32:
8552:
5437:, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.
294:
11171:
6227:
5382:. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.
4262:
2937:(since, formally, a rational number is an equivalence class of pairs of integers, and a real number is an equivalence class of Cauchy series), and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case in
6408:. Real numbers were called "proportions", being the ratios of two lengths, or equivalently being measures of a length in terms of another length, called unit length. Two lengths are "commensurable", if there is a unit in which they are both measured by integers, that is, in modern terminology, if their ratio is a
400:
between any two
Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this is what mathematicians and physicists did during several centuries
5720:
arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The
Dedekind cuts construction uses the order topology presentation, while the Cauchy
4000:
6816:, which is described hereinafter. There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their
7765:) rather than their rational or decimal approximation. But exact and symbolic arithmetic also have limitations: for instance, they are computationally more expensive; it is not in general possible to determine whether two symbolic expressions are equal (the
7392:
In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as
3640:
6604:
The concept that many points existed between rational numbers, such as the square root of 2, was well known to the ancient Greeks. The existence of a continuous number line was considered self-evident, but the nature of this continuity, presently called
7497:
set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in
Robinson's theory).
9683:(1986). "Hermann Weyl and the Unity of Knowledge: In the linkage of four mysteries—the "how come" of existence, time, the mathematical continuum, and the discontinuous yes-or-no of quantum physics—may lie the key to deep new insight".
9021:
isomorphism between them. This implies that the identity is the unique field automorphism of the reals that is compatible with the ordering. In fact, the identity is the unique field automorphism of the reals, since
8489:
in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally
Archimedean. As with the previous two examples, this set is no longer a field or additive
4257:{\displaystyle {\begin{aligned}D_{n}&=b_{k}10^{k}+b_{k-1}10^{k-1}+\cdots +b_{0}+{\frac {a_{1}}{10}}+\cdots +{\frac {a_{n}}{10^{n}}}\\&=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{j=1}^{n}a_{j}10^{-j}\end{aligned}}}
2928:
7164:(which is not implied by other definitions of completeness), which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound
3905:
5492:
is "complete" in the sense that nothing further can be added to it without making it no longer an
Archimedean field. This sense of completeness is most closely related to the construction of the reals from
5231:
7567:
proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.
3477:
6806:
7826:
is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".
4005:
7583:
cannot operate on arbitrary real numbers, because finite computers cannot directly store infinitely many digits or other infinite representations. Nor do they usually even operate on arbitrary
5301:
7125:
7773:
explosion in the size of representation of a single number (for instance, squaring a rational number roughly doubles the number of digits in its numerator and denominator, and squaring a
8069:
7428:
Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.
6824:), and then define the real number system geometrically. All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems are
5132:
The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive
4648:
4571:
2372:
2227:
6459:, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra). Arabic mathematicians merged the concepts of "
9097:
3275:
3492:
1869:
2412:
2267:
8213:
8025:
8176:
3675:
7763:
8142:
7440:
axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in
4786:
4354:
8244:
8110:
7993:
7964:
7935:
7906:
7322:
7293:
7260:
7231:
6820:
or as
Dedekind cuts, which are certain subsets of rational numbers. Another approach is to start from some rigorous axiomatization of Euclidean geometry (say of Hilbert or of
5553:
8785:
8747:
8709:
8671:
8633:
5646:
1968:
1138:
753:
6340:
6312:
6284:
6256:
5884:
implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of
8310:
2884:
2833:
2782:
2670:
10983:
10906:
10867:
10829:
10801:
10773:
10745:
10658:
10625:
10597:
10569:
8273:
7858:
7347:
7064:
7038:
7012:
6923:
6886:
6856:
6189:
6163:
6106:
6080:
6058:
6032:
6004:
5982:
5956:
5930:
5908:
5836:
5810:
5788:
5490:
5468:
5425:
3970:
2859:
2808:
2757:
2701:
2641:
1780:
875:
189:
7561:
7530:
7154:
5622:
5584:
4414:
1737:
981:
803:
5151:, are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.
2710:
The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties (
1642:
1577:
1451:
587:
7686:
1177:
4915:
3784:
3749:
3714:
3342:
2301:
2156:
1063:
558:
7714:
4842:
4474:
3937:
3045:
2608:
2122:
2057:
1539:
1484:
1385:
1356:
1250:
1213:
644:
9046:
4718:
4684:
4299:
3201:
3166:
3071:
2091:
2006:
1608:
1510:
1413:
1327:
1301:
1108:
6199:
does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If
5876:
The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with
5339:
4869:
4813:
4741:
4501:
4445:
4381:
934:
676:
10363:
4961:
1911:
480:
2574:
1808:
1275:
1022:
533:
5497:, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
3118:
2543:
2520:
2477:
4935:
3306:
3221:
3091:
3019:
2995:
2497:
6132:
is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.
8441:. It is also a compact space. Again, it is no longer a field, or even an additive group. However, it allows division of a nonzero element by zero. It has
5682:
in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
8482:) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of
6215:
8144:
In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted
9180:
325:
involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of
2892:
293:
7365:
of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; ...)
3354:(if the leading coefficient is positive, take the least upper bound of real numbers for which the value of the polynomial is negative).
127:
means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite
6203:
is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.
3810:
3386:, extending to finitely many positive powers of ten to the left and infinitely many negative powers of ten to the right. For a number
10446:
3366:, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two.
5163:
10519:
9947:
6345:
10408:
9641:
9330:
7350:
5401:
is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for
3397:
7204:
The real numbers are uniquely specified by the above properties. More precisely, given any two
Dedekind-complete ordered fields
10313:
7619:
6729:
6753:
5136:
of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive
2886:
The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write
10257:
10098:
9988:
9806:
9781:
9607:
9502:
9469:
9406:
9304:
9268:
6677:, as equivalence classes of Cauchy sequences. Several problems were left open by these definitions, which contributed to the
5405:, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that
1825:
388:
Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an
10683:
10028:
8411:
9008:
without reference to real numbers, but these generalizations are relatively recent, and used only in very specific cases.
6367:
include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early
10678:
8523:) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their
5353:
The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.
5246:
9866:
7631:
6678:
7077:
5660:(ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.
10227:
10197:
10171:
10142:
10014:
9910:
9442:
7627:
6609:, was not understood. The rigor developed for geometry did not cross over to the concept of numbers until the 1800s.
1916:
75:
53:
6468:
46:
11200:
10358:
10353:
9398:
9296:
8979:
8565:
7375:
6745:
6606:
5125:
Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the
4971:
402:
10663:
11205:
11056:
8030:
6635:
made calculus rigorous, but he used the real numbers without defining them, and assumed without proof that every
6211:
6109:
5753:. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable
5653:
4576:
4506:
3127:
Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences.
2322:
2177:
1692:
321:
The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs of
11134:
6416:(c. 390−340 BC) provided a definition of the equality of two irrational proportions in a way that is similar to
5881:
3635:{\displaystyle x=b_{k}10^{k}+b_{k-1}10^{k-1}+\cdots +b_{0}+{\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2}}}+\cdots .}
7607:
5843:
3363:
9125:
9051:
3241:
11017:
10439:
10284:
9930:
8506:
6733:
6456:
9224:
2377:
2232:
326:
10512:
9350:
9325:
8181:
7998:
7799:
7445:
6519:
notation, and insisted that there is no difference between rational and irrational numbers in this regard.
5962:
work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in
5556:
8147:
4882:
between the real numbers and the decimal representations that do not end with infinitely many trailing 9.
3648:
10668:
10279:
8343:
6669:, leading to the publication in 1872 of two independent definitions of real numbers, one by Dedekind, as
307:
8426:. It is no longer a field, or even an additive group, but it still has a total order; moreover, it is a
7719:
7421:, that are based on the real numbers, although actual measurements of physical quantities are of finite
7071:
10306:
9973:
Redefining
Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction
9636:
8404:
8115:
7564:
4319:
8220:
8086:
7969:
7940:
7911:
7882:
7298:
7269:
7236:
7207:
5531:
4754:
11051:
11007:
10368:
10348:
9348:
Matvievskaya, Galina (1987), "The Theory of
Quadratic Irrationals in Medieval Oriental Mathematics",
9150:
8764:
8726:
8688:
8650:
8612:
6576:
5627:
1113:
696:
138:(and more generally in all mathematics), in particular by their role in the classical definitions of
6317:
6289:
6261:
6233:
6195:
in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an
11174:
11046:
10432:
9646:
9434:
9335:
8290:
7823:
4992:, which is a different sense than the Dedekind completeness of the order in the previous section):
2938:
2864:
2813:
2762:
2650:
2271:
40:
20:
10966:
10889:
10850:
10812:
10784:
10756:
10728:
10641:
10608:
10580:
10552:
9822:
8256:
7841:
7330:
7047:
7021:
6995:
6906:
6869:
6839:
6172:
6146:
6089:
6063:
6041:
6015:
5987:
5965:
5939:
5913:
5891:
5819:
5793:
5771:
5473:
5451:
5408:
3942:
2842:
2791:
2740:
2714:). So, the identification of natural numbers with some real numbers is justified by the fact that
2679:
2624:
827:
172:
10505:
10274:
9902:
9659:
9251:
7654:
7539:
7508:
7135:
6582:
6534:
6464:
5600:
5562:
5394:
4981:
4386:
3479:
in descending order by power of ten, with non-negative and negative powers of ten separated by a
1755:
954:
948:
758:
311:
10158:
8361:
is used as an adjective, meaning that the underlying field is the field of the real numbers (or
1614:
1550:
1418:
563:
9715:
9680:
9288:
9282:
T. K. Puttaswamy, "The
Accomplishments of Ancient Indian Mathematicians", pp. 410–11. In:
9246:
7819:
7643:
7603:
7468:
7422:
6702:
6444:
6117:
4417:
4313:
3375:
2644:
1815:
1741:
1143:
1077:
377:
267:
228:
57:
9515:
Cantor found a remarkable shortcut to reach Liouville's conclusion with a fraction of the work
9492:
9459:
7664:
6533:
In the 18th and 19th centuries, there was much work on irrational and transcendental numbers.
5314:
sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that
4891:
3753:
3718:
3683:
3311:
2277:
2132:
1027:
537:
401:
before the first formal definitions were provided in the second half of the 19th century. See
11195:
11119:
10955:
10398:
10299:
10078:
9552:
8918:
8580:
8575:
8512:
8351:
8276:
7611:
7599:
7591:
7584:
7362:
6710:
6542:
6136:
5850:
5148:
5056:
4818:
4450:
3913:
3347:
3024:
2966:
2673:
2587:
2126:
2098:
2033:
1515:
1460:
1361:
1332:
1226:
1189:
821:
611:
418:
154:
10215:
10186:
10185:
10157:
10130:
10066:
10050:
10002:
9894:
9422:
9256:
9250:
9179:
9025:
7690:
4700:
4659:
4274:
3171:
3145:
3050:
2070:
1985:
1587:
1489:
1392:
1306:
1280:
1083:
10872:
10630:
10393:
9734:
9694:
9359:
8520:
8453:
8434:
8368:
7794:
real numbers fail to be computable. Moreover, the equality of two computable numbers is an
7658:
7502:
7494:
7437:
7161:
6421:
5959:
5649:
5317:
5155:
4847:
4791:
4723:
4479:
4423:
4359:
3351:
3131:
2942:
907:
649:
9586:
Stefan Drobot "Real numbers". Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 vii+102 pp.
9423:
7324:. This uniqueness allows us to think of them as essentially the same mathematical object.
5597:
The statement that there is no subset of the reals with cardinality strictly greater than
5360:. It is easy to see that no ordered field can be lattice-complete, because it can have no
4940:
1883:
456:
8:
11083:
10993:
10950:
10932:
10710:
10486:
10338:
10033:
9632:
9321:
8528:
8446:
7869:
7795:
7639:
7623:
7595:
7490:
7441:
7394:
7366:
6889:
6622:
6452:
6010:
5742:
5559:. It is strictly greater than the cardinality of the set of all natural numbers (denoted
5521:
4977:
2551:
1785:
1254:
1001:
512:
143:
139:
10205:
10051:
10003:
9938:
9895:
9738:
9698:
9363:
9257:
6657:
in 1858, several mathematicians worked on the definition of the real numbers, including
5587:
3975:
Such a decimal representation specifies the real number as the least upper bound of the
3100:
2525:
2502:
2459:
10988:
10700:
10216:
10067:
9998:
9750:
9724:
9702:
9685:
9569:
9375:
9371:
8570:
8557:
7787:
7770:
7635:
7406:
6714:
6706:
6627:
6564:
6479:
6448:
6368:
6353:
6128:
of elements of this set, using rational coefficients only, and such that no element of
6125:
5869:
has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g.
4920:
3291:
3206:
3076:
3004:
2980:
2723:
2482:
346:
239:
1656:
Several other operations are commonly used, which can be deduced from the above ones.
11146:
11109:
11073:
11012:
10998:
10693:
10673:
10466:
10378:
10253:
10223:
10193:
10167:
10153:
10138:
10131:
10094:
10010:
10005:
The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass
9984:
9975:. Sources and Studies in the History of Mathematics and Physical Sciences. Springer.
9906:
9862:
9802:
9777:
9754:
9613:
9603:
9573:
9498:
9465:
9438:
9402:
9379:
9300:
9264:
8902:
8873:
8551:
8532:
8280:
7835:
7782:
7402:
7070:
The last property applies to the real numbers but not to the rational numbers (or to
6859:
6425:
6413:
6390:
6207:
5933:
5877:
5862:
5839:
5746:
5433:
5126:
3802:
3359:
2974:
2934:
2704:
901:
393:
365:
358:
303:
235:
128:
10046:
198:
11164:
11093:
11068:
11002:
10911:
10877:
10718:
10688:
10635:
10538:
10476:
10245:
10086:
10062:
9976:
9890:
9837:
9772:, Grundlehren der Mathematischen Wissenschaften , vol. 279, Berlin, New York:
9742:
9561:
9367:
9005:
8941:
8911:
8831:
8816:
8494:
8427:
8347:
8324:
7803:
7766:
7460:
7452:
7414:
7398:
6809:
6654:
6643:
6550:
6498:). In Europe, such numbers, not commensurable with the numerical unit, were called
6398:
6349:
5885:
5854:
5730:
5448:
Archimedean field in the sense that every other Archimedean field is a subfield of
5357:
4748:
3976:
1676:
995:
422:
254:
248:
202:
8395:
The real numbers can be generalized and extended in several different directions:
6662:
380:. All these definitions satisfy the axiomatic definition and are thus equivalent.
11041:
10945:
10602:
10388:
9951:
9856:
9773:
9228:
9202:
8975:
8866:
8683:
8536:
8339:
7861:
7533:
6817:
6682:
6572:
6440:
6409:
6113:
6035:
5734:
5726:
5675:
5657:
5506:
5361:
5010:
3484:
3124:
is bounded above, it has an upper bound that is less than any other upper bound.
2962:
2946:
2546:
369:
274:
217:
164:
6589:
is transcendental. Lindemann's proof was much simplified by Weierstrass (1885),
6124:
of real numbers such that every real number can be written uniquely as a finite
5444:, who meant still something else by it. He meant that the real numbers form the
11088:
11078:
11063:
10882:
10750:
10546:
10455:
10373:
10237:
10122:
10118:
9968:
9220:
8859:
8759:
8607:
8498:
8407:
unlike the real numbers. However, the complex numbers are not an ordered field.
8400:
7418:
7410:
6897:
6686:
6658:
6647:
6405:
5738:
5721:
sequences construction uses the metric topology presentation. The reals form a
5713:
5698:
5510:
5494:
4743:
in the preceding construction. These two representations are identical, unless
2734:
2431:
2093:
2020:
1873:
430:
405:
for details about these formal definitions and the proof of their equivalence.
