(m) and
2876:exponere
2861:exponent
2806:matrices
1295:", "the
1160:". When
1136:, where
1124:exponent
1122:and the
924:radicand
823:exponent
752:quotient
741:fraction
658:dividend
629:Division
240:Addition
52:notation
29188:related
29152:related
29116:related
29114:Sorting
28999:Vampire
28984:Harshad
28926:related
28898:Repunit
28812:Lychrel
28787:Dudeney
28639:Størmer
28634:Sphenic
28619:Regular
28557:divisor
28496:Perfect
28392:Sublime
28362:Perfect
28089:Motzkin
28044:Catalan
27585:Padovan
27519:Leyland
27514:Idoneal
27509:Hilbert
27481:Woodall
27052:Inverse
27005:Primary
26832:(ed.).
26771:(ed.).
26419:Algèbre
26128:(ed.).
26086:2723248
25414:"Power"
24234:a^(b*c)
24230:(a^b)^c
24226:a^(b^c)
24164:Mercury
24104:Gnuplot
24096:Fortran
24076:Z shell
24056:Fortran
24038:Haskell
23969:). The
23056:minimal
22561:, when
22545:0 <
22543:, when
22525:, when
22503:1 <
22501:, when
22445:(1, −∞)
22441:(1, +∞)
22437:(+∞, 0)
22352:. Then
22025:
22017:" is a
22014:
21767:In the
21700:subsets
21492:modules
21149:-tuple
21131:, etc.
21068:(where
20648:of two
19021:. Also
18980:over a
18908:radical
18873:reduced
18789:. In a
18498:, then
18453:to the
18437:is the
18365:So, if
17878:of any
17862:from a
17286:equals
16675:(using
16529:(using
16231:Clausen
16227:paradox
16197:{1, −1}
14776:0.2079.
13082:unless
12313:of the
12297:in its
11966:of the
11951:with a
11939:as the
11847:coprime
11083:is its
11055:is the
10697:of the
10456:unless
10157:complex
9861:inverse
9859:is the
9748:complex
8757:3.14160
8744:3.14159
8218:complex
7458:, with
7375:th root
5941:, when
5847:integer
5712:below.
5699:below.
5674:(1 + 1/
5662:grows.
5642:< –1
5634:grows.
5465:pairs.
5331:), the
5309:), the
5219:2 = 256
5178:subsets
5152:quarter
5009:decimal
4857:from a
4805:, many
3458:to the
3450:to the
3442:to the
3409:is the
3321:" and "
3232:indices
3224:indices
3000:Islamic
2914:History
2896:dúnamis
2825:physics
2817:biology
2802:complex
1319:(th)".
1315:to the
1307:to the
1186:bases:
1180:product
1178:is the
1166:integer
1148:is the
1140:is the
903:th root
665:divisor
609:product
444:minuend
297:summand
287:summand
159:
147:
29054:Odious
28979:Frugal
28933:Cyclic
28922:Digit-
28629:Smooth
28614:Pronic
28574:Cyclic
28551:Other
28524:Euclid
28174:Wilson
28148:Primes
27807:Square
27676:Polite
27638:Riesel
27633:Knödel
27595:Perrin
27476:Thabit
27461:Fermat
27451:Cullen
27374:Square
27342:Powers
27266:second
27207:of ten
27205:powers
26953:
26924:
26916:(45).