10249:
10090:
9980:
9841:
9746:
7590:
Instead, computers typically work with finite-precision approximations called
4416:(this integer exists because of the Archimedean property). Then, supposing by
3358:
The last two properties are summarized by saying that the real numbers form a
11189:
11151:
11124:
11033:
10471:
10383:
10181:
9617:
9284:
8600:
8585:
8516:
8502:
8423:
7615:
7486:
7464:
6821:
6594:
6590:
6357:
6196:
6192:
5866:
5750:
5441:
5390:
5386:
4989:
3480:
3394:
places to the left, the standard notation is the juxtaposition of the digits
3379:
2836:
414:
389:
361:
334:
330:
7879:
The sets of positive real numbers and negative real numbers are often noted
7786:
if there exists an algorithm that yields its digits. Because there are only
7777:
roughly doubles its number of terms), overwhelming finite computer storage.
6697:(Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor's
1215:
being that it is a total order means two properties: given two real numbers
11114:
10916:
10126:
10031:" [On a property of the collection of all real algebraic numbers].
10024:
9597:
8809:
8442:
8320:
7456:
6694:
6690:
6674:
6670:
6666:
6568:
6512:
6417:
6083:
5858:
5758:
5722:
5686:
5517:
5402:
4985:
4886:
3383:
2785:
2730:
2715:
2027:
1389:
The order is compatible with addition and multiplication, which means that
373:
7876:
is frequently used when its algebraic properties are under consideration.
5385:
These two notions of completeness ignore the field structure. However, an
3382:
each representing the product of an integer between zero and nine times a
10940:
8524:
8379:
7802:
accept the existence of only those reals that are computable. The set of
7263:
7132:
7015:
6926:
6813:
6721:
6598:
6483:
6432:
6420:(introduced more than 2,000 years later), except that he did not use any
6383:
5765:
5754:
5525:
5137:
5133:
3992:
3285:
2970:
1979:
1660:
1544:
Many other properties can be deduced from the above ones. In particular:
690:
605:
434:
397:
354:
315:
278:
206:
120:
101:
89:
10029:Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen
9706:
9144:
6386:
of certain numbers, such as 2 and 61, could not be exactly determined.
5431:
uniformly complete ordered field, but it is the only uniformly complete
153:
The set of real numbers, sometimes called "the reals", is traditionally
10921:
10778:
10413:
10343:
9565:
7815:
7791:
7774:
7576:
6825:
6698:
6650:, highlighting the need for a rigorous definition of the real numbers.
6527:
6467:" into a more general idea of real numbers. The Egyptian mathematician
6394:
6140:
5870:
5816:
of odd degree admits at least one real root: these two properties make
5813:
5440:
But the original use of the phrase "complete Archimedean field" was by
3980:
243:
147:
9713:
Bengtsson, Ingemar (2017). "The Number Behind the Simplest SIC-POVM".
9017:
More precisely, given two complete totally ordered fields, there is a
8991:
The terminating rational numbers may have two decimal expansions (see
2610:) is identified with the division of the real numbers identified with
10403:
9602:. Oxford handbooks. Oxford ; New York: Oxford University Press.
8336:
8317:
7647:
7580:
6725:
6523:
6491:
5679:
5669:
5591:
4879:
4652:
One can use the defining properties of the real numbers to show that
2923:{\displaystyle \mathbb {N} \subset \mathbb {Q} \subset \mathbb {R} .}
2456:
This identification can be pursued by identifying a negative integer
2013:
282:
105:
9543:
8974:
This is not sufficient for distinguishing the real numbers from the
6701:
was published several years later. Thirdly, these definitions imply
4976:
A main reason for using real numbers is so that many sequences have
11029:
10960:
10806:
10333:
9729:
9395:
Mathematics Across Cultures: The History of Non-western Mathematics
9293:
Mathematics Across Cultures: The History of Non-western Mathematics
8992:
8850:
7868:(upright bold). As it is naturally endowed with the structure of a
7041:
6988:
The order is Dedekind-complete, meaning that every nonempty subset
6893:
6618:
6503:
6495:
6487:
6428:). This may be viewed as the first definition of the real numbers.
5717:
5144:
5023:
4996:
4872:
2950:
426:
338:
135:
112:
108:
10424:
238:. Some irrational numbers (as well as all the rationals) are the
10574:
10497:
8721:
7472:
6705:
on infinite sets, and this cannot be formalized in the classical
6516:
6226:
6143:
if the axiom of choice is assumed: there exists a total order on
5790:, although no negative number does. This shows that the order on
322:
221:
9799:
Computer algebra and symbolic computation: elementary algorithms
6567:, and then established the existence of transcendental numbers;
5910:. Ordered fields that satisfy the same first-order sentences as
5087:
if its elements eventually come and remain arbitrarily close to
3900:{\displaystyle b_{k},b_{k-1},\ldots ,b_{0},a_{1},a_{2},\ldots .}
396:. Here, "completely characterized" means that there is a unique
10528:
10291:
8403:
contain solutions to all polynomial equations and hence are an
7480:
7172:– 1 would not be an upper bound, and there would be an integer
6636:
6632:
6460:
6424:
other than multiplication of a length by a natural number (see
6372:
6166:
5712:. By virtue of being a totally ordered set, they also carry an
5678:. This is because the set of rationals, which is countable, is
5652:(CH). It is neither provable nor refutable using the axioms of
97:
7505:
posits that the cardinality of the set of the real numbers is
5226:{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}
9934:
8995:); the other real numbers have exactly one decimal expansion.
8247:
7476:
7463:
and others extend the set of the real numbers by introducing
6689:
are rigorously defined; this was done a few years later with
5143:
The completeness property of the reals is the basis on which
2945:. In the latter case, these homomorphisms are interpreted as
2711:
350:
342:
297:
Real numbers can be thought of as all points on a number line
10242:
The Real Numbers: An Introduction to Set Theory and Analysis
9527:
Hurwitz, Adolf (1893). "Beweis der Transendenz der Zahl e".
7197:, which is a contradiction with the upper-bound property of
6639:
sequence has a limit and that this limit is a real number.
5068:
eventually come and remain arbitrarily close to each other.
4697:
Another decimal representation can be obtained by replacing
2718:
are satisfied by these real numbers, with the addition with
9099:
and the second formula is stable under field automorphisms.
8793:
8384:
8350:
of the Euclidean space is identified with the tuple of its
8080:
7373:. For details and other constructions of real numbers, see
6720:
In 1874, Cantor showed that the set of all real numbers is
6478:
was the first to accept irrational numbers as solutions to
6436:
6200:
3472:{\displaystyle b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots ,}
3374:
The most common way of describing a real number is via its
3350:
of odd degree with real coefficients has at least one real
893:
116:
9926:
9924:
9922:
9861:(3 ed.), Sudbury, MA: Jones and Bartlett Publishers,
9494:
The Calculus Gallery: Masterpieces from Newton to Lebesgue
7995:
are also used. The non-negative real numbers can be noted
5389:(in this case, the additive group of the field) defines a
9897:
Chapter Zero: Fundamental Notions of Abstract Mathematics
9823:"Computing numerically with functions instead of numbers"
8801:
8346:
has been chosen in the latter. In this identification, a
7356:
6646:
highlighted the limitations of calculus in the method of
5348:
2961:
Previous properties do not distinguish real numbers from
2417:
19:
For the real numbers used in descriptive set theory, see
8387:, meaning a real number (as in "the set of all reals").
6801:{\displaystyle (\mathbb {R} ;{}+{};{}\cdot {};{}<{})}
5984:), we know that the same statement must also be true of
4271:
defined by the sequence is the least upper bound of the
2676:
described below implies that some real numbers, such as
10116:
10085:. Springer Undergraduate Mathematics Series. Springer.
10069:
The Number Systems: Foundations of Algebra and Analysis
9919:
7413:
are described using mathematical structures, typically
4788:
In this case, in the first decimal representation, all
3234:
is a positive real number, there is a positive integer
2969:, which states that every set of real numbers with an
259:
9596:
Robson, Eleanor; Stedall, Jacqueline A., eds. (2009).
7722:
7693:
7667:
6717:
were developed in the first half of the 20th century.
5812:
is determined by its algebraic structure. Also, every
4885:
The preceding considerations apply directly for every
4757:
1758:
257:. There are also real numbers which are not, such as
10969:
10892:
10853:
10815:
10787:
10759:
10731:
10644:
10611:
10583:
10555:
9630:
9319:
9186:
Clash Of Symbols: A Ride Through The Riches Of Glyphs
9054:
9028:
8767:
8729:
8691:
8653:
8615:
8323:
over the field of the real numbers, often called the
8293:
8259:
8223:
8184:
8150:
8118:
8089:
8033:
8001:
7972:
7943:
7914:
7885:
7844:
7790:
many algorithms, but an uncountable number of reals,
7542:
7511:
7436:
The real numbers are most often formalized using the
7333:
7301:
7272:
7239:
7210:
7138:
7080:
7050:
7024:
6998:
6909:
6872:
6842:
6756:
6320:
6292:
6264:
6236:
6175:
6149:
6092:
6066:
6044:
6018:
5990:
5968:
5942:
5916:
5894:
5842:. Proving this is the first half of one proof of the
5822:
5796:
5774:
5630:
5603:
5565:
5534:
5476:
5454:
5411:
5378:
Additionally, an order can be Dedekind-complete, see
5320:
5249:
5166:
4943:
4923:
4894:
4850:
4821:
4794:
4726:
4703:
4662:
4579:
4509:
4482:
4453:
4426:
4389:
4362:
4322:
4277:
4003:
3945:
3916:
3813:
3756:
3721:
3686:
3651:
3495:
3400:
3314:
3294:
3244:
3209:
3174:
3148:
3103:
3079:
3053:
3027:
3007:
2983:
2895:
2867:
2845:
2816:
2794:
2765:
2743:
2682:
2653:
2627:
2590:
2554:
2528:
2505:
2485:
2462:
2380:
2325:
2280:
2235:
2180:
2135:
2101:
2073:
2036:
1988:
1919:
1886:
1828:
1788:
1695:
1617:
1590:
1553:
1518:
1492:
1463:
1421:
1395:
1364:
1335:
1309:
1283:
1257:
1229:
1192:
1146:
1116:
1086:
1030:
1004:
957:
910:
830:
761:
699:
652:
614:
566:
540:
515:
459:
175:
8547:
817:, and that parentheses may be omitted in both cases.
417:. Intuitively, this means that methods and rules of
9392:
9391:Jacques Sesiano, "Islamic mathematics", p. 148, in
9283:
6571:(1873) extended and greatly simplified this proof.
5393:structure, and uniform structures have a notion of
3223:is the least upper bound of the integers less than
1913:measures its distance from zero, and is defined as
364:. Other common definitions of real numbers include
246:with integer coefficients, such as the square root
10977:
10900:
10861:
10823:
10795:
10767:
10739:
10652:
10619:
10591:
10563:
9143:
9091:
9040:
8779:
8741:
8703:
8665:
8627:
8467:copies of the real line plus a single point (here
8304:
8267:
8238:
8207:
8170:
8136:
8104:
8063:
8019:
7987:
7958:
7929:
7900:
7852:
7757:
7708:
7680:
7555:
7524:
7341:
7316:
7287:
7254:
7225:
7148:
7119:
7058:
7032:
7006:
6917:
6880:
6850:
6800:
6526:introduced the term "real" to describe roots of a
6334:
6306:
6278:
6250:
6183:
6157:
6100:
6074:
6052:
6026:
5998:
5976:
5950:
5924:
5902:
5830:
5804:
5782:
5640:
5616:
5578:
5547:
5524:from the real numbers to the natural numbers. The
5484:
5462:
5419:
5333:
5296:{\displaystyle \sum _{n=N}^{M}{\frac {x^{n}}{n!}}}
5295:
5225:
4955:
4929:
4909:
4863:
4836:
4807:
4780:
4735:
4712:
4678:
4642:
4565:
4495:
4468:
4439:
4408:
4375:
4348:
4293:
4256:
3964:
3931:
3899:
3778:
3743:
3708:
3669:
3634:
3471:
3336:
3300:
3269:
3215:
3195:
3160:
3112:
3085:
3065:
3039:
3013:
2989:
2977:. This means the following. A set of real numbers
2922:
2878:
2853:
2827:
2802:
2776:
2751:
2695:
2664:
2635:
2602:
2568:
2537:
2514:
2491:
2471:
2406:
2366:
2295:
2261:
2221:
2150:
2116:
2085:
2051:
2000:
1962:
1905:
1863:
1802:
1774:
1731:
1636:
1602:
1571:
1533:
1504:
1478:
1445:
1407:
1379:
1350:
1321:
1295:
1269:
1244:
1207:
1171:
1132:
1102:
1057:
1016:
975:
928:
869:
797:
747:
670:
638:
581:
552:
527:
474:
306:is the association of points on lines (especially
273:Real numbers can be thought of as all points on a
183:
9599:The Oxford handbook of the history of mathematics
8390:
5099:(possibly depending on ε) such that the distance
4686:So, the resulting sequence of digits is called a
4356:as decimal representation of the largest integer
11187:
10244:. Undergraduate Texts in Mathematics. Springer.
7657:can operate on irrational quantities exactly by
7127:has a rational upper bound (e.g., 1.42), but no
7120:{\displaystyle \{x\in \mathbb {Q} :x^{2}<2\}}
6530:, distinguishing them from "imaginary" numbers.
6404:For Greek mathematicians, numbers were only the
5306:can be made arbitrarily small (independently of
1936:
9885:
9883:
8535:operators correspond to the positive reals and
8505:and infinitely large numbers and are therefore
7382:
2499:is a natural number) with the additive inverse
7471:in a way closer to the original intuitions of
5509:, in the sense that while both the set of all
3987:, the truncation of the sequence at the place
3288:, that is, there exist a positive real number
285:, where the points corresponding to integers (
10513:
10440:
10307:
9858:Discrete Structures, Logic, and Computability
9767:
9595:
9589:
9393:Selin, Helaine; D'Ambrosio, Ubiratan (2000),
7598:. The achievable precision is limited by the
6549:is not the square root of a rational number.
5674:As a topological space, the real numbers are
5528:of the set of all real numbers is denoted by
4308:, one can define a decimal representation of
2949:that can often be done automatically by the
2441:. This allows identifying any natural number
421:apply to them. More precisely, there are two
9940:
9880:
9347:
9197:
9195:
8055:
8049:
7467:and infinite numbers, allowing for building
7114:
7081:
6314:, which in turn include the natural numbers
5888:satisfies the same first order sentences as
4304:Conversely, given a nonnegative real number
1973:
383:
292:
9497:, Princeton University Press, p. 127,
9457:
9326:"Arabic mathematics: forgotten brilliance?"
9245:
8493:Ordered fields extending the reals are the
8064:{\displaystyle \mathbb {R} ^{+}\cup \{0\}.}
6545:(1794) completed the proof and showed that
4965:
4643:{\displaystyle D_{n}=D_{n-1}+a_{n}/10^{n}.}
4566:{\displaystyle D_{n-1}+a_{n}/10^{n}\leq a,}
2367:{\displaystyle (b<a){\text{ or }}(a=b),}
2222:{\displaystyle (a<b){\text{ or }}(a=b),}
11170:
10520:
10506:
10447:
10433:
10314:
10300:
9997:
9889:
9351:Annals of the New York Academy of Sciences
8771:
8733:
8695:
8657:
8619:
8519:(for example, self-adjoint square complex
7829:
6900:are defined and have the usual properties.