26735:
26704:
26628:
26622:107384
26620:
26534:
26448:
26345:
26299:
26270:
26233:
26208:
26172:
26146:
26084:
26074:
26035:
25986:
25950:
25923:
25515:
25480:
24317:Erlang
24242:MATLAB
24196:x ^^ y
24170:, and
24168:Turing
24136:Python
24108:Groovy
24100:FoxPro
24068:a xx b
24054:. The
24052:x ** y
24002:MATLAB
23858:arcsin
23601:. So,
23360:before
23298:means
23099:domain
22832:2 = 4
22650:0 = +∞
22619:< 0
22600:< 0
22567:< 0
22555:0 = +∞
22531:< 1
22433:(0, 0)
22228:(0, 0)
22183:(3, 3)
21771:, the
21745:|
21739:|
21735:|
21723:|
21431:where
20944:tuples
20505:cyclic
20347:linear
20053:where
19669:where
19602:, the
19387:. The
18883:of an
18480:finite
18259:while
17874:, and
17852:fields
17844:groups
17543:, and
17529:has a
17391:monoid
17248:(as a
17224:, and
15606:⋅ log
15320:⋅ log
13798:where
13744:. The
13632:where
13044:where
12912:where
12766:where
12617:where
12479:it is
12218:where
11063:, and
11035:where
10920:where
10233:where
9943:> 0
8928:where
8721:3.1416
8708:3.1415
8499:, the
7302:, and
6260:. For
5845:is an
5573:|
5522:> 1
5482:, see
5383:> 0
5329:< 0
5307:> 0
5111:means
5036:0.0001
4976:5 = 1
4968:4 = 4
4960:3 = 9
4952:2 = 8
4944:1 = 1
4936:0 = 0
4879:tuples
4112:monoid
3364:" or "
3291:" or "
3283:square
2991:Kaʿbah
2986:square
2970:kaʿbah
2904:Euclid
2870:, the
2827:, and
2592:. The
2184:gives
1793:gives
1116:: the
1108:is an
930:degree
548:factor
538:factor
339:addend
329:augend
318:addend
308:addend
167:(0, 1)
142:
129:
127:
112:
110:
97:
95:
29095:Prime
29090:Lucky
29079:sieve
29008:Other
28994:Smith
28874:Digit
28832:Happy
28807:Keith
28780:Other
28624:Rough
28594:Giuga
28059:Euler
27921:Cubic
27575:Lucas
27471:Proth
26850:(PDF)
26839:(PDF)
26789:(PDF)
26778:(PDF)
26626:S2CID
26618:JSTOR
26553:(PDF)
26520:(PDF)
26027:(and
26019:[
25871:
25819:, or
25813:(And
25786:, ou
25246:Notes
25170:Limit
24238:Algol
24222:a^b^c
24178:x ↑ y
24152:Seed7
24116:OCaml
24088:COBOL
23994:BASIC
23986:x ^ y
23963:caret
23364:after
23085:is a
22614:with
22595:when
22563:−∞ ≤
22537:0 = 0
22378:with
21568:basis
21490:, or
21209:from
21129:rings
21004:When
20759:up to
20605:from
20562:then
20466:) of
20389:. If
20349:over
20191:. As
19629:or a
19584:field
18982:field
18795:ideal
18744:In a
18576:order
18484:order
17889:is a
17848:rings
17389:is a
15611:as a
14417:with
13927:with
13729:atan2
13726:(see
13696:atan2
13507:. If
13382:graph
13340:. If
13272:then
13243:with
13086:is a
12142:z^{w}
12111:z^{w}
11849:with
11842:with
11089:up to
9686:when
8942:2.718
8856:as a
8685:3.142
8672:3.141
8327:when
7843:0 = 0
7454:is a
6670:1024
6167:When
5571:when
5517:when
5377:with
5315:0 = 0
5271:1 = 1
5091:2.998
5073:2.997
5063:(the
4939:none
4411:right
3413:, or
3155:, or
2900:Greek
2884:Latin
2880:power
2865:Latin
2752:, so
1150:power
1128:power
983:(log)
881:power
841:power
763:ratio
177:. At
58:base
29049:Evil
28729:Self
28679:and
28569:Blum
28280:and
28084:Lobb
28039:Cake
28034:Bell
27784:Star
27691:Ulam
27590:Pell
27379:Cube
26951:ISBN
26922:ISSN
26733:ISBN
26702:ISBN
26446:ISBN
26343:ISBN
26297:ISBN
26268:ISBN
26231:ISBN
26206:ISBN
26170:ISBN
26144:ISBN
26072:ISBN
26033:ISBN
25984:ISBN
25948:ISBN
25921:ISBN
25513:ISBN
25478:ISBN
25239:Yes
25214:Yes
25189:Yes
25164:Yes
25139:Yes
25114:Yes
25084:Yes
25059:Yes
25034:Yes
25009:Yes
24980:Yes
24955:Yes
24930:Yes
24905:Yes
24880:Yes
24855:Yes
24830:Yes
24805:Yes
24780:Yes
24755:Yes
24730:Yes
24705:Yes
24680:Yes
24655:Yes
24630:Yes
24605:Yes
24580:Yes
24390:for
24380:and
24376:for
24366:and
24362:for
24353:Rust
24327:Java
24297:math
24295:(in
24172:VHDL
24160:ABAP
24144:Ruby
24140:Rexx
24132:PL/I
24124:Perl
24084:Bash
23634:and
23549:and
23429:and
22849:4096
22669:→ +∞
22579:for
22557:and
22549:≤ +∞
22539:and
22527:0 ≤
22523:= +∞
22518:and
22507:≤ +∞
22494:and
22492:= +∞
22477:and
22471:(+∞)
22459:≤ +∞
22455:0 ≤
22443:and
22334:>
22177:and
22159:and
21779:and
21624:So,
21471:the
21257:and
20860:and
20753:nor
20720:and
20655:and
20650:sets
20644:The
20514:The
20454:has
20112:the
18918:. A
18746:ring
18675:and
18612:and
18076:and
17614:and
17606:and
17350:and
17278:are
17268:and
17121:) =
16221:and
16212:) =
16195:are
16032:and
15908:and
15895:) =
15885:and
15877:) =
15631:and
15625:log(
15617:Log(
15595:log(
15316:) =
15312:log(
14176:and
13964:are
13931:and
13682:and
13550:and
13490:and
13427:The
13254:>
13236:and
12804:and
12417:<
12309:The
11976:-gon
11927:The
11899:and
11771:The
11614:The
11584:The
11413:<
10155:and
10093:So,
8888:The
8864:and
8649:3.15
8636:3.14
8178:and
8172:See
7462:and
7156:1000
7080:6561
7001:4096
6928:2401
6862:7776
6858:1296
6796:3125
6741:4096
6737:1024
6701:6561
6696:2187
6529:<
6518:For
6271:>
5926:>
5849:and
5693:See
5678:) →
5618:and
5496:The
5233:and
5215:byte
5149:and
5146:half
5118:1000
5109:kilo
5029:and
5026:1000
4914:The
4897:and
4843:and
4793:and
4415:left
4405:for
3677:and
3605:and
3438:", "
3356:cube
3129:for
3072:for
3053:and
2996:cube
2968:and
2845:wave
1303:", "
1144:and
1142:base
1119:base
1003:base
962:root
836:base
818:base
433:term
423:term
276:term
266:term
29167:Ban
28555:or
28074:Lah
27203:by
26610:doi
26602:103
26542:doi
26260:doi
25976:doi
25782:Et
25774:".
25697:344
25593:doi
25081:Yes
25056:Yes
25031:Yes
25006:Yes
24977:Yes
24974:Yes
24952:Yes
24949:Yes
24927:Yes
24924:Yes
24902:Yes
24899:Yes
24877:Yes
24874:Yes
24852:Yes
24849:Yes
24827:Yes
24824:Yes
24802:Yes
24799:Yes
24796:Yes
24777:Yes
24774:Yes
24771:Yes
24752:Yes
24749:Yes
24746:Yes
24727:Yes
24724:Yes
24721:Yes
24718:Yes
24702:Yes
24699:Yes
24696:Yes
24693:Yes
24677:Yes
24674:Yes
24671:Yes
24665:Yes
24652:Yes
24649:Yes
24646:Yes
24643:Yes
24640:Yes
24627:Yes
24624:Yes
24621:Yes
24618:Yes
24615:Yes
24602:Yes
24599:Yes
24596:Yes
24590:Yes
24577:Yes
24574:Yes
24571:Yes
24568:Yes
24565:Yes
24293:C++
24210:APL
24206:x⋆y
24156:Tcl
24148:SAS
24128:PHP
24072:Ada
24042:Lua
24026:TeX
24012:),
23990:AWK
23923:sin
23899:sin
23831:sin
23809:sin
23740:sin
23731:sin
23699:sin
23681:sin
23643:sin
23610:sin
23046:of
22975:log
22937:376
22934:205
22931:703
22928:496
22925:401
22922:229
22919:228
22916:600
22913:650
22910:267
22899:624
22896:842
22893:906
22890:899
22887:125
22876:432
22873:554
22862:216
22859:777
22721:100
22700:− 1
22671:as
22516:= 0
22499:= 0
22425:of
22198:987
22195:484
22192:597
22189:625
22028:".
21968:hom
21937:hom
21834:hom
21795:to
21787:to
21710:of
21702:of
21663:to
21655:of
21582:).
21482:of
21300:):
21265:to
20632:if
20503:is
20474:of
20345:is
20224:in
20080:In
20073:is
19852:− 1
19814:in
18999:If
18991:).
18910:of
18567:).
18563:or
18494:is
18486:of
17864:set
17494:If
17381:An
17290:or
17256:is
17216:If
17170:log
17087:exp
17081:log
17031:exp
16993:exp
16987:log
16952:exp
16910:log
16898:exp
16836:exp
15807:Log
15778:log
15696:Log
15657:log
15565:mod
15551:log
15529:log
15437:log
15413:log
15380:log
15346:log
15289:of
15179:sin
15149:cos
15035:sin
14978:cos
14839:is
14653:log
14565:log
14539:log
14484:is
14352:sin
14298:cos
13975:log
13946:log
13892:If
13870:log
13756:log
13739:of
13608:sin
13593:cos
13502:of
13459:of
13199:If
12980:log
12891:log
12850:log
12842:If
12786:log
12698:If
12625:log
12594:log
12503:log
12420:Arg
12396:of
12360:log
12325:log
12264:log
12226:log
12192:log
12052:of
11611:).