6724:, but the set of all algebraic numbers is
5055:. This definition, originally provided by
3801:and integers between zero and nine in the
2703:are not rational numbers; they are called
234:. The rest of the real numbers are called
10971:
10894:
10855:
10817:
10789:
10761:
10733:
10646:
10613:
10585:
10557:
10236:
10213:
10045:
9820:
9768:Bishop, Errett; Bridges, Douglas (1985),
9728:
9712:
9192:
8773:
8735:
8697:
8659:
8621:
8295:
8261:
8226:
8187:
8153:
8121:
8096:
8092:
8036:
8004:
7975:
7946:
7917:
7888:
7846:
7748:
7650:implemented with approximate arithmetic.
7361:The real numbers can be constructed as a
7335:
7308:
7304:
7275:
7242:
7213:
7091:
7052:
7026:
7000:
6911:
6874:
6844:
6761:
6625:without defining them rigorously. In his
6325:
6297:
6269:
6241:
6177:
6151:
6120:of this vector space: there exists a set
6094:
6068:
6046:
6020:
5992:
5970:
5944:
5918:
5896:
5824:
5798:
5776:
5478:
5456:
5413:
3369:
3120:So, Dedekind completeness means that, if
2913:
2905:
2897:
2869:
2847:
2818:
2796:
2767:
2745:
2655:
2629:
345:definition is that real numbers form the
177:
76:Learn how and when to remove this message
16:Number representing a continuous quantity
10152:
10061:
9420:
9219:
9092:{\displaystyle \exists z\mid x-y=z^{2},}
9004:Limits and continuity can be defined in
8083:, and these sets are noted respectively
7587:, which are inconvenient to manipulate.
6681:. Firstly both definitions suppose that
6225:
5154:For example, the standard series of the
5022:(possibly depending on ε) such that the
3270:{\displaystyle 0<{\frac {1}{n}}<x}
2956:
39:This article includes a list of general
9801:, vol. 1, A K Peters, p. 32,
9679:
9642:MacTutor History of Mathematics Archive
9526:
9458:Arndt, Jörg; Haenel, Christoph (2001),
9331:MacTutor History of Mathematics Archive
9233:The Stanford Encyclopedia of Philosophy
7769:); and arithmetic operations can cause
6713:. This is one of the reasons for which
5379:
4844:and, in the second representation, all
4301:which exists by Dedekind completeness.
3983:the sequence: given a positive integer
3362:. This implies the real version of the
1864:{\displaystyle {\frac {a}{b}}=ab^{-1}.}
1651:
11188:
10180:
10023:
9541:
9490:
8331:; this space may be identified to the
7806:is broader, but still only countable.
7602:allocated for each number, whether as
7357:Construction from the rational numbers
6165:with the property that every nonempty
2418:Integers and fractions as real numbers
2407:{\displaystyle {\text{not }}(a<b).}
2262:{\displaystyle {\text{not }}(b<a).}
1876:: the absolute value of a real number
1663:: the subtraction of two real numbers
310:) to real numbers such that geometric
10501:
10428:
10295:
10077:
9796:
9637:"The real numbers: Stevin to Hilbert"
8208:{\displaystyle \mathbb {R} _{-}^{*}.}
8020:{\displaystyle \mathbb {R} _{\geq 0}}
7610:, or some other representation. Most
7369:to a unique real number—in this case
6831:
6739:
6693:. Secondly, both definitions involve
6139:implies that the real numbers can be
5590:), and equals the cardinality of the
5236:converges to a real number for every
3645:For example, for the circle constant
3390:whose decimal representation extends
1810:and defined as the multiplication of
689:Addition and multiplication are both
604:Addition and multiplication are both
10364:Decidability of first-order theories
10107:
9854:
9667:18.095 Lecture Series in Mathematics
9177:
8412:affinely extended real number system
8171:{\displaystyle \mathbb {R} _{+}^{*}}
7351:Tarski's axiomatization of the reals
6925:is ordered, meaning that there is a
6490:(often in the form of square roots,
5764:Every nonnegative real number has a
5516:and the set of all real numbers are
3670:{\displaystyle \pi =3.14159\cdots ,}
1982:that is considered above is denoted
437:that have the following properties.
134:The real numbers are fundamental in
25:
10684:Set-theoretically definable numbers
10454:
9967:
9263:, North-Holland, pp. 274–285,
7758:{\textstyle \int _{0}^{1}x^{x}\,dx}
7618:floating-point arithmetic, often a
7563:, the cardinality of the integers.
6730:Cantor's first uncountability proof
5663:
5633:
5594:of the set of the natural numbers.
5537:
2933:These identifications are formally
2643:of the rational numbers an ordered
2621:These identifications make the set
2522:of the real number identified with
13:
10527:
10133:Foundations of Measurement, Vol. 1
9372:10.1111/j.1749-6632.1987.tb37206.x
9055:
8539:correspond to the complex numbers.
8279:), which can be identified to the
8027:but one often sees this set noted
7544:
7513:
6679:foundational crisis of mathematics
6612:
5605:
5567:
5196:
3797:consists of a nonnegative integer
2965:. This distinction is provided by
201:, distinguishes real numbers from
45:it lacks sufficient corresponding
14:
11217:
10267:
9855:Hein, James L. (2010), "14.1.1",
8475:denotes the reversed ordering of
8137:{\displaystyle \mathbb {R} _{-}.}
6929:≥ such that for all real numbers
5398:
5129:of the real numbers is complete.
4781:{\textstyle {\frac {m}{10^{h}}}.}
4349:{\displaystyle b_{k}\cdots b_{0}}
2835:and an injective homomorphism of
2430:are commonly identified with the
11169:
10321:
10214:Stevenson, Frederick W. (2000).
8566:Completeness of the real numbers
8550:
8239:{\displaystyle \mathbb {R} ^{n}}
8105:{\displaystyle \mathbb {R_{+}} }
7988:{\displaystyle \mathbb {R} _{-}}
7959:{\displaystyle \mathbb {R} _{+}}
7930:{\displaystyle \mathbb {R} ^{-}}
7901:{\displaystyle \mathbb {R} ^{+}}
7376:Construction of the real numbers
7317:{\displaystyle \mathbb {R_{2}} }
7288:{\displaystyle \mathbb {R} _{1}}
7255:{\displaystyle \mathbb {R} _{2}}
7226:{\displaystyle \mathbb {R} _{1}}
7072:other more exotic ordered fields
6746:Construction of the real numbers
6537:(1761) gave a flawed proof that
6435:brought about the acceptance of
5548:{\displaystyle {\mathfrak {c}}.}
4972:Completeness of the real numbers
4656:is the least upper bound of the
4503:as the largest digit such that
1744:: the division of a real number
824:over addition, which means that
403:Construction of the real numbers
30:
9848:
9830:Mathematics in Computer Science
9814:
9790:
9761:
9673:
9652:
9624:
9580:
9535:
9520:
9484:
9451:
9414:
9385:
9341:
9313:
9276:
9252:"5. The Constructible Universe"
9239:
8780:{\displaystyle :\;\mathbb {N} }
8742:{\displaystyle :\;\mathbb {Z} }
8704:{\displaystyle :\;\mathbb {Q} }
8666:{\displaystyle :\;\mathbb {R} }
8628:{\displaystyle :\;\mathbb {C} }
7838:of all real numbers is denoted
5641:{\displaystyle {\mathfrak {c}}}
5505:The set of all real numbers is
5059:, formalizes the fact that the
4980:. More formally, the reals are
2729:Formally, one has an injective
1963:{\displaystyle |a|=\max(a,-a).}
1133:{\displaystyle {\frac {1}{a}}.}
943:There is a real number denoted
748:{\displaystyle (a+b)+c=a+(b+c)}
318:between corresponding numbers.
10164:Mathematics: Form and Function
10053:Foundations of Modern Analysis
9433:. St. Martin's Press. p.
9213:
9171:
9154:(3rd ed.). 2008. 'real',
9136:
9118:
9011:
8998:
8985:
8968:
8527:are real and they form a real
8507:non-Archimedean ordered fields
8391:Generalizations and extensions
7659:manipulating symbolic formulas
7594:, a representation similar to
7571:
7327:For another axiomatization of
7262:, there exists a unique field
7131:rational upper bound, because
6795:
6757:
6732:was different from his famous
6335:{\displaystyle (\mathbb {N} )}
6329:
6321:
6307:{\displaystyle (\mathbb {Z} )}
6301:
6293:
6279:{\displaystyle (\mathbb {Q} )}
6273:
6265:
6251:{\displaystyle (\mathbb {R} )}
6245:
6237:
6116:guarantees the existence of a
5844:fundamental theorem of algebra
5745: 1. The real numbers are
5500:
5008:) of real numbers is called a
3793:for a nonnegative real number
3364:fundamental theorem of algebra
2398:
2386:
2358:
2346:
2338:
2326:
2253:
2241:
2213:
2201:
2193:
2181:
1954:
1939:
1929:
1921:
1896:
1888:
1723:
1714:
1644:for every nonzero real number
1179:for every nonzero real number
1046:
1037:
892:There is a real number called
846:
834:
792:
783:
771:
762:
742:
730:
712:
700:
509:produce a real number denoted
453:produce a real number denoted
197:, used in the 17th century by
1:
11018:Plane-based geometric algebra
9188:. Springer. pp. 198–199.
9106:
8383:. The word is also used as a
8305:{\displaystyle \mathbb {R} .}
7809:
7632:floating-point numbers do not
7532:; i.e. the smallest infinite
6515:created the basis for modern
6473:
6377:
6362:
6360:" ("The rules of chords") in
6286:, which include the integers
6258:include the rational numbers
2879:{\displaystyle \mathbb {R} .}
2828:{\displaystyle \mathbb {Q} ,}
2784:an injective homomorphism of
2777:{\displaystyle \mathbb {Z} ,}
2665:{\displaystyle \mathbb {R} .}
408:
216:The real numbers include the
10978:{\displaystyle \mathbb {S} }
10901:{\displaystyle \mathbb {C} }
10862:{\displaystyle \mathbb {R} }
10824:{\displaystyle \mathbb {O} }
10796:{\displaystyle \mathbb {H} }
10768:{\displaystyle \mathbb {C} }
10740:{\displaystyle \mathbb {R} }
10653:{\displaystyle \mathbb {A} }
10620:{\displaystyle \mathbb {Q} }
10592:{\displaystyle \mathbb {Z} }
10564:{\displaystyle \mathbb {N} }
9821:Trefethen, Lloyd N. (2007).
9111:
8268:{\displaystyle \mathbb {R} }
7853:{\displaystyle \mathbb {R} }
7383:Applications and connections
7342:{\displaystyle \mathbb {R} }
7059:{\displaystyle \mathbb {R} }
7033:{\displaystyle \mathbb {R} }
7007:{\displaystyle \mathbb {R} }
6918:{\displaystyle \mathbb {R} }
6881:{\displaystyle \mathbb {R} }
6851:{\displaystyle \mathbb {R} }
6563:can be a root of an integer
6206:A real number may be either
6184:{\displaystyle \mathbb {R} }
6158:{\displaystyle \mathbb {R} }
6101:{\displaystyle \mathbb {Q} }
6075:{\displaystyle \mathbb {R} }
6053:{\displaystyle \mathbb {Q} }
6027:{\displaystyle \mathbb {R} }
5999:{\displaystyle \mathbb {R} }
5977:{\displaystyle \mathbb {R} }
5951:{\displaystyle \mathbb {R} }
5925:{\displaystyle \mathbb {R} }
5903:{\displaystyle \mathbb {R} }
5849:The reals carry a canonical
5831:{\displaystyle \mathbb {R} }
5805:{\displaystyle \mathbb {R} }
5783:{\displaystyle \mathbb {R} }
5557:cardinality of the continuum
5485:{\displaystyle \mathbb {R} }
5463:{\displaystyle \mathbb {R} }
5420:{\displaystyle \mathbb {R} }
5349:"The complete ordered field"
4917:simply by replacing 10 with
3965:{\displaystyle b_{k}\neq 0.}
2854:{\displaystyle \mathbb {Q} }
2803:{\displaystyle \mathbb {Z} }
2752:{\displaystyle \mathbb {N} }
2696:{\displaystyle {\sqrt {2}},}
2636:{\displaystyle \mathbb {Q} }
2307:is greater than or equal to
1775:{\textstyle {\frac {a}{b}},}
870:{\displaystyle a(b+c)=ab+ac}
184:{\displaystyle \mathbb {R} }
7:
10280:Encyclopedia of Mathematics
10204:Translated from the German
10149:Vol. 2, 1989. Vol. 3, 1990.
9901:. Addison-Wesley. pp.
8543:
8344:Cartesian coordinate system
8071:In French mathematics, the
7626:. Real numbers satisfy the
7608:arbitrary-precision numbers
7556:{\displaystyle \aleph _{0}}
7525:{\displaystyle \aleph _{1}}
7160:These properties imply the
7149:{\displaystyle {\sqrt {2}}}
6862:of all real numbers. Then:
6553:(1840) showed that neither
6110:Zermelo–Fraenkel set theory
6082:can therefore be seen as a
5654:Zermelo–Fraenkel set theory
5617:{\displaystyle \aleph _{0}}
5579:{\displaystyle \aleph _{0}}
5051:that are both greater than
4409:{\displaystyle D_{0}\leq x}
3280:Every positive real number
1732:{\displaystyle a-b=a+(-b).}
1186:The total order is denoted
976:{\displaystyle a\times 1=a}
798:{\displaystyle (ab)c=a(bc)}
10:
11222:
10218:Exploring the Real Numbers
9961:
9225:"The Continuum Hypothesis"
9181:"Set of Natural Numbers ℕ"
9163:A real number. Usually in
8405:algebraically closed field
7387:
6743:
6221:
5667:
5624:and strictly smaller than
5341:is well defined for every
4969:
4420:that the decimal fraction
3001:if there is a real number
1637:{\displaystyle 0<a^{2}}
1572:{\displaystyle 0\cdot a=0}
1446:{\displaystyle a+c<b+c}
1072:Every nonzero real number
582:{\displaystyle a\times b,}
18:
11160:
11102:
11028:
11008:Algebra of physical space
10930:
10838:
10709:
10536:
10462:
10369:Extended real number line
10329:
10250:10.1007/978-3-319-01577-4
10192:(3rd ed.). Chelsea.
10091:10.1007/978-1-4471-0341-7
9981:10.1007/978-1-4613-0087-8
9842:10.1007/s11786-007-0001-y
9747:10.1007/s10701-017-0078-3
9464:, Springer, p. 192,
9151:Oxford English Dictionary
8246:refers to the set of the
7681:{\textstyle {\sqrt {2}},}
7628:usual rules of arithmetic
6469:Abū Kāmil Shujā ibn Aslam
6382:, who was aware that the
6352:around 1000 BC; the
6060:of rational numbers, and
5865:normalized such that the
5838:the premier example of a
5380:§ Axiomatic approach
4966:Topological completeness
2737:from the natural numbers
2162:is less than or equal to
1974:Auxiliary order relations
1748:by a nonzero real number
1172:{\displaystyle aa^{-1}=1}
413:The real numbers form an
384:Characterizing properties
329:and is the foundation of
287:..., −2, −1, 0, 1, 2, ...
11064:Extended complex numbers
11047:Extended natural numbers
9931:École Normale Supérieure
9647:University of St Andrews
9491:Dunham, William (2015),
9336:University of St Andrews
8961:
7780:A real number is called
7655:computer algebra systems
7446:constructive mathematics
7431:
6389:Around 500 BC, the
6210:or uncomputable; either
5882:Löwenheim–Skolem theorem
5861:on their structure as a
5685:The real numbers form a
5095:there exists an integer
5018:there exists an integer
4910:{\displaystyle B\geq 2,}
3779:{\displaystyle a_{2}=4,}
3744:{\displaystyle a_{1}=1,}
3709:{\displaystyle b_{0}=3,}
3337:{\displaystyle r^{2}=x.}
3134:: for every real number
2939:constructive mathematics
2810:to the rational numbers
2296:{\displaystyle a\geq b,}
2272:Greater than or equal to
2151:{\displaystyle a\leq b,}
2023:are also commonly used:
1058:{\displaystyle a+(-a)=0}
553:{\displaystyle a\cdot b}
327:19th-century mathematics
21:Baire space (set theory)
11201:Real algebraic geometry
10207:Grundlagen der Analysis
10188:Foundations of Analysis
9797:Cohen, Joel S. (2002),
9681:Wheeler, John Archibald
9421:Beckmann, Petr (1971).
9247:Moschovakis, Yiannis N.
9207:Encyclopedia Britannica
8501:; both of them contain
7830:Vocabulary and notation
7709:{\textstyle \arctan 5,}
7622:with around 16 decimal
6750:The real number system
6673:, and the other one by
6581:is transcendental, and
6397:also realized that the
5689:: the distance between
5356:First, an order can be
5043:is less than ε for all
4878:In summary, there is a
4837:{\displaystyle n>h,}
4469:{\displaystyle i<n,}
3932:{\displaystyle k>0,}
3040:{\displaystyle s\leq u}
2603:{\displaystyle q\neq 0}
2117:{\displaystyle b<a.}
2052:{\displaystyle a>b,}
1534:{\displaystyle 0<b.}
1479:{\displaystyle 0<ab}
1380:{\displaystyle a<c.}
1351:{\displaystyle b<c,}
1245:{\displaystyle a<b,}
1208:{\displaystyle a<b.}
949:multiplicative identity
877:for every real numbers
805:for every real numbers
678:for every real numbers
639:{\displaystyle a+b=b+a}
378:decimal representations
372:(of rational numbers),
60:more precise citations.
11206:Elementary mathematics
11120:Transcendental numbers
10979:
10956:Hyperbolic quaternions
10902:
10863:
10825:
10797:
10769:
10741:
10654:
10621:
10593:
10565:
9941:
9716:Foundations of Physics
9259:Descriptive Set Theory
9235:. Stanford University.
9178:Webb, Stephen (2018).
9093:
9042:
9041:{\displaystyle x>y}
8867:Dyadic (finite binary)
8781:
8743:
8705:
8667:
8629:
8576:Definable real numbers
8513:Self-adjoint operators
8306:
8269:
8240:
8209:
8172:
8138:
8106:
8065:
8021:
7989:
7960:
7931:
7902:
7854:
7820:descriptive set theory
7759:
7710:
7682:
7612:scientific computation
7592:floating-point numbers
7585:definable real numbers
7577:Electronic calculators
7557:
7526:
7469:infinitesimal calculus
7423:accuracy and precision
7343:
7318:
7289:
7256:
7227:
7150:
7121:
7060:
7044:(a.k.a., supremum) in
7034:
7008:
6919:
6882:
6852:
6802:
6711:first-order predicates
6621:used real numbers and
6453:Chinese mathematicians
6342:
6336:
6308:
6280:
6252:
6212:algorithmically random
6185:
6159:
6102:
6076:
6054:
6034:of real numbers is an
6028:
6000:
5978:
5952:
5926:
5904:
5832:
5806:
5784:
5642:
5618:
5580:
5549:
5486:
5464:
5421:
5335:
5297:
5270:
5227:
5200:
5091:, that is, if for any
5082:converges to the limit
4957:
4931:
4911:
4865:
4838:
4809:
4782:
4737:
4714:
4713:{\displaystyle \leq x}
4688:decimal representation
4680:
4679:{\displaystyle D_{n}.}
4644:
4567:
4497:
4470:
4441:
4410:
4377:
4350:
4295:
4294:{\displaystyle D_{n},}
4258:
4226:
4182:
3966:
3933:
3901:
3791:decimal representation
3780:
3745:
3710:
3671:
3636:
3473:
3376:decimal representation
3370:Decimal representation
3338:
3302:
3271:
3217:
3197:
3196:{\displaystyle n=u+1,}
3162:
3161:{\displaystyle x<n}
3138:, there is an integer
3114:
3087:
3067:
3066:{\displaystyle s\in S}
3041:
3015:
2991:
2924:
2880:
2855:
2829:
2804:
2778:
2753:
2697:
2666:
2637:
2604:
2570:
2539:
2516:
2493:
2473:
2449:real numbers equal to
2408:
2368:
2297:
2263:
2223:
2152:
2118:
2087:
2086:{\displaystyle a>b}
2053:
2002:
2001:{\displaystyle a<b}
1964:
1907:
1865:
1816:multiplicative inverse
1804:
1776:
1733:
1671:results in the sum of
1638:
1604:
1603:{\displaystyle 0<1}
1579:for every real number
1573:
1535:
1506:
1505:{\displaystyle 0<a}
1480:
1453:for every real number
1447:
1409:
1408:{\displaystyle a<b}
1381:
1352:
1323:
1322:{\displaystyle a<b}
1297:
1296:{\displaystyle b<a}
1271:
1246:
1209:
1173:
1134:
1104:
1103:{\displaystyle a^{-1}}
1078:multiplicative inverse
1065:for every real number
1059:
1018:
983:for every real number
977:
936:for every real number
930:
871:
799:
749:
672:
640:
583:
554:
529:
476:
298:
289:) are equally spaced.
268:transcendental numbers
185:
11052:Extended real numbers
10980:
10903:
10873:Split-complex numbers
10864:
10826:
10798:
10770:
10742:
10655:
10631:Constructible numbers
10622:
10594:
10566:
10409:Tarski axiomatization
10399:Real coordinate space
10349:Cantor–Dedekind axiom
10108:Katz, Robert (1964).
9770:Constructive analysis
9553:Mathematische Annalen
9542:Gordan, Paul (1893).
9529:Mathematische Annalen
9094:
9043:
8782:
8744:
8706:
8668:
8630:
8581:Positive real numbers
8352:Cartesian coordinates
8307:
8277:real coordinate space
8270:
8241:
8210:
8173:
8139:
8107:
8077:negative real numbers
8073:positive real numbers
8066:
8022:
7990:
7961:
7932:
7903:
7874:field of real numbers
7855:
7760:
7711:
7683:
7620:64-bit representation
7606:, floating-point, or
7558:
7527:
7344:
7319:
7290:
7257:
7228:
7151:
7122:
7061:
7035:
7009:
6920:
6883:
6853:
6803:
6522:In the 17th century,
6511:In the 16th century,
6457:Arabic mathematicians
6369:Indian mathematicians
6337:
6309:
6281:
6253:
6229:
6216:arithmetically random
6186:
6160:
6137:well-ordering theorem
6103:
6077:
6055:
6029:
6001:
5979:
5958:. This is what makes
5953:
5927:
5905:
5833:
5807:
5785:
5643:
5619:
5581:
5550:
5487:
5465:
5422:
5397:; the description in
5336:
5334:{\displaystyle e^{x}}
5298:
5250:
5228:
5180:
5149:mathematical analysis
5147:, and more generally
4958:
4932:
4912:
4866:
4864:{\displaystyle a_{n}}
4839:
4810:
4808:{\displaystyle a_{n}}
4783:
4738:
4736:{\displaystyle <x}
4715:
4681:
4645:
4568:
4498:
4496:{\displaystyle a_{n}}
4471:
4447:has been defined for
4442:
4440:{\displaystyle D_{i}}
4411:
4378:
4376:{\displaystyle D_{0}}
4351:
4316:, as follows. Define
4296:
4259:
4206:
4162:
3979:that are obtained by
3967:
3934:
3902:
3781:
3746:
3711:
3672:
3637:
3474:
3348:univariate polynomial
3339:
3303:
3272:
3218:
3198:
3163:
3115:
3088:
3068:
3042:
3016:
2992:
2967:Dedekind completeness
2957:Dedekind completeness
2925:
2881:
2856:
2830:
2805:
2779:
2754:
2698:
2674:Dedekind completeness
2667:
2638:
2605:
2571:
2540:
2517:
2494:
2474:
2409:
2369:
2298:
2264:
2224:
2153:
2127:Less than or equal to
2119:
2088:
2054:
2003:
1965:
1908:
1866:
1805:
1777:
1734:
1639:
1605:
1574:
1536:
1507:
1481:
1448:
1410:
1382:
1353:
1324:
1298:
1272:
1247:
1210:
1174:
1135:
1105:
1060:
1019:
978:
931:
929:{\displaystyle a+0=a}
872:
800:
750:
673:
671:{\displaystyle ab=ba}
641:
584:
555:
530:
477:
419:elementary arithmetic
296:
186:
11084:Supernatural numbers
10994:Multicomplex numbers
10967:
10951:Dual-complex numbers
10890:
10851:
10813:
10785:
10757:
10729:
10711:Composition algebras
10679:Arithmetical numbers
10642:
10609:
10581:
10553:
10394:Rational zeta series
9633:Robertson, Edmund F.
9322:Robertson, Edmund F.
9289:D'Ambrosio, Ubiratan
9052:
9026:
8912:Algebraic irrational
8765:
8727:
8689:
8651:
8613:
8437:adds only one value
8435:real projective line
8291:
8257:
8221:
8182:
8148:
8116:
8087:
8031:
7999:
7970:
7941:
7912:
7883:
7842:
7720:
7691:
7665:
7540:
7509:
7503:continuum hypothesis
7331:
7299:
7270:
7237:
7208:
7162:Archimedean property
7136:
7078:
7048:
7022:
6996:
6907:
6870:
6840:
6754:
6722:uncountably infinite
6585:(1882), showed that
6541:cannot be rational;
6422:arithmetic operation
6391:Greek mathematicians
6318:
6290:
6262:
6234:
6173:
6147:
6090:
6064:
6042:
6016:
5988:
5966:
5960:nonstandard analysis
5940:
5914:
5892:
5820:
5794:
5772:
5650:continuum hypothesis
5628:
5601:
5563:
5532:
5474:
5452:
5409:
5318:
5247:
5164:
5156:exponential function
4956:{\displaystyle B-1.}
4941:
4921:
4892:
4848:
4819:
4792:
4755:
4724:
4701:
4660:
4577:
4507:
4480:
4451:
4424:
4387:
4360:
4320:
4275:
4001:
3943:
3914:
3811:
3754:
3719:
3684:
3649:
3493:
3398:
3312:
3292:
3242:
3207:
3172:
3146:
3132:Archimedean property
3101:
3077:
3051:
3025:
3005:
2981:
2943:computer programming
2893:
2865:
2861:to the real numbers
2843:
2814:
2792:
2763:
2741:
2680:
2651:
2647:of the real numbers
2625:
2588:
2552:
2526:
2503:
2483:
2460:
2378:
2323:
2278:
2233:
2178:
2170:is not greater than
2133:
2099:
2071:
2034:
1986:
1917:
1906:{\displaystyle |a|,}
1884:
1826:
1786:
1756:
1693:
1652:Auxiliary operations
1615:
1588:
1551:
1516:
1490:
1461:
1419:
1393:
1362:
1333:
1307:
1281:
1255:
1227:
1190:
1144:
1114:
1084:
1028:
1002:
955:
908:
828:
759:
697:
650:
612:
564:
538:
513:
501:of two real numbers
475:{\displaystyle a+b,}
457:
445:of two real numbers
314:are proportional to
173:
100:that can be used to
10989:Split-biquaternions
10701:Eisenstein integers
10664:Closed-form numbers
10487:Unit complex number
10339:Absolute difference
9999:Bottazzini, Umberto
9739:2017FoPh...47.1031B
9699:1986AmSci..74..366W
9631:O'Connor, John J.;
9364:1987NYASA.500..253M
9320:O'Connor, John J.;
8603:
8529:associative algebra
8447:separation relation
8201:
8167:
7796:undecidable problem
7737:
7624:digits of precision
7596:scientific notation
7491:internal set theory
7442:reverse mathematics
7395:classical mechanics
6736:published in 1891.
6715:higher-order logics
6575:(1873) proved that
6480:quadratic equations
6214:or not; and either
5743:Hausdorff dimension
5522:one-to-one function
5364:(given any element
5240:, because the sums
5114:is less than ε for
3939:then by convention
3483:, representing the
2569:{\displaystyle p/q}
2374:or equivalently as
2229:or equivalently as
1803:{\displaystyle a/b}
1270:{\displaystyle a=b}
1017:{\displaystyle -a.}
951:, which means that
904:, which means that
693:, which means that
608:, which means that
528:{\displaystyle ab,}
366:equivalence classes
266:; these are called
253:; these are called
11147:Profinite integers
11110:Irrational numbers
10975:
10898:
10859:
10821:
10793:
10765:
10737:
10694:Gaussian rationals
10674:Computable numbers
10650:
10617:
10589:
10561:
10154:Mac Lane, Saunders
10137:. Academic Press.
10117:Krantz, David H.;
10110:Axiomatic Analysis
9950:2014-05-08 at the
9945:" ("Real numbers")
9686:American Scientist
9566:10.1007/bf01443647
9544:"Transcendenz von
9089:
9038:
8777:
8739:
8701:
8663:
8625:
8599:
8571:Continued fraction
8558:Mathematics portal
8414:adds two elements
8302:
8265:
8236:
8205:
8185:
8168:
8151:
8134:
8102:
8061:
8017:
7985:
7956:
7927:
7898:
7850:
7755:
7723:
7706:
7678:
7661:for them (such as
7636:numerical analysis
7600:data storage space
7553:
7522:
7407:general relativity
7339:
7314:
7285:
7252:
7223:
7146:
7117:
7056:
7030:
7004:
6915:
6878:
6848:
6832:Axiomatic approach
6798:
6740:Formal definitions
6726:countably infinite
6617:The developers of
6565:quadratic equation
6447:numbers, first by
6343:
6332:
6304:
6276:
6248:
6181:
6155:
6126:linear combination
6098:
6072:
6050:
6024:
5996:
5974:
5948:
5934:nonstandard models
5922:
5900:
5828:
5802:
5780:
5697:is defined as the
5638:
5614:
5576:
5545:
5520:, there exists no
5514:{1, 2, 3, 4, ...}
5482:
5460:
5417:
5331:
5293:
5223:
4953:
4927:
4907:
4861:
4834:
4805:
4778:
4733:
4710:
4676:
4640:
4563:
4493:
4466:
4437:
4406:
4373:
4346:
4291:
4254:
4252:
3962:
3929:
3897:
3776:
3741:
3706:
3667:
3632:
3469:
3334:
3298:
3267:
3213:
3193:
3158:
3113:{\displaystyle S.}
3110:
3083:
3063:
3037:
3011:
2987:
2935:abuses of notation
2920:
2876:
2851:
2825:
2800:
2774:
2749:
2724:successor function
2705:irrational numbers
2693:
2662:
2633:
2600:
2566:
2538:{\displaystyle n.}
2535:
2515:{\displaystyle -n}
2512:
2489:
2472:{\displaystyle -n}
2469:
2404:
2364:
2293:
2259:
2219:
2148:
2114:
2083:
2049:
1998:
1960:
1903:
1861:
1800:
1772:
1729:
1634:
1600:
1569:
1531:
1502:
1476:
1443:
1405:
1377:
1358:then one has also
1348:
1319:
1293:
1267:
1242:
1205:
1169:
1130:
1100:
1055:
1014:
990:Every real number
973:
926:
867:
820:Multiplication is
795:
745:
668:
636:
579:
550:
525:
472:
299:
236:irrational numbers
181:
11183:
11182:
11094:Superreal numbers
11074:Levi-Civita field
11069:Hyperreal numbers
11013:Spacetime algebra
10999:Geometric algebra
10912:Bicomplex numbers
10878:Split-quaternions
10719:Division algebras
10689:Gaussian integers
10636:Algebraic numbers
10539:definable numbers
10495:
10494:
10467:Complex conjugate
10422:
10421:
10379:Irrational number
10259:978-3-319-01576-7
10222:. Prentice Hall.