11597:= 1
11285:If
11059:of
11008:sin
10993:cos
10951:as
10838:sin
10808:cos
10651:sin
10621:cos
10571:of
10540:of
10512:sin
10497:cos
10263:of
10147:If
9886:exp
9853:ln(
9762:exp
9702:exp
9649:exp
9617:exp
9570:exp
9552:exp
9537:exp
9394:lim
9309:lim
9290:exp
9275:exp
9240:exp
9225:exp
9201:exp
9157:exp
9086:lim
9067:exp
9009:exp
8970:exp
8954:is
8860:of
8613:3.2
8600:3.1
8380:lim
8325:= 1
8320:is
7917:odd
7915:is
7837:If
7423:If
7377:of
7341:or
7294:If
7231:000
7228:000
7225:000
7217:000
7214:000
7211:000
7203:000
7200:000
7198:100
7192:000
7189:000
7181:000
7178:000
7170:000
7168:100
7162:000
7153:100
7141:401
7138:784
7135:486
7127:489
7124:420
7122:387
7116:721
7113:046
7105:969
7102:782
7094:441
7092:531
7086:049
7077:729
7062:824
7059:741
7056:073
7048:728
7045:217
7043:134
7037:216
7034:777
7026:152
7023:097
7015:144
7013:262
7007:768
6998:512
6983:249
6980:475
6978:282
6972:607
6969:353
6961:801
6958:764
6950:543
6948:823
6942:649
6940:117
6934:807
6925:343
6910:176
6907:466
6899:696
6896:077
6888:616
6885:679
6877:936
6875:279
6869:656
6855:216
6840:625
6837:765
6829:125
6826:953
6818:625
6816:390
6810:125
6802:625
6793:625
6790:125
6775:576
6772:048
6764:144
6762:262
6756:536
6748:384
6734:256
6716:049
6708:683
6692:729
6689:243
6667:512
6664:256
6661:128
6216:'s
5688:→ ∞
5683:as
5637:If
5622:as
5605:= 1
5600:if
5592:= 1
5569:→ ∞
5564:as
5562:→ 0
5539:as
5515:→ ∞
5510:as
5508:→ ∞
5447:If
5434:If
5207:bit
5089:as
5075:924
5059:458
5056:792
5054:299
4979:()
4861:of
4859:set
4103:).
4012:).
3540:is
3462:".
3407:243
3376:is
3358:of
3303:is
3285:of
3200:bxx
3065:iii
3015:kāf
3009:mīm
2982:māl
2966:Māl
2934:In
2874:of
1369:is
1126:or
1100:In
999:log
905:(√)
793:(^)
631:(÷)
514:(×)
399:(−)
378:sum
242:(+)
182:= 1
173:is
29237::
26910:.
26875:.
26763:;
26650:.
26624:.
26616:.
26600:.
26540:.
26528:27
26526:.
26522:.
26444:.
26442:45
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26266:.
26254:.
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26202:28
26184:^
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26080:.
26070:.
26066:,
26058:.
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25859:.
25816:aa
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25650:.
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25604:^
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25548:.
25542:.
25527:^
25511:.
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25468:.
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25416:.
25388:.
25370:^
25262::
25236:No
25233:No
25230:No
25227:No
25224:No
25211:No
25208:No
25205:No
25202:No
25199:No
25186:No
25183:No
25180:No
25177:No
25174:No
25158:No
25155:No
25152:No
25149:No
25133:No
25130:No
25127:No
25124:No
25108:No
25105:No
25102:No
25099:No
25096:)
25078:No
25075:No
25072:No
25069:No
25053:No
25050:No
25047:No
25044:No
25028:No
25025:No
25022:No
25019:No
25003:No
25000:No
24997:No
24994:No
24971:No
24968:No
24965:No
24946:No
24943:No
24940:No
24921:No
24918:No
24915:No
24896:No
24893:No
24890:No
24871:No
24868:No
24865:No
24846:No
24843:No
24840:No
24821:No
24818:No
24815:No
24793:No
24790:No
24768:No
24765:No
24743:No
24740:No
24715:No
24690:No
24668:No
24484:=
24335::
24325::
24315::
24307:C#
24305::
24291:,
24287::
24280::
24270:F#
24268::
24258::
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24034:bc
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24024:,
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22857:16
22627:.