10159:"4. Real Numbers"
10100:978-1-85233-314-0
10073:. Addison-Wesley.
10063:Feferman, Solomon
10057:. Academic Press.
9990:978-1-4612-6521-4
9891:Schumacher, Carol
9808:978-1-56881-158-1
9783:978-3-540-15066-4
9609:978-0-19-921312-2
9504:978-1-4008-6679-3
9471:978-3-540-66572-4
9408:978-1-4020-0260-1
9306:978-1-4020-0260-1
9270:978-0-444-85305-9
9048:is equivalent to
8982:is also required.
8959:
8958:
8955:
8954:
8951:
8950:
8947:
8946:
8936:
8935:
8932:
8931:
8928:
8927:
8924:
8923:
8893:
8892:
8889:
8888:
8885:
8884:
8881:
8880:
8874:Repeating decimal
8841:
8840:
8837:
8836:
8832:Negative integers
8826:
8825:
8822:
8821:
8817:Composite numbers
8533:Positive-definite
8495:hyperreal numbers
8281:Cartesian product
8079:commonly include
7872:, the expression
7804:definable numbers
7673:
7453:hyperreal numbers
7403:quantum mechanics
7157:is not rational.
7144:
7042:least upper bound
6734:diagonal argument
6426:Eudoxus of Cnidus
6414:Eudoxus of Cnidus
6348:were used by the
5886:hyperreal numbers
5878:first-order logic
5863:topological group
5840:real closed field
5434:Archimedean field
5291:
5221:
5127:topological space
4984:(in the sense of
4930:{\displaystyle B}
4773:
4150:
4117:
3977:decimal fractions
3803:infinite sequence
3789:More formally, a
3621:
3594:
3360:real closed field
3301:{\displaystyle r}
3259:
3230:Equivalently, if
3216:{\displaystyle u}
3086:{\displaystyle u}
3014:{\displaystyle u}
2990:{\displaystyle S}
2975:least upper bound
2688:
2584:are integers and
2492:{\displaystyle n}
2422:The real numbers
2384:
2344:
2319:", is defined as
2315:is not less than
2239:
2199:
2174:", is defined as
2067:", is defined as
1837:
1767:
1223:, exactly one of
1125:
902:additive identity
423:binary operations
394:Dedekind complete
359:Dedekind-complete
304:analytic geometry
255:algebraic numbers
232:4 / 3
203:imaginary numbers
129:decimal expansion
104:a continuous one-
86:
85:
78:
11213:
11173:
11172:
11140:
11130:
11042:Cardinal numbers
11003:Clifford algebra
10984:
10982:
10981:
10976:
10974:
10946:Dual quaternions
10907:
10905:
10904:
10899:
10897:
10868:
10866:
10865:
10860:
10858:
10830:
10828:
10827:
10822:
10820:
10802:
10800:
10799:
10794:
10792:
10774:
10772:
10771:
10766:
10764:
10746:
10744:
10743:
10738:
10736:
10659:
10657:
10656:
10651:
10649:
10626:
10624:
10623:
10618:
10616:
10603:Rational numbers
10598:
10596:
10595:
10590:
10588:
10570:
10568:
10567:
10562:
10560:
10522:
10515:
10508:
10499:
10498:
10477:Imaginary number
10449:
10442:
10435:
10426:
10425:
10316:
10309:
10302:
10293:
10292:
10288:
10263:
10233:
10221:
10203:
10191:
10177:
10161:
10148:
10136:
10113:
10104:
10074:
10072:
10058:
10056:
10042:
10034:Crelle's Journal
10020:
10008:
9994:
9955:
9944:
9928:
9917:
9916:
9900:
9887:
9878:
9877:
9876:
9875:
9852:
9846:
9845:
9827:
9818:
9812:
9811:
9794:
9788:
9786:
9765:
9759:
9758:
9732:
9710:
9677:
9671:
9670:
9664:
9656:
9650:
9649:
9635:(October 2005),
9628:
9622:
9621:
9593:
9587:
9584:
9578:
9577:
9560:(2–3): 222–224.
9539:
9533:
9532:
9524:
9518:
9517:
9512:
9511:
9488:
9482:
9480:
9479:
9478:
9455:
9449:
9448:
9432:
9428:
9418:
9412:
9411:
9389:
9383:
9382:
9345:
9339:
9338:
9317:
9311:
9309:
9280:
9274:
9273:
9262:
9254:
9243:
9237:
9236:
9229:Zalta, Edward N.
9217:
9211:
9210:
9199:
9190:
9189:
9183:
9175:
9169:
9168:
9147:
9140:
9134:
9133:
9130:Oxford Reference
9122:
9100:
9098:
9096:
9095:
9090:
9085:
9084:
9047:
9045:
9044:
9039:
9015:
9009:
9006:general topology
9002:
8996:
8989:
8983:
8978:; a property of
8976:rational numbers
8972:
8908:
8907:
8899:
8898:
8856:
8855:
8847:
8846:
8790:
8789:
8786:
8784:
8783:
8778:
8776:
8756:
8755:
8752:
8751:
8748:
8746:
8745:
8740:
8738:
8718:
8717:
8714:
8713:
8710:
8708:
8707:
8702:
8700:
8680:
8679:
8676:
8675:
8672:
8670:
8669:
8664:
8662:
8642:
8641:
8638:
8637:
8634:
8632:
8631:
8626:
8624:
8604:
8598:
8595:
8594:
8591:
8590:
8560:
8555:
8554:
8537:normal operators
8488:
8481:
8474:
8466:
8456:pastes together
8440:
8428:complete lattice
8421:
8417:
8365:). For example,
8357:In mathematics,
8334:
8330:
8325:coordinate space
8315:
8311:
8309:
8308:
8303:
8298:
8286:
8274:
8272:
8271:
8266:
8264:
8250:
8245:
8243:
8242:
8237:
8235:
8234:
8229:
8214:
8212:
8211:
8206:
8200:
8195:
8190:
8177:
8175:
8174:
8169:
8166:
8161:
8156:
8143:
8141:
8140:
8135:
8130:
8129:
8124:
8111:
8109:
8108:
8103:
8101:
8100:
8099:
8070:
8068:
8067:
8062:
8045:
8044:
8039:
8026:
8024:
8023:
8018:
8016:
8015:
8007:
7994:
7992:
7991:
7986:
7984:
7983:
7978:
7965:
7963:
7962:
7957:
7955:
7954:
7949:
7937:, respectively;
7936:
7934:
7933:
7928:
7926:
7925:
7920:
7907:
7905:
7904:
7899:
7897:
7896:
7891:
7859:
7857:
7856:
7851:
7849:
7767:constant problem
7764:
7762:
7761:
7756:
7747:
7746:
7736:
7731:
7715:
7713:
7712:
7707:
7687:
7685:
7684:
7679:
7674:
7669:
7562:
7560:
7559:
7554:
7552:
7551:
7531:
7529:
7528:
7523:
7521:
7520:
7495:Zermelo–Fraenkel
7461:Abraham Robinson
7455:as developed by
7438:Zermelo–Fraenkel
7415:smooth manifolds
7399:electromagnetism
7372:
7348:
7346:
7345:
7340:
7338:
7323:
7321:
7320:
7315:
7313:
7312:
7311:
7294:
7292:
7291:
7286:
7284:
7283:
7278:
7261:
7259:
7258:
7253:
7251:
7250:
7245:
7232:
7230:
7229:
7224:
7222:
7221:
7216:
7196:
7186:
7155:
7153:
7152:
7147:
7145:
7140:
7126:
7124:
7123:
7118:
7107:
7106:
7094:
7074:). For example,
7065:
7063:
7062:
7057:
7055:
7039:
7037:
7036:
7031:
7029:
7013:
7011:
7010:
7005:
7003:
6924:
6922:
6921:
6916:
6914:
6887:
6885:
6884:
6879:
6877:
6857:
6855:
6854:
6849:
6847:
6818:Cauchy sequences
6807:
6805:
6804:
6799:
6794:
6789:
6784:
6779:
6774:
6769:
6764:
6683:rational numbers
6655:Richard Dedekind
6644:Bernhard Riemann
6588:
6579:
6562:
6556:
6548:
6540:
6477:
6475:
6443:, integers, and
6441:negative numbers
6401:is irrational.
6399:square root of 2
6381:
6379:
6366:
6364:
6346:Simple fractions
6341:
6339:
6338:
6333:
6328:
6313:
6311:
6310:
6305:
6300:
6285:
6283:
6282:
6277:
6272:
6257:
6255:
6254:
6249:
6244:
6190:
6188:
6187:
6182:
6180:
6164:
6162:
6161:
6156:
6154:
6107:
6105:
6104:
6099:
6097:
6081:
6079:
6078:
6073:
6071:
6059:
6057:
6056:
6051:
6049:
6033:
6031:
6030:
6025:
6023:
6005:
6003:
6002:
5997:
5995:
5983:
5981:
5980:
5975:
5973:
5957:
5955:
5954:
5949:
5947:
5931:
5929:
5928:
5923:
5921:
5909:
5907:
5906:
5901:
5899:
5855:Lebesgue measure
5837:
5835:
5834:
5829:
5827:
5811:
5809:
5808:
5803:
5801:
5789:
5787:
5786:
5781:
5779:
5757:are necessarily
5755:order topologies
5741:metric space of
5731:simply connected
5711:
5664:Other properties
5648:is known as the
5647:
5645:
5644:
5639:
5637:
5636:
5623:
5621:
5620:
5615:
5613:
5612:
5585:
5583:
5582:
5577:
5575:
5574:
5554:
5552:
5551:
5546:
5541:
5540:
5515:
5491:
5489:
5488:
5483:
5481:
5469:
5467:
5466:
5461:
5459:
5426:
5424:
5423:
5418:
5416:
5374:
5358:lattice-complete
5340:
5338:
5337:
5332:
5330:
5329:
5302:
5300:
5299:
5294:
5292:
5290:
5282:
5281:
5272:
5269:
5264:
5232:
5230:
5229:
5224:
5222:
5220:
5212:
5211:
5202:
5199:
5194:
5176:
5175:
5113:
5094:
5042:
5017:
4962:
4960:
4959:
4954:
4936:
4934:
4933:
4928:
4916:
4914:
4913:
4908:
4870:
4868:
4867:
4862:
4860:
4859:
4843:
4841:
4840:
4835:
4814:
4812:
4811:
4806:
4804:
4803:
4787:
4785:
4784:
4779:
4774:
4772:
4771:
4759:
4749:decimal fraction
4746:
4742:
4740:
4739:
4734:
4719:
4717:
4716:
4711:
4693:
4685:
4683:
4682:
4677:
4672:
4671:
4655:
4649:
4647:
4646:
4641:
4636:
4635:
4626:
4621:
4620:
4608:
4607:
4589:
4588:
4572:
4570:
4569:
4564:
4553:
4552:
4543:
4538:
4537:
4525:
4524:
4502:
4500:
4499:
4494:
4492:
4491:
4475:
4473:
4472:
4467:
4446:
4444:
4443:
4438:
4436:
4435:
4415:
4413:
4412:
4407:
4399:
4398:
4382:
4380:
4379:
4374:
4372:
4371:
4355:
4353:
4352:
4347:
4345:
4344:
4332:
4331:
4311:
4307:
4300:
4298:
4297:
4292:
4287:
4286:
4270:
4267:The real number
4263:
4261:
4260:
4255:
4253:
4249:
4248:
4236:
4235:
4225:
4220:
4202:
4201:
4192:
4191:
4181:
4176:
4155:
4151:
4149:
4148:
4139:
4138:
4129:
4118:
4113:
4112:
4103:
4098:
4097:
4079:
4078:
4063:
4062:
4044:
4043:
4034:
4033:
4017:
4016:
3990:
3986:
3971:
3969:
3968:
3963:
3955:
3954:
3938:
3936:
3935:
3930:
3906:
3904:
3903:
3898:
3887:
3886:
3874:
3873:
3861:
3860:
3842:
3841:
3823:
3822:
3800:
3796:
3785:
3783:
3782:
3777:
3766:
3765:
3750:
3748:
3747:
3742:
3731:
3730:
3715:
3713:
3712:
3707:
3696:
3695:
3679:
3676:
3674:
3673:
3668:
3641:
3639:
3638:
3633:
3622:
3620:
3619:
3610:
3609:
3600:
3595:
3590:
3589:
3580:
3575:
3574:
3556:
3555:
3540:
3539:
3521:
3520:
3511:
3510:
3478:
3476:
3475:
3470:
3462:
3461:
3452:
3451:
3439:
3438:
3426:
3425:
3410:
3409:
3393:
3389:
3378:, a sequence of
3343:
3341:
3340:
3335:
3324:
3323:
3307:
3305:
3304:
3299:
3283:
3276:
3274:
3273:
3268:
3260:
3252:
3237:
3233:
3226:
3222:
3220:
3219:
3214:
3202:
3200:
3199:
3194:
3167:
3165:
3164:
3159:
3141:
3137:
3123:
3119:
3117:
3116:
3111:
3092:
3090:
3089:
3084:
3072:
3070:
3069:
3064:
3046:
3044:
3043:
3038:
3020:
3018:
3017:
3012:
2996:
2994:
2993:
2988:
2963:rational numbers
2947:type conversions
2929:
2927:
2926:
2921:
2916:
2908:
2900:
2885:
2883:
2882:
2877:
2872:
2860:
2858:
2857:
2852:
2850:
2834:
2832:
2831:
2826:
2821:
2809:
2807:
2806:
2801:
2799:
2783:
2781:
2780:
2775:
2770:
2759:to the integers
2758:
2756:
2755:
2750:
2748:
2721:
2702:
2700:
2699:
2694:
2689:
2684:
2671:
2669:
2668:
2663:
2658:
2642:
2640:
2639:
2634:
2632:
2617:
2613:
2609:
2607:
2606:
2601:
2583:
2579:
2575:
2573:
2572:
2567:
2562:
2544:
2542:
2541:
2536:
2521:
2519:
2518:
2513:
2498:
2496:
2495:
2490:
2478:
2476:
2475:
2470:
2452:
2448:
2445:with the sum of
2444:
2440:
2436:
2429:
2425:
2413:
2411:
2410:
2405:
2385:
2382:
2373:
2371:
2370:
2365:
2345:
2342:
2318:
2314:
2310:
2306:
2302:
2300:
2299:
2294:
2268:
2266:
2265:
2260:
2240:
2237:
2228:
2226:
2225:
2220:
2200:
2197:
2173:
2169:
2165:
2161:
2157:
2155:
2154:
2149:
2123:
2121:
2120:
2115:
2092:
2090:
2089:
2084:
2066:
2063:is greater than
2062:
2058:
2056:
2055:
2050:
2018:
2011:
2007:
2005:
2004:
1999:
1969:
1967:
1966:
1961:
1932:
1924:
1912:
1910:
1909:
1904:
1899:
1891:
1879:
1870:
1868:
1867:
1862:
1857:
1856:
1838:
1830:
1821:
1813:
1809:
1807:
1806:
1801:
1796:
1781:
1779:
1778:
1773:
1768:
1760:
1751:
1747:
1738:
1736:
1735:
1730:
1688:
1684:
1677:additive inverse
1674:
1670:
1666:
1647:
1643:
1641:
1640:
1635:
1633:
1632:
1609:
1607:
1606:
1601:
1582:
1578:
1576:
1575:
1570:
1540:
1538:
1537:
1532:
1511:
1509:
1508:
1503:
1485:
1483:
1482:
1477:
1456:
1452:
1450:
1449:
1444:
1414:
1412:
1411:
1406:
1386:
1384:
1383:
1378:
1357:
1355:
1354:
1349:
1328:
1326:
1325:
1320:
1303:is true; and if
1302:
1300:
1299:
1294:
1276:
1274:
1273:
1268:
1251:
1249:
1248:
1243:
1222:
1218:
1214:
1212:
1211:
1206:
1182:
1178:
1176:
1175:
1170:
1162:
1161:
1140:This means that
1139:
1137:
1136:
1131:
1126:
1118:
1109:
1107:
1106:
1101:
1099:
1098:
1075:
1068:
1064:
1062:
1061:
1056:
1024:This means that
1023:
1021:
1020:
1015:
996:additive inverse
993:
986:
982:
980:
979:
974:
946:
939:
935:
933:
932:
927:
899:
888:
884:
880:
876:
874:
873:
868:
816:
812:
808:
804:
802:
801:
796:
754:
752:
751:
746:
685:
681:
677:
675:
674:
669:
645:
643:
642:
637:
600:
596:
588:
586:
585:
580:
559:
557:
556:
551:
534:
532:
531:
526:
508:
504:
493:
489:
481:
479:
478:
473:
452:
448:
370:Cauchy sequences
337:and real-valued
288:
265:
262:
252:
233:
226:
218:rational numbers
212:
193:. The adjective
192:
190:
188:
187:
182:
180:
162:
81:
74:
70:
67:
61:
56:this article by
47:inline citations
34:
33:
26:
11221:
11220:
11216:
11215:
11214:
11212:
11211:
11210:
11186:
11185:
11184:
11179:
11156:
11135:
11125:
11098:
11089:Surreal numbers
11079:Ordinal numbers
11024:
10970:
10968:
10965:
10964:
10926:
10893:
10891:
10888:
10887:
10885:
10883:Split-octonions
10854:
10852:
10849:
10848:
10840:
10834:
10816:
10814:
10811:
10810:
10788:
10786:
10783:
10782:
10760:
10758:
10755:
10754:
10751:Complex numbers
10732:
10730:
10727:
10726:
10705:
10645:
10643:
10640:
10639:
10612:
10610:
10607:
10606:
10584:
10582:
10579:
10578:
10556:
10554:
10551:
10550:
10547:Natural numbers
10532:
10526:
10496:
10491:
10458:
10456:Complex numbers
10453:
10423:
10418:
10389:Rational number
10325:
10320:
10273:
10270:
10260:
10238:Stillwell, John
10230:
10200:
10174:
10145:
10123:Suppes, Patrick
10119:Luce, R. Duncan
10101:
10047:Dieudonné, Jean
10017:
9991:
9964:
9959:
9958:
9952:Wayback Machine
9929:
9920:
9913:
9888:
9881:
9873:
9871:
9869:
9853:
9849:
9825:
9819:
9815:
9809:
9795:
9791:
9784:
9774:Springer-Verlag
9766:
9762:
9711:
9678:
9674:
9662:
9658:
9657:
9653:
9629:
9625:
9610:
9594:
9590:
9585:
9581:
9540:
9536:
9525:
9521:
9509:
9507:
9505:
9489:
9485:
9476:
9474:
9472:
9456:
9452:
9445:
9426:
9419:
9415:
9409:
9390:
9386:
9346:
9342:
9318:
9314:
9307:
9291:, eds. (2000),
9281:
9277:
9271:
9244:
9240:
9221:Koellner, Peter
9218:
9214:
9201:
9200:
9193:
9176:
9172:
9142:
9141:
9137:
9124:
9123:
9119:
9114:
9109:
9104:
9103:
9080:
9076:
9053:
9050:
9049:
9027:
9024:
9023:
9016:
9012:
9003:
8999:
8990:
8986:
8973:
8969:
8964:
8772:
8766:
8763:
8762:
8734:
8728:
8725:
8724:
8696:
8690:
8687:
8686:
8658:
8652:
8649:
8648:
8620:
8614:
8611:
8610:
8556:
8549:
8546:
8499:surreal numbers
8487:
8483:
8480:
8476:
8472:
8468:
8465:
8461:
8457:
8445:described by a
8438:
8419:
8415:
8401:complex numbers
8393:
8374:real polynomial
8340:Euclidean space
8332:
8328:
8313:
8294:
8292:
8289:
8288:
8284:
8260:
8258:
8255:
8254:
8253:of elements of
8248:
8230:
8225:
8224:
8222:
8219:
8218:
8196:
8191:
8186:
8183:
8180:
8179:
8162:
8157:
8152:
8149:
8146:
8145:
8125:
8120:
8119:
8117:
8114:
8113:
8095:
8091:
8090:
8088:
8085:
8084:
8040:
8035:
8034:
8032:
8029:
8028:
8008:
8003:
8002:
8000:
7997:
7996:
7979:
7974:
7973:
7971:
7968:
7967:
7950:
7945:
7944:
7942:
7939:
7938:
7921:
7916:
7915:
7913:
7910:
7909:
7892:
7887:
7886:
7884:
7881:
7880:
7862:blackboard bold
7845:
7843:
7840:
7839:
7832:
7818:, specifically
7812:
7800:constructivists
7742:
7738:
7732:
7727:
7721:
7718:
7717:
7692:
7689:
7688:
7668:
7666:
7663:
7662:
7634:. The field of
7574:
7547:
7543:
7541:
7538:
7537:
7534:cardinal number
7516:
7512:
7510:
7507:
7506:
7434:
7390:
7385:
7370:
7359:
7334:
7332:
7329:
7328:
7307:
7303:
7302:
7300:
7297:
7296:
7279:
7274:
7273:
7271:
7268:
7267:
7246:
7241:
7240:
7238:
7235:
7234:
7217:
7212:
7211:
7209:
7206:
7205:
7188:
7177:
7139:
7137:
7134:
7133:
7102:
7098:
7090:
7079:
7076:
7075:
7051:
7049:
7046:
7045:
7025:
7023:
7020:
7019:
6999:
6997:
6994:
6993:
6910:
6908:
6905:
6904:
6892:, meaning that
6873:
6871:
6868:
6867:
6843:
6841:
6838:
6837:
6834:
6808:can be defined
6793:
6788:
6783:
6778:
6773:
6768:
6760:
6755:
6752:
6751:
6748:
6742:
6687:natural numbers
6653:Beginning with
6628:Cours d'Analyse
6615:
6613:Modern analysis
6586:
6577:
6558:
6554:
6546:
6538:
6471:
6410:rational number
6406:natural numbers
6375:
6361:
6324:
6319:
6316:
6315:
6296:
6291:
6288:
6287:
6268:
6263:
6260:
6259:
6240:
6235:
6232:
6231:
6224:
6176:
6174:
6171:
6170:
6150:
6148:
6145:
6144:
6114:axiom of choice
6093:
6091:
6088:
6087:
6067:
6065:
6062:
6061:
6045:
6043:
6040:
6039:
6036:extension field
6019:
6017:
6014:
6013:
5991:
5989:
5986:
5985:
5969:
5967:
5964:
5963:
5943:
5941:
5938:
5937:
5917:
5915:
5912:
5911:
5895:
5893:
5890:
5889:
5857:, which is the
5823:
5821:
5818:
5817:
5797:
5795:
5792:
5791:
5775:
5773:
5770:
5769:
5747:locally compact
5701:
5672:
5666:
5658:axiom of choice
5632:
5631:
5629:
5626:
5625:
5608:
5604:
5602:
5599:
5598:
5570:
5566:
5564:
5561:
5560:
5555:and called the
5536:
5535:
5533:
5530:
5529:
5513:
5511:natural numbers
5503:
5495:surreal numbers
5477:
5475:
5472:
5471:
5455:
5453:
5450:
5449:
5412:
5410:
5407:
5406:
5369:
5362:largest element
5351:
5325:
5321:
5319:
5316:
5315:
5283:
5277:
5273:
5271:
5265:
5254:
5248:
5245:
5244:
5213:
5207:
5203:
5201:
5195:
5184:
5171:
5167:
5165:
5162:
5161:
5106:
5100:
5092:
5079:
5067:
5039:
5032:
5026:
5015:
5011:Cauchy sequence
5007:
4974:
4968:
4942:
4939:
4938:
4922:
4919:
4918:
4893:
4890:
4889:
4855:
4851:
4849:
4846:
4845:
4820:
4817:
4816:
4799:
4795:
4793:
4790:
4789:
4767:
4763:
4758:
4756:
4753:
4752:
4744:
4725:
4722:
4721:
4702:
4699:
4698:
4691:
4667:
4663:
4661:
4658:
4657:
4653:
4631:
4627:
4622:
4616:
4612:
4597:
4593:
4584:
4580:
4578:
4575:
4574:
4548:
4544:
4539:
4533:
4529:
4514:
4510:
4508:
4505:
4504:
4487:
4483:
4481:
4478:
4477:
4452:
4449:
4448:
4431:
4427:
4425:
4422:
4421:
4394:
4390:
4388:
4385:
4384:
4367:
4363:
4361:
4358:
4357:
4340:
4336:
4327:
4323:
4321:
4318:
4317:
4309:
4305:
4282:
4278:
4276:
4273:
4272:
4268:
4251:
4250:
4241:
4237:
4231:
4227:
4221:
4210:
4197:
4193:
4187:
4183:
4177:
4166:
4153:
4152:
4144:
4140:
4134:
4130:
4128:
4108:
4104:
4102:
4093:
4089:
4068:
4064:
4052:
4048:
4039:
4035:
4029:
4025:
4018:
4012:
4008:
4004:
4002:
3999:
3998:
3988:
3984:
3950:
3946:
3944:
3941:
3940:
3915:
3912:
3911:
3882:
3878:
3869:
3865:
3856:
3852:
3831:
3827:
3818:
3814:
3812:
3809:
3808:
3798:
3794:
3761:
3757:
3755:
3752:
3751:
3726:
3722:
3720:
3717:
3716:
3691:
3687:
3685:
3682:
3681:
3677:
3650:
3647:
3646:
3615:
3611:
3605:
3601:
3599:
3585:
3581:
3579:
3570:
3566:
3545:
3541:
3529:
3525:
3516:
3512:
3506:
3502:
3494:
3491:
3490:
3485:infinite series
3457:
3453:
3447:
3443:
3434:
3430:
3415:
3411:
3405:
3401:
3399:
3396:
3395:
3391:
3387:
3372:
3319:
3315:
3313:
3310:
3309:
3293:
3290:
3289:
3284:has a positive
3281:
3251:
3243:
3240:
3239:
3235:
3231:
3224:
3208:
3205:
3204:
3173:
3170:
3169:
3147:
3144:
3143:
3139:
3135:
3121:
3102:
3099:
3098:
3078:
3075:
3074:
3052:
3049:
3048:
3026:
3023:
3022:
3006:
3003:
3002:
2982:
2979:
2978:
2959:
2912:
2904:
2896:
2894:
2891:
2890:
2868:
2866:
2863:
2862:
2846:
2844:
2841:
2840:
2817:
2815:
2812:
2811:
2795:
2793:
2790:
2789:
2766:
2764:
2761:
2760:
2744:
2742:
2739:
2738:
2735:ordered monoids
2719:
2683:
2681:
2678:
2677:
2654:
2652:
2649:
2648:
2628:
2626:
2623:
2622:
2615:
2611:
2589:
2586:
2585:
2581:
2577:
2558:
2553:
2550:
2549:
2547:rational number
2527:
2524:
2523:
2504:
2501:
2500:
2484:
2481:
2480:
2461:
2458:
2457:
2450:
2446:
2442:
2438:
2434:
2432:natural numbers
2427:
2423:
2420:
2381:
2379:
2376:
2375:
2341:
2324:
2321:
2320:
2316:
2312:
2308:
2304:
2279:
2276:
2275:
2236:
2234:
2231:
2230:
2196:
2179:
2176:
2175:
2171:
2167:
2163:
2159:
2134:
2131:
2130:
2100:
2097:
2096:
2072:
2069:
2068:
2064:
2060:
2035:
2032:
2031:
2021:order relations
2019:". Three other
2016:
2009:
1987:
1984:
1983:
1976:
1928:
1920:
1918:
1915:
1914:
1895:
1887:
1885:
1882:
1881:
1877:
1849:
1845:
1829:
1827:
1824:
1823:
1819:
1811:
1792:
1787:
1784:
1783:
1759:
1757:
1754:
1753:
1749:
1745:
1694:
1691:
1690:
1686:
1679:
1672:
1668:
1664:
1654:
1645:
1628:
1624:
1616:
1613:
1612:
1589:
1586:
1585:
1580:
1552:
1549:
1548:
1517:
1514:
1513:
1491:
1488:
1487:
1462:
1459:
1458:
1454:
1420:
1417:
1416:
1394:
1391:
1390:
1363:
1360:
1359:
1334:
1331:
1330:
1308:
1305:
1304:
1282:
1279:
1278:
1256:
1253:
1252:
1228:
1225:
1224:
1220:
1216:
1191:
1188:
1187:
1180:
1154:
1150:
1145:
1142:
1141:
1117:
1115:
1112:
1111:
1091:
1087:
1085:
1082:
1081:
1073:
1066:
1029:
1026:
1025:
1003:
1000:
999:
991:
984:
956:
953:
952:
944:
937:
909:
906:
905:
897:
886:
882:
878:
829:
826:
825:
814:
810:
806:
760:
757:
756:
698:
695:
694:
683:
679:
651:
648:
647:
613:
610:
609:
598:
594:
565:
562:
561:
539:
536:
535:
514:
511:
510:
506:
502:
491:
487:
458:
455:
454:
450:
446:
411:
386:
376:, and infinite
333:, the study of
300:
286:
260:
258:
247:
231:
224:
210:
176:
174:
171:
170:
168:
165:blackboard bold
158:
82:
71:
65:
62:
52:Please help to
51:
35:
31:
24:
17:
12:
11:
5:
11219:
11209:
11208:
11203:
11198:
11181:
11180:
11178:
11177:
11167:
11165:Classification
11161:
11158:
11157:
11155:
11154:
11152:Normal numbers
11149:
11144:
11122:
11117:
11112:
11106:
11104:
11100:
11099:
11097:
11096:
11091:
11086:
11081:
11076:
11071:
11066:
11061:
11060:
11059:
11049:
11044:
11038:
11036:
11034:infinitesimals
11026:
11025:
11023:
11022:
11021:
11020:
11015:
11010:
10996:
10991:
10986:
10973:
10958:
10953:
10948:
10943:
10937:
10935:
10928:
10927:
10925:
10924:
10919:
10914:
10909:
10896:
10880:
10875:
10870:
10857:
10844:
10842:
10836:
10835:
10833:
10832:
10819:
10804:
10791:
10776:
10763:
10748:
10735:
10715:
10713:
10707:
10706:
10704:
10703:
10698:
10697:
10696:
10686:
10681:
10676:
10671:
10666:
10661:
10648:
10633:
10628:
10615:
10600:
10587:
10572:
10559:
10543:
10541:
10534:
10533:
10525:
10524:
10517:
10510:
10502:
10493:
10492:
10490:
10489:
10484:
10479:
10474:
10469:
10463:
10460:
10459:
10452:
10451:
10444:
10437:
10429:
10420:
10419:
10417:
10416:
10411:
10406:
10401:
10396:
10391:
10386:
10381:
10376:
10374:Gregory number
10371:
10366:
10361:
10356:
10351:
10346:
10341:
10336:
10330:
10327:
10326:
10319:
10318:
10311:
10304:
10296:
10290:
10289:
10269:
10268:External links
10266:
10265:
10264:
10258:
10234:
10228:
10211:
10198:
10182:Landau, Edmund
10178:
10172:
10150:
10143:
10114:
10105:
10099:
10079:Howie, John M.