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22469:,
22461:,
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22435:,
22400:=
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21486:,
21475:.
21145:A
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19137:Ax
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22965:n
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18435:x
18424:n
18410:G
18404:x
18382:n
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18367:G
18331:.
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18325:x
18322:(
18317:n
18313:f
18292:)
18289:x
18286:(
18283:)
18278:n
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18267:(
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18242:n
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18234:x
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18205:x
18202:(
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18186:(
18163:.
18160:)
18157:)
18151:)
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18145:x
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18127:(
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18115:=
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18109:x
18106:(
18103:)
18098:n
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18087:(
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18058:)
18055:x
18052:(
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18040:x
18037:(
18034:f
18030:)
18027:x
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18018:=
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18002:x
17999:(
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17984:x
17981:(
17978:)
17973:n
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17965:(
17936:n
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17906:n
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17887:f
17818:,
17815:x
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17809:=
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17782:n
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17774:=
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17757:y
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17742:n
17739:m
17735:x
17731:=
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17685:m
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17677:=
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17588:.
17583:n
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17535:x
17527:x
17511:n
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17490:.
17488:n
17472:n
17468:x
17464:x
17461:=
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17453:+
17450:n
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17424:,
17421:1
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17330:,
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17218:b
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17159:=
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17117:(
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17099:z
17096:=
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17061:.
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17040:1
17037:(
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16999:(
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16981:n
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16919:)
16916:x
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16904:(
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16848:)
16845:y
16842:(
16814:y
16810:e
16799:n
16794:)
16792:e
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16774:=
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16659:=
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16627:e
16621:n
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16602:1
16598:e
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16570:x
16566:e
16562:=
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16539:(
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16513:=
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16502:n
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16395:=
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16272:=
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16219:x
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16210:e
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16169:=
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16102:=
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15966:(
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15957:=
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15924:(
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15899:/
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15887:(
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15873:(
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15847:Z
15840:n
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15825:2
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15816:+
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15804:z
15800:{
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15788:}
15784:w
15775:z
15771:{
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15757:Z
15750:n
15747:,
15744:m
15738:m
15735:i
15729:2
15726:+
15723:n
15720:i
15714:2
15708:z
15705:+
15702:w
15690:z
15686:{
15682:=
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15668:z
15664:w
15653:{
15637:n
15633:m
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15627:w
15621:)
15619:w
15608:w
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15599:)
15597:w
15578:)
15575:i
15569:2
15562:(
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15548:z
15540:z
15536:w
15504:i
15498:=
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15476:2
15473:=
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15465:2
15461:/
15454:i
15447:e
15443:(
15434:2
15431:=
15428:)
15425:i
15419:(
15410:2
15401:i
15398:=
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15392:1
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15369:2
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15318:x
15314:b
15291:w
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15206:.
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15141:)
15135:k
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15113:e
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15096:=
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15083:)
15080:)
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15068:+
15062:(
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15053:2
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15041:(
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14975:(
14970:)
14964:k
14961:2
14958:+
14952:(
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14942:e
14936:3
14932:2
14928:=
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14916:4
14913:+
14910:3
14906:)
14902:2
14896:(
14872:.
14864:i
14860:e
14856:2
14853:=
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14820:i
14817:4
14814:+
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14807:)
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14731:i
14727:i
14706:.
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14664:=
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14650:i
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14642:=
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14633:i
14612:.
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14601:k
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14590:2
14581:(
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14574:=
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14545:i
14519:,
14514:2
14510:/
14503:i
14499:e
14495:=
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14465:i
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14431:0
14428:=
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14270:(
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14260:e
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14246:=
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14211:y
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14144:=
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14100:=
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14077:,
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14068:k
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14050:+
14038:d
14035:(
14032:i
14029:+
14026:)
14020:k
14017:d
14014:2
14005:d
13990:c
13987:(
13984:=
13981:z
13972:w
13952:z
13943:w
13933:d
13929:c
13915:i
13912:d
13909:+
13906:c
13903:=
13900:w
13879:.
13876:z
13867:w
13855:.
13853:k
13836:k
13833:i
13830:2
13786:,
13780:i
13777:+
13765:=
13762:z
13741:z
13714:)
13711:b
13708:,
13705:a
13702:(
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13658:+
13653:2
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13643:=
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13617:)
13605:i
13602:+
13590:(
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13565:=
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13552:b
13548:a
13546:(
13544:z
13530:b
13527:i
13524:+
13521:a
13518:=
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13488:z
13472:w
13468:z
13447:y
13444:i
13441:+
13438:x
13405:w
13401:z
13390:0
13386:z
13368:,
13365:0
13359:z
13342:w
13336:n
13320:,
13317:1
13314:=
13311:m
13301:n
13285:w
13281:z
13260:,
13257:0
13251:n
13238:n
13234:m
13218:n
13215:m
13210:=
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13185:.
13182:i
13176:2
13156:b
13150:a
13128:b
13124:e
13120:=
13115:a
13111:e
13092:d
13084:w
13068:w
13064:z
13053:k
13046:k
13029:,
13024:w
13018:k
13015:i
13012:2
13008:e
13002:w
12998:z
12994:=
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12986:z
12977:+
12971:k
12968:i
12965:2
12962:(
12959:w
12955:e
12929:w
12925:z
12914:k
12900:,
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12888:+
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12856:z
12833:z
12817:w
12813:z
12792:z
12768:n
12754:,
12751:n
12747:/
12743:1
12740:=
12737:w
12715:w
12711:z
12700:z
12693:z
12677:w
12673:z
12666:)
12663:w
12660:,
12657:z
12654:(
12631:z
12605:,
12600:z
12591:w
12587:e
12583:=
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12551:w
12547:z
12524:.
12521:z
12512:=
12509:z
12493:z
12485:z
12467:,
12464:0
12461:=
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12435:.
12426:z
12398:z
12377:,
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12371:=
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12356:e
12342:z
12328:,
12295:z
12278:z
12275:=
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12260:e
12232:z
12203:,
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12189:w
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12181:=
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12137:w
12133:z
12124:z
12106:w
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12073:n
12069:/
12065:1
12061:1
12044:n
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12026:n
12009:n
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11999:k
11996:2
11990:e
11974:n
11960:n
11953:n
11949:z
11945:n
11941:n
11937:z
11933:n
11929:n
11913:.
11910:i
11887:i
11867:;
11864:1
11851:n
11844:k
11830:,
11824:n
11820:i
11814:k
11811:2
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11801:=
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11779:n
11773:n
11759:.
11754:1
11748:n
11740:,
11735:2
11727:,
11722:1
11714:=
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11703:n
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11679:1
11656:n
11652:i
11646:2
11640:e
11636:=
11623:n
11619:n
11616:n
11601:n
11595:w
11590:n
11586:n
11579:1
11557:n
11543:n
11539:/
11532:i
11529:2
11525:e
11512:n
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11439:n
11425:,
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11364:n
11350:n
11346:/
11339:i
11336:2
11293:2
11270:.
11264:n
11257:i
11251:e
11243:n
11234:=
11228:n
11225:1
11219:)
11210:i
11206:e
11198:(
11183:n
11179:n
11175:n
11158:k
11135:k
11132:2
11129:+
11095:π
11093:2
11061:z
11020:,
11017:)
11005:i
11002:+
10990:(
10984:=
10976:i
10972:e
10965:=
10962:z
10945:z
10937:n
10926:n
10922:n
10908:,
10905:n
10901:/
10897:1
10887:n
10865:.