10075:
10059:
10043:
10021:
10015:
9995:
9989:
9969:Bos, Henk J.M.
9963:
9960:
9957:
9956:
9918:
9911:
9879:
9868:97-80763772062
9867:
9847:
9813:
9807:
9789:
9782:
9760:
9723:(8): 1031–41.
9672:
9651:
9623:
9608:
9588:
9579:
9534:
9519:
9503:
9483:
9470:
9450:
9443:
9413:
9407:
9384:
9358:(1): 253–77 ,
9340:
9312:
9305:
9285:Selin, Helaine
9275:
9269:
9238:
9212:
9191:
9170:
9135:
9116:
9115:
9113:
9110:
9108:
9105:
9102:
9101:
9088:
9083:
9079:
9075:
9072:
9069:
9066:
9063:
9060:
9057:
9037:
9034:
9031:
9010:
8997:
8984:
8966:
8965:
8963:
8960:
8957:
8956:
8953:
8952:
8949:
8948:
8945:
8944:
8938:
8937:
8934:
8933:
8930:
8929:
8926:
8925:
8922:
8921:
8919:Transcendental
8915:
8914:
8905:
8895:
8894:
8891:
8890:
8887:
8886:
8883:
8882:
8879:
8878:
8876:
8870:
8869:
8863:
8862:
8860:Finite decimal
8853:
8843:
8842:
8839:
8838:
8835:
8834:
8828:
8827:
8824:
8823:
8820:
8819:
8813:
8812:
8806:
8805:
8798:
8797:
8787:
8775:
8770:
8749:
8737:
8732:
8711:
8699:
8694:
8673:
8661:
8656:
8635:
8623:
8618:
8601:Number systems
8589:
8588:
8583:
8578:
8573:
8568:
8562:
8561:
8545:
8542:
8541:
8540:
8510:
8491:
8485:
8478:
8470:
8463:
8459:
8454:long real line
8450:
8431:
8408:
8392:
8389:
8363:the real field
8301:
8297:
8263:
8233:
8228:
8204:
8199:
8194:
8189:
8165:
8160:
8155:
8133:
8128:
8123:
8098:
8094:
8060:
8057:
8054:
8051:
8048:
8043:
8038:
8014:
8011:
8006:
7982:
7977:
7953:
7948:
7924:
7919:
7895:
7890:
7848:
7831:
7828:
7811:
7808:
7754:
7751:
7745:
7741:
7735:
7730:
7726:
7705:
7702:
7699:
7696:
7677:
7672:
7573:
7570:
7550:
7546:
7519:
7515:
7433:
7430:
7419:Hilbert spaces
7411:standard model
7389:
7386:
7384:
7381:
7358:
7355:
7337:
7310:
7306:
7282:
7277:
7249:
7244:
7220:
7215:
7143:
7116:
7113:
7110:
7105:
7101:
7097:
7093:
7089:
7086:
7083:
7068:
7067:
7054:
7028:
7002:
6986:
6985:
6984:
6969:
6913:
6901:
6898:multiplication
6876:
6846:
6833:
6830:
6797:
6792:
6787:
6782:
6777:
6772:
6767:
6763:
6759:
6744:Main article:
6741:
6738:
6703:quantification
6659:Hermann Hankel
6648:Fourier series
6614:
6611:
6455:, and then by
6331:
6327:
6323:
6303:
6299:
6295:
6275:
6271:
6267:
6247:
6243:
6239:
6223:
6220:
6179:
6153:
6096:
6070:
6048:
6022:
5994:
5972:
5946:
5920:
5898:
5826:
5800:
5778:
5761:to the reals.
5714:order topology
5699:absolute value
5665:
5662:
5656:including the
5635:
5611:
5607:
5588:'aleph-naught'
5573:
5569:
5544:
5539:
5502:
5499:
5480:
5458:
5415:
5399:§ Completeness
5350:
5347:
5328:
5324:
5310:) by choosing
5304:
5303:
5289:
5286:
5280:
5276:
5268:
5263:
5260:
5257:
5253:
5234:
5233:
5219:
5216:
5210:
5206:
5198:
5193:
5190:
5187:
5183:
5179:
5174:
5170:
5104:
5075:
5063:
5037:
5030:
5003:
4990:uniform spaces
4970:Main article:
4967:
4964:
4952:
4949:
4946:
4926:
4906:
4903:
4900:
4897:
4875:for details).
4858:
4854:
4833:
4830:
4827:
4824:
4802:
4798:
4777:
4770:
4766:
4762:
4732:
4729:
4709:
4706:
4675:
4670:
4666:
4639:
4634:
4630:
4625:
4619:
4615:
4611:
4606:
4603:
4600:
4596:
4592:
4587:
4583:
4562:
4559:
4556:
4551:
4547:
4542:
4536:
4532:
4528:
4523:
4520:
4517:
4513:
4490:
4486:
4465:
4462:
4459:
4456:
4434:
4430:
4405:
4402:
4397:
4393:
4370:
4366:
4343:
4339:
4335:
4330:
4326:
4290:
4285:
4281:
4265:
4264:
4247:
4244:
4240:
4234:
4230:
4224:
4219:
4216:
4213:
4209:
4205:
4200:
4196:
4190:
4186:
4180:
4175:
4172:
4169:
4165:
4161:
4158:
4156:
4154:
4147:
4143:
4137:
4133:
4127:
4124:
4121:
4116:
4111:
4107:
4101:
4096:
4092:
4088:
4085:
4082:
4077:
4074:
4071:
4067:
4061:
4058:
4055:
4051:
4047:
4042:
4038:
4032:
4028:
4024:
4021:
4019:
4015:
4011:
4007:
4006:
3991:is the finite
3961:
3958:
3953:
3949:
3928:
3925:
3922:
3919:
3908:
3907:
3896:
3893:
3890:
3885:
3881:
3877:
3872:
3868:
3864:
3859:
3855:
3851:
3848:
3845:
3840:
3837:
3834:
3830:
3826:
3821:
3817:
3775:
3772:
3769:
3764:
3760:
3740:
3737:
3734:
3729:
3725:
3705:
3702:
3699:
3694:
3690:
3666:
3663:
3660:
3657:
3654:
3643:
3642:
3631:
3628:
3625:
3618:
3614:
3608:
3604:
3598:
3593:
3588:
3584:
3578:
3573:
3569:
3565:
3562:
3559:
3554:
3551:
3548:
3544:
3538:
3535:
3532:
3528:
3524:
3519:
3515:
3509:
3505:
3501:
3498:
3468:
3465:
3460:
3456:
3450:
3446:
3442:
3437:
3433:
3429:
3424:
3421:
3418:
3414:
3408:
3404:
3380:decimal digits
3371:
3368:
3356:
3355:
3344:
3333:
3330:
3327:
3322:
3318:
3297:
3278:
3266:
3263:
3258:
3255:
3250:
3247:
3228:
3212:
3192:
3189:
3186:
3183:
3180:
3177:
3157:
3154:
3151:
3109:
3106:
3082:
3062:
3059:
3056:
3036:
3033:
3030:
3010:
2986:
2958:
2955:
2931:
2930:
2919:
2915:
2911:
2907:
2903:
2899:
2875:
2871:
2849:
2837:ordered fields
2824:
2820:
2798:
2773:
2769:
2747:
2692:
2687:
2661:
2657:
2631:
2599:
2596:
2593:
2565:
2561:
2557:
2534:
2531:
2511:
2508:
2488:
2468:
2465:
2419:
2416:
2415:
2414:
2403:
2400:
2397:
2394:
2391:
2388:
2363:
2360:
2357:
2354:
2351:
2348:
2343: or
2340:
2337:
2334:
2331:
2328:
2292:
2289:
2286:
2283:
2269:
2258:
2255:
2252:
2249:
2246:
2243:
2218:
2215:
2212:
2209:
2206:
2203:
2198: or
2195:
2192:
2189:
2186:
2183:
2147:
2144:
2141:
2138:
2124:
2113:
2110:
2107:
2104:
2094:if and only if
2082:
2079:
2076:
2048:
2045:
2042:
2039:
1997:
1994:
1991:
1975:
1972:
1971:
1970:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1931:
1927:
1923:
1902:
1898:
1894:
1890:
1874:Absolute value
1871:
1860:
1855:
1852:
1848:
1844:
1841:
1836:
1833:
1799:
1795:
1791:
1771:
1766:
1763:
1739:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1653:
1650:
1649:
1648:
1631:
1627:
1623:
1620:
1610:
1599:
1596:
1593:
1583:
1568:
1565:
1562:
1559:
1556:
1542:
1541:
1530:
1527:
1524:
1521:
1501:
1498:
1495:
1486:is implied by
1475:
1472:
1469:
1466:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1404:
1401:
1398:
1387:
1376:
1373:
1370:
1367:
1347:
1344:
1341:
1338:
1318:
1315:
1312:
1292:
1289:
1286:
1266:
1263:
1260:
1241:
1238:
1235:
1232:
1204:
1201:
1198:
1195:
1184:
1168:
1165:
1160:
1157:
1153:
1149:
1129:
1124:
1121:
1097:
1094:
1090:
1070:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1013:
1010:
1007:
988:
972:
969:
966:
963:
960:
941:
925:
922:
919:
916:
913:
890:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
818:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
708:
705:
702:
687:
667:
664:
661:
658:
655:
635:
632:
629:
626:
623:
620:
617:
602:
578:
575:
572:
569:
549:
546:
543:
524:
521:
518:
499:multiplication
495:
471:
468:
465:
462:
431:multiplication
410:
407:
385:
382:
335:real functions
291:
220:, such as the
199:René Descartes
179:
163:, often using
84:
83:
38:
36:
29:
15:
9:
6:
4:
3:
2:
11218:
11207:
11204:
11202:
11199:
11197:
11194:
11193:
11191:
11176:
11168:
11166:
11163:
11162:
11159:
11153:
11150:
11148:
11145:
11142:
11138:
11132:
11128:
11123:
11121:
11118:
11116:
11115:Fuzzy numbers
11113:
11111:
11108:
11107:
11105:
11101:
11095:
11092:
11090:
11087:
11085:
11082:
11080:
11077:
11075:
11072:
11070:
11067:
11065:
11062:
11058:
11055:
11054:
11053:
11050:
11048:
11045:
11043:
11040:
11039:
11037:
11035:
11031:
11027:
11019:
11016:
11014:
11011:
11009:
11006:
11005:
11004:
11000:
10997:
10995:
10992:
10990:
10987:
10962:
10959:
10957:
10954:
10952:
10949:
10947:
10944:
10942:
10939:
10938:
10936:
10934:
10929:
10923:
10920:
10918:
10917:Biquaternions
10915:
10913:
10910:
10884:
10881:
10879:
10876:
10874:
10871:
10846:
10845:
10843:
10837:
10808:
10805:
10780:
10777:
10752:
10749:
10724:
10720:
10717:
10716:
10714:
10712:
10708:
10702:
10699:
10695:
10692:
10691:
10690:
10687:
10685:
10682:
10680:
10677:
10675:
10672:
10670:
10667:
10665:
10662:
10637:
10634:
10632:
10629:
10604:
10601:
10576:
10573:
10548:
10545:
10544:
10542:
10540:
10535:
10530:
10523:
10518:
10516:
10511:
10509:
10504:
10503:
10500:
10488:
10485:
10483:
10480:
10478:
10475:
10473:
10472:Complex plane
10470:
10468:
10465:
10464:
10461:
10457:
10450:
10445:
10443:
10438:
10436:
10431:
10430:
10427:
10415:
10412:
10410:
10407:
10405:
10402:
10400:
10397:
10395:
10392:
10390:
10387:
10385:
10384:Normal number
10382:
10380:
10377:
10375:
10372:
10370:
10367:
10365:
10362:
10360:
10357:
10355:
10352:
10350:
10347:
10345:
10342:
10340:
10337:
10335:
10332:
10331:
10328:
10324:
10317:
10312:
10310:
10305:
10303:
10298:
10297:
10294:
10286:
10282:
10281:
10276:
10275:"Real number"
10272:
10271:
10261:
10255:
10251:
10247:
10243:
10239:
10235:
10231:
10229:9780130402615
10225:
10220:
10219:
10212:
10209:
10208:
10201:
10199:9780828400794
10195:
10190:
10189:
10183:
10179:
10175:
10173:9780387962177
10169:
10165:
10160:
10155:
10151:
10146:
10144:9780124254015
10140:
10135:
10134:
10128:
10127:Tversky, Amos
10124:
10120:
10115:
10111:
10106:
10102:
10096:
10092:
10088:
10084:
10083:Real Analysis
10080:
10076:
10071:
10070:
10064:
10060:
10055:
10054:
10048:
10044:
10040:
10037:(in German).
10036:
10035:
10030:
10026:
10025:Cantor, Georg
10022:
10018:
10016:9780387963020
10012:
10007:
10006:
10000:
9996:
9992:
9986:
9982:
9978:
9974:
9970:
9966:
9965:
9953:
9949:
9946:
9943:
9942:Nombres réels
9936:
9932:
9927:
9925:
9923:
9914:
9912:9780201826531
9908:
9904:
9899:
9898:
9892:
9886:
9884:
9870:
9864:
9860:
9859:
9851:
9843:
9839:
9835:
9831:
9824:
9817:
9810:
9804:
9800:
9793:
9785:
9779:
9775:
9771:
9764:
9756:
9752:
9748:
9744:
9740:
9736:
9731:
9726:
9722:
9718:
9717:
9708:
9704:
9700:
9696:
9693:(4): 366–75.
9692:
9688:
9687:
9682:
9676:
9669:. 2015-01-05.
9668:
9661:
9655:
9648:
9644:
9643:
9638:
9634:
9627:
9619:
9615:
9611:
9605:
9601:
9600:
9592:
9583:
9575:
9571:
9567:
9563:
9559:
9555:
9554:
9549:
9547:
9538:
9531:(43): 134–35.
9530:
9523:
9516:
9506:
9500:
9496:
9495:
9487:
9473:
9467:
9463:
9462:
9454:
9446:
9444:9780312381851
9440:
9436:
9431:
9430:
9425:A History of
9417:
9410:
9404:
9400:
9396:
9388:
9381:
9377:
9373:
9369:
9365:
9361:
9357:
9353:
9352:
9344:
9337:
9333:
9332:
9327:
9323:
9316:
9308:
9302:
9298:
9294:
9290:
9286:
9279:
9272:
9266:
9261:
9260:
9253:
9248:
9242:
9234:
9230:
9226:
9222:
9216:
9208:
9204:
9203:"Real number"
9198:
9196:
9187:
9182:
9174:
9167:
9166:
9162:
9157:
9153:
9152:
9146:
9139:
9132:. 2011-08-03.