10862:)
10859:)
10856:b
10847:y
10844:(
10835:i
10832:+
10829:)
10826:b
10817:y
10814:(
10805:(
10800:x
10796:b
10792:=
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10778:y
10775:i
10771:e
10765:x
10761:b
10757:=
10752:y
10749:i
10745:b
10739:x
10735:b
10731:=
10726:y
10723:i
10720:+
10717:x
10713:b
10678:,
10675:)
10672:)
10669:b
10660:y
10657:(
10648:i
10645:+
10642:)
10639:b
10630:y
10627:(
10618:(
10613:x
10609:b
10605:=
10600:y
10597:i
10594:+
10591:x
10587:b
10573:z
10553:z
10549:b
10521:,
10518:y
10509:i
10506:+
10503:y
10494:=
10489:y
10486:i
10482:e
10462:t
10458:z
10441:,
10436:t
10433:z
10429:b
10420:t
10415:)
10410:z
10406:b
10402:(
10380:b
10363:t
10358:)
10353:z
10349:b
10345:(
10320:,
10315:t
10311:b
10305:z
10301:b
10297:=
10292:t
10289:+
10286:z
10282:b
10265:b
10247:b
10218:,
10213:)
10210:b
10201:z
10198:(
10194:e
10190:=
10185:z
10181:b
10161:z
10153:b
10149:b
10137:b
10132:b
10115:b
10106:x
10102:e
10076:b
10067:x
10063:e
10059:=
10054:x
10049:)
10044:b
10034:e
10030:(
10025:=
10020:x
10016:b
9992:,
9987:y
9984:x
9980:e
9976:=
9971:y
9967:)
9961:x
9957:e
9953:(
9941:b
9921:b
9911:e
9907:=
9904:)
9901:b
9892:(
9883:=
9880:b
9866:e
9857:)
9855:x
9842:b
9837:b
9831:e
9818:z
9804:,
9799:z
9795:e
9774:)
9771:z
9768:(
9752:x
9741:x
9725:x
9721:e
9717:=
9714:)
9711:x
9708:(
9692:x
9688:x
9672:x
9668:e
9664:=
9661:)
9658:x
9655:(
9629:)
9626:1
9623:(
9614:=
9611:e
9588:)
9585:y
9582:+
9579:x
9576:(
9567:=
9564:)
9561:y
9558:(
9549:)
9546:x
9543:(
9513:2
9509:n
9504:y
9501:x
9475:,
9470:n
9465:)
9457:2
9453:n
9448:y
9445:x
9439:+
9434:n
9430:y
9427:+
9424:x
9418:+
9415:1
9411:(
9398:n
9390:=
9385:n
9380:)
9374:n
9371:y
9366:+
9363:1
9359:(
9352:n
9347:)
9341:n
9338:x
9333:+
9330:1
9326:(
9313:n
9305:=
9302:)
9299:y
9296:(
9287:)
9284:x
9281:(
9252:)
9249:y
9246:(
9237:)
9234:x
9231:(
9222:=
9219:)
9216:y
9213:+
9210:x
9207:(
9178:,
9175:1
9172:=
9169:)
9166:0
9163:(
9134:.
9129:n
9124:)
9118:n
9115:x
9110:+
9107:1
9103:(
9090:n
9082:=
9079:)
9076:x
9073:(
9037:.
9032:x
9028:e
9024:=
9021:)
9018:x
9015:(
8985:,
8982:)
8979:x
8976:(
8936:e
8916:,
8911:x
8907:e
8900:x
8866:x
8862:b
8854:x
8850:b
8834:x
8830:b
8807:.
8798:b
8767:,
8763:]
8753:b
8749:,
8740:b
8735:[
8731:,
8727:]
8717:b
8713:,
8704:b
8699:[
8695:,
8691:]
8681:b
8677:,
8668:b
8663:[
8659:,
8655:]
8645:b
8641:,
8632:b
8627:[
8623:,
8619:]
8609:b
8605:,
8596:b
8591:[
8587:,
8583:]
8577:4
8573:b
8569:,
8564:3
8560:b
8555:[
8532::
8523:b
8506:π
8496:π
8492:x
8484:x
8480:b
8476:r
8459:,
8456:)
8452:R
8445:x
8441:,
8436:+
8431:R
8423:b
8420:(
8414:r
8410:b
8404:x
8398:)
8394:Q
8387:(
8384:r
8376:=
8371:x
8367:b
8349:x
8345:b
8329:n
8323:e
8317:e
8275:s
8272:r
8268:b
8264:=
8259:s
8254:)
8249:r
8245:b
8241:(
8206:(
8151:=
8146:1
8142:)
8138:1
8132:(
8129:=
8122:2
8119:1
8111:2
8107:)
8103:1
8097:(
8091:1
8088:=
8082:2
8079:1
8074:1
8070:=
8064:2
8061:1
8055:)
8049:2
8045:)
8041:1
8035:(
8031:(
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7997:x
7993:=
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7978:a
7974:x
7970:(
7950:,
7944:n
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7890:s
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7839:r
7823:.
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7803:p
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7771:p
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7667:q
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7617:=
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6134:=
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6128:x
6122:(
6119:f
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6094:n
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5981:=
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4930:}
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4751:i
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4724:!
4721:)
4718:i
4712:n
4709:(
4706:!
4703:i
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4695:n
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4631:i
4628:n
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4540:.
4535:q
4532:p
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4524:=
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4514:)
4509:p
4505:b
4501:(
4473:,
4467:)
4462:q
4458:p
4454:(
4449:b
4445:=
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4360:n
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4339:=
4330:n
4326:)
4322:c
4316:b
4313:(
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4290:=
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4271:m
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4263:(
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4239:m
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4231:=
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4219:+
4216:m
4212:b
4181:.
4158:.
4153:1
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4135:x
4124:1
4091:n
4085:=
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4047:m
4043:b
4039:=
4034:n
4031:+
4028:m
4024:b
3988:.
3972:n
3968:b
3964:1
3959:=
3954:n
3947:b
3933:b
3929:n
3918:.
3910:0
3906:1
3902:0
3885:0
3882:=
3879:n
3854:n
3850:b
3841:m
3837:b
3833:=
3828:n
3825:+
3822:m
3818:b
3776:=
3771:0
3767:b
3753:1
3749:0
3727:.
3722:n
3719:m
3715:b
3711:=
3706:n
3702:)
3696:m
3692:b
3688:(
3662:,
3657:n
3653:b
3644:m
3640:b
3636:=
3631:n
3628:+
3625:m
3621:b
3607:n
3603:m
3586:.
3583:b
3575:n
3571:b
3567:=
3562:1
3559:+
3556:n
3552:b
3521:b
3518:=
3513:1
3509:b
3460:n
3456:b
3452:n
3448:b
3444:n
3440:b
3436:n
3432:b
3427:b
3422:3
3403:5
3399:5
3395:3
3379:b
3373:b
3367:b
3361:b
3351:b
3347:b
3343:b
3339:b
3330:b
3324:b
3318:b
3312:b
3306:b
3300:b
3294:b
3288:b
3278:b
3274:b
3270:b
3208:d
3176:a
3170:a
3164:a
3158:a
3132:A
3126:A
3076:x
3074:4
3068:4
3046:x
2882:(
2783:2
2779:/
2775:1
2771:b
2767:=
2762:b
2737:2
2734:1
2728:=
2725:r
2705:1
2702:=
2699:r
2696:+
2693:r
2671:1
2667:b
2663:=
2658:r
2655:+
2652:r
2648:b
2625:1
2621:b
2600:b
2580:b
2577:=
2572:r
2569:+
2566:r
2562:b
2541:b
2538:=
2533:r
2529:b
2520:r
2516:b
2495:r
2475:b
2472:=
2467:b
2457:b
2433:b
2428:=
2423:r
2419:b
2398:r
2376:b
2349:n
2345:b
2340:/
2336:1
2333:=
2328:n
2321:b
2300:b
2297:=
2292:1
2288:b
2267:b
2263:/
2259:1
2256:=
2251:1
2244:b
2221:1
2217:b
2212:/
2208:1
2205:=
2200:1
2193:b
2170:1
2166:b
2145:1
2142:=
2137:0
2133:b
2129:=
2124:1
2121:+
2118:1
2111:b
2107:=
2102:1
2098:b
2089:1
2082:b
2059:1
2052:b
2028:b
2025:=
2020:1
2016:b
1993:3
1989:b
1985:=
1980:1
1977:+
1974:1
1971:+
1968:1
1964:b
1960:=
1955:1
1951:b
1942:1
1938:b
1929:1
1925:b
1921:=
1916:3
1912:)
1906:1
1902:b
1898:(
1878:b
1875:=
1870:1
1866:b
1842:1
1839:=
1834:n
1830:b
1825:/
1819:n
1815:b
1811:=
1806:0
1802:b
1779:n
1775:b
1752:n
1748:b
1744:=
1739:n
1736:+
1733:0
1729:b
1725:=
1720:n
1716:b
1707:0
1703:b
1682:n
1662:0
1656:b
1634:0
1630:b
1601:m
1597:b
1588:n
1584:b
1580:=
1563:m
1553:b
1541:b
1524:n
1514:b
1502:b
1495:=
1478:m
1475:+
1472:n
1462:b
1450:b
1443:=
1434:m
1431:+
1428:n
1424:b
1397:b
1377:n
1355:n
1351:b
1330:n
1317:n
1313:b
1309:n
1305:b
1301:b
1297:n
1293:n
1289:b
1285:n
1281:b
1276:b
1255:.
1245:n
1235:b
1229:b
1217:b
1211:b
1204:=
1199:n
1195:b
1184:n
1175:b
1162:n
1158:n
1154:b
1146:n
1138:b
1133:b
1089:e
1082:t
1075:v
1023:=
1019:)
1011:(
937:=
901:n
856:=
851:}
732:{
706:=
701:}
583:=
578:}
468:=
463:}
353:=
348:}
334:+
313:+
292:+
271:+
225:e
218:t
211:v
192:1
187:y
180:x
175:1
171:0
162:.
156:2
153:/
150:1
136:,
133:2
121:,
117:e
104:,
88:b
82:b
78:y
64:n
60:b
44:b
34:.
20:)
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