9131:
9127:
9126:"Real number"
9121:
9117:
9086:
9081:
9077:
9073:
9070:
9067:
9064:
9061:
9058:
9035:
9032:
9029:
9020:
9014:
9007:
9001:
8994:
8988:
8981:
8977:
8971:
8967:
8943:
8940:
8939:
8920:
8917:
8916:
8913:
8910:
8909:
8906:
8904:
8901:
8900:
8897:
8896:
8877:
8875:
8872:
8871:
8868:
8865:
8864:
8861:
8858:
8857:
8854:
8852:
8849:
8848:
8845:
8844:
8833:
8830:
8829:
8818:
8815:
8814:
8811:
8810:Prime numbers
8808:
8807:
8803:
8800:
8799:
8795:
8792:
8791:
8788:
8768:
8761:
8758:
8757:
8754:
8753:
8750:
8730:
8723:
8720:
8719:
8716:
8715:
8712:
8692:
8685:
8682:
8681:
8678:
8677:
8674:
8654:
8647:
8644:
8643:
8640:
8639:
8636:
8616:
8609:
8606:
8605:
8602:
8597:
8596:
8593:
8592:
8587:
8586:Real analysis
8584:
8582:
8579:
8577:
8574:
8572:
8569:
8567:
8564:
8563:
8559:
8553:
8548:
8538:
8534:
8530:
8526:
8522:
8518:
8517:Hilbert space
8514:
8511:
8508:
8504:
8503:infinitesimal
8500:
8496:
8492:
8455:
8451:
8448:
8444:
8436:
8432:
8429:
8425:
8424:compact space
8413:
8409:
8406:
8402:
8398:
8397:
8396:
8388:
8386:
8382:
8381:
8375:
8371:
8370:
8364:
8360:
8355:
8353:
8349:
8345:
8342:as soon as a
8341:
8338:
8327:of dimension
8326:
8322:
8319:
8299:
8282:
8278:
8252:
8231:
8217:The notation
8215:
8202:
8197:
8192:
8163:
8158:
8131:
8126:
8082:
8078:
8074:
8058:
8052:
8046:
8041:
8012:
8009:
7980:
7951:
7922:
7893:
7877:
7875:
7871:
7867:
7863:
7837:
7827:
7825:
7821:
7817:
7807:
7805:
7801:
7797:
7793:
7789:
7785:
7784:
7778:
7776:
7772:
7768:
7752:
7749:
7743:
7739:
7733:
7728:
7724:
7703:
7700:
7697:
7694:
7675:
7670:
7660:
7656:
7653:Alternately,
7651:
7649:
7646:of numerical
7645:
7641:
7637:
7633:
7629:
7625:
7621:
7617:
7613:
7609:
7605:
7601:
7597:
7593:
7588:
7586:
7582:
7578:
7569:
7566:
7548:
7535:
7517:
7504:
7499:
7496:
7493:enriches the
7492:
7488:
7487:Edward Nelson
7484:
7482:
7478:
7474:
7470:
7466:
7465:infinitesimal
7462:
7458:
7454:
7449:
7447:
7443:
7439:
7429:
7426:
7424:
7420:
7416:
7412:
7408:
7404:
7400:
7396:
7380:
7378:
7377:
7368:
7364:
7354:
7352:
7325:
7280:
7265:
7247:
7218:
7202:
7200:
7195:
7191:
7184:
7180:
7175:
7171:
7167:
7163:
7158:
7156:
7141:
7130:
7111:
7108:
7103:
7099:
7095:
7087:
7084:
7073:
7043:
7017:
6991:
6987:
6982:
6978:
6974:
6970:
6967:
6963:
6959:
6955:
6951:
6947:
6943:
6942:
6940:
6936:
6932:
6928:
6902:
6899:
6895:
6891:
6865:
6864:
6863:
6861:
6829:
6827:
6823:
6819:
6815:
6811:
6810:axiomatically
6790:
6785:
6780:
6775:
6770:
6765:
6747:
6737:
6735:
6731:
6727:
6723:
6718:
6716:
6712:
6708:
6704:
6700:
6696:
6695:infinite sets
6692:
6688:
6684:
6680:
6676:
6672:
6671:Dedekind cuts
6668:
6664:
6663:Charles Méray
6660:
6656:
6651:
6649:
6645:
6640:
6638:
6634:
6630:
6629:
6624:
6620:
6610:
6608:
6602:
6600:
6596:
6592:
6584:
6580:
6574:
6570:
6566:
6561:
6552:
6544:
6536:
6531:
6529:
6525:
6520:
6518:
6514:
6509:
6507:
6506:
6501:
6497:
6493:
6489:
6485:
6481:
6470:
6466:
6462:
6458:
6454:
6450:
6446:
6442:
6438:
6434:
6429:
6427:
6423:
6419:
6418:Dedekind cuts
6415:
6411:
6407:
6402:
6400:
6396:
6392:
6387:
6385:
6374:
6370:
6359:
6358:Shulba Sutras
6355:
6351:
6347:
6230:Real numbers
6228:
6219:
6217:
6213:
6209:
6204:
6202:
6198:
6197:open interval
6194:
6193:least element
6168:
6142:
6138:
6133:
6131:
6127:
6123:
6119:
6115:
6111:
6085:
6038:of the field
6037:
6012:
6007:
5961:
5935:
5887:
5883:
5879:
5874:
5872:
5868:
5867:unit interval
5864:
5860:
5856:
5852:
5847:
5845:
5841:
5815:
5767:
5762:
5760:
5756:
5752:
5748:
5744:
5740:
5736:
5732:
5728:
5724:
5719:
5715:
5709:
5705:
5700:
5696:
5692:
5688:
5683:
5681:
5677:
5671:
5661:
5659:
5655:
5651:
5609:
5595:
5593:
5589:
5571:
5558:
5542:
5527:
5523:
5519:
5518:infinite sets
5512:
5508:
5498:
5496:
5447:
5443:
5442:David Hilbert
5438:
5436:
5435:
5430:
5404:
5403:metric spaces
5400:
5396:
5392:
5388:
5387:ordered group
5383:
5381:
5376:
5372:
5367:
5363:
5359:
5354:
5346:
5344:
5326:
5322:
5313:
5309:
5287:
5284:
5278:
5274:
5266:
5261:
5258:
5255:
5251:
5243:
5242:
5241:
5239:
5217:
5214:
5208:
5204:
5191:
5188:
5185:
5181:
5177:
5172:
5168:
5160:
5159:
5158:
5157:
5152:
5150:
5146:
5141:
5139:
5135:
5130:
5128:
5123:
5121:
5118:greater than
5117:
5111:
5107:
5098:
5090:
5086:
5083:
5078:
5074:
5069:
5066:
5062:
5058:
5054:
5050:
5046:
5040:
5033:
5025:
5021:
5013:
5012:
5006:
5002:
4998:
4993:
4991:
4987:
4986:metric spaces
4983:
4979:
4973:
4963:
4950:
4947:
4944:
4924:
4904:
4901:
4898:
4895:
4888:
4883:
4881:
4876:
4874:
4856:
4852:
4831:
4828:
4825:
4822:
4815:are zero for
4800:
4796:
4775:
4768:
4764:
4760:
4750:
4730:
4727:
4707:
4704:
4695:
4689:
4673:
4668:
4664:
4650:
4637:
4632:
4628:
4623:
4617:
4613:
4609:
4604:
4601:
4598:
4594:
4590:
4585:
4581:
4573:and one sets
4560:
4557:
4554:
4549:
4545:
4540:
4534:
4530:
4526:
4521:
4518:
4515:
4511:
4488:
4484:
4463:
4460:
4457:
4454:
4432:
4428:
4419:
4403:
4400:
4395:
4391:
4368:
4364:
4341:
4337:
4333:
4328:
4324:
4315:
4302:
4288:
4283:
4279:
4245:
4242:
4238:
4232:
4228:
4222:
4217:
4214:
4211:
4207:
4203:
4198:
4194:
4188:
4184:
4178:
4173:
4170:
4167:
4163:
4159:
4157:
4145:
4141:
4135:
4131:
4125:
4122:
4119:
4114:
4109:
4105:
4099:
4094:
4090:
4086:
4083:
4080:
4075:
4072:
4069:
4065:
4059:
4056:
4053:
4049:
4045:
4040:
4036:
4030:
4026:
4022:
4020:
4013:
4009:
3997:
3996:
3995:
3994:
3982:
3978:
3973:
3959:
3956:
3951:
3947:
3926:
3923:
3920:
3917:
3894:
3891:
3888:
3883:
3879:
3875:
3870:
3866:
3862:
3857:
3853:
3849:
3846:
3843:
3838:
3835:
3832:
3828:
3824:
3819:
3815:
3807:
3806:
3805:
3804:
3792:
3787:
3773:
3770:
3767:
3762:
3758:
3738:
3735:
3732:
3727:
3723:
3703:
3700:
3697:
3692:
3688:
3664:
3661:
3658:
3655:
3652:
3629:
3626:
3623:
3616:
3612:
3606:
3602:
3596:
3591:
3586:
3582:
3576:
3571:
3567:
3563:
3560:
3557:
3552:
3549:
3546:
3542:
3536:
3533:
3530:
3526:
3522:
3517:
3513:
3507:
3503:
3499:
3496:
3489:
3488:
3487:
3486:
3482:
3481:decimal point
3466:
3463:
3458:
3454:
3448:
3444:
3440:
3435:
3431:
3427:
3422:
3419:
3416:
3412:
3406:
3402:
3385:
3381:
3377:
3367:
3365:
3361:
3353:
3349:
3345:
3331:
3328:
3325:
3320:
3316:
3295:
3287:
3279:
3264:
3261:
3256:
3253:
3248:
3245:
3229:
3210:
3190:
3187:
3184:
3181:
3178:
3175:
3155:
3152:
3149:
3133:
3130:
3129:
3128:
3125:
3107:
3104:
3096:
3093:is called an
3080:
3060:
3057:
3054:
3034:
3031:
3028:
3008:
3000:
2999:bounded above
2984:
2976:
2972:
2968:
2964:
2954:
2952:
2948:
2944:
2940:
2936:
2917:
2909:
2901:
2889:
2888:
2887:
2873:
2838:
2822:
2787:
2786:ordered rings
2771:
2736:
2732:
2727:
2725:
2722:taken as the
2717:
2713:
2708:
2706:
2690:
2685:
2675:
2659:
2646:
2619:
2597:
2594:
2591:
2563:
2559:
2555:
2548:
2532:
2529:
2509:
2506:
2486:
2466:
2463:
2454:
2433:
2401:
2395:
2392:
2389:
2361:
2355:
2352:
2349:
2335:
2332:
2329:
2290:
2287:
2284:
2281:
2273:
2270:
2256:
2250:
2247:
2244:
2216:
2210:
2207:
2204:
2190:
2187:
2184:
2145:
2142:
2139:
2136:
2128:
2125:
2111:
2108:
2105:
2102:
2095:
2080:
2077:
2074:
2046:
2043:
2040:
2037:
2029:
2026:
2025:
2024:
2022:
2015:
2008:and read as "
1995:
1992:
1989:
1981:
1957:
1951:
1948:
1945:
1942:
1933:
1925:
1900:
1892:
1875:
1872:
1858:
1853:
1850:
1846:
1842:
1839:
1834:
1831:
1817:
1797:
1793:
1789:
1769:
1764:
1761:
1743:
1740:
1726:
1720:
1717:
1711:
1708:
1705:
1702:
1699:
1696:
1683:
1678:
1662:
1659:
1658:
1657:
1629:
1625:
1621:
1618:
1611:
1597:
1594:
1591:
1584:
1566:
1563:
1560:
1557:
1554:
1547:
1546:
1545:
1528:
1525:
1522:
1519:
1499:
1496:
1493:
1473:
1470:
1467:
1464:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1402:
1399:
1396:
1388:
1374:
1371:
1368:
1365:
1345:
1342:
1339:
1336:
1316:
1313:
1310:
1290:
1287:
1284:
1264:
1261:
1258:
1239:
1236:
1233:
1230:
1202:
1199:
1196:
1193:
1185:
1166:
1163:
1158:
1155:
1151:
1147:
1127:
1122:
1119:
1095:
1092:
1088:
1079:
1071:
1052:
1049:
1043:
1040:
1034:
1031:
1011:
1008:
1005:
997:
989:
970:
967:
964:
961:
958:
950:
942:
923:
920:
917:
914:
911:
903:
895:
891:
864:
861:
858:
855:
852:
849:
843:
840:
837:
831:
823:
819:
789:
786:
780:
777:
774:
768:
765:
739:
736:
733:
727:
724:
721:
718:
715:
709:
706:
703:
692:
688:
665:
662:
659:
656:
653:
633:
630:
627:
624:
621:
618:
615:
607:
603:
592:
589:which is the
576:
573:
570:
567:
547:
544:
541:
522:
519:
516:
500:
496:
485:
482:which is the
469:
466:
463:
460:
444:
440:
439:
438:
436:
432:
428:
424:
420:
416:
415:ordered field
406:
404:
399:
395:
391:
390:ordered field
381:
379:
375:
374:Dedekind cuts
371:
367:
363:
362:ordered field
360:
356:
352:
348:
344:
340:
336:
332:
331:real analysis
328:
324:
319:
317:
313:
312:displacements
309:
305:
295:
290:
284:
280:
276:
271:
269:
263:
256:
250:
245:
241:
237:
230:
223:
219:
214:
208:
204:
200:
196:
166:
161:
156:
151:
149:
145:
141:
137:
132:
130:
126:
122:
118:
114:
110:
107:
103:
99:
95:
91:
80:
77:
69:
59:
55:
49:
48:
42:
37:
28:
27:
22:
11196:Real numbers
11136:
11126:
10941:Dual numbers
10933:hypercomplex
10723:Real numbers
10722:
10481:
10359:Construction
10354:Completeness
10323:Real numbers
10322:
10278:
10241:
10217:
10206:
10187:
10166:. Springer.
10163:
10132:
10109:
10082:
10068:
10052:
10038:
10032:
10009:. Springer.
10004:
9972:
9896:
9872:, retrieved
9857:
9850:
9833:
9829:
9816:
9798:
9792:
9787:, chapter 2.
9769:
9763:
9720:
9714:
9690:
9684:
9675:
9666:
9660:"Lecture #1"
9654:
9640:
9626:
9598:
9591:
9582:
9557:
9551:
9545:
9537:
9528:
9522:
9514:
9508:, retrieved
9493:
9486:
9475:, retrieved
9461:Pi Unleashed
9460:
9453:
9424:
9416:
9394:
9387:
9355:
9349:
9343:
9329:
9315:
9292:
9278:
9258:
9241:
9232:
9215:
9206:
9185:
9173:
9164:
9161:Mathematics.
9160:
9159:
9155:
9149:
9138:
9129:
9120:
9018:
9013:
9000:
8987:
8980:completeness
8970:
8645:
8443:cyclic order
8394:
8377:
8373:
8366:
8362:
8358:
8356:
8321:vector space
8216:
8076:
8072:
7878:
7873:
7865:
7833:
7813:
7781:
7779:
7652:
7638:studies the
7589:
7575:
7500:
7485:
7483:and others.
7457:Edwin Hewitt
7450:
7435:
7427:
7391:
7374:
7360:
7326:
7203:
7198:
7193:
7189:
7182:
7178:
7173:
7169:
7165:
7159:
7128:
7069:
6989:
6980:
6976:
6972:
6965:
6961:
6957:
6953:
6949:
6945:
6938:
6934:
6930:
6835:
6749:
6719:
6691:Peano axioms
6675:Georg Cantor
6667:Eduard Heine
6652:
6641:
6626:
6616:
6607:completeness
6603:
6559:
6532:
6521:
6513:Simon Stevin
6510:
6504:
6499:
6496:fourth roots
6484:coefficients
6430:
6403:
6388:
6384:square roots
6344:
6205:
6141:well-ordered
6134:
6129:
6121:
6084:vector space
6008:
5875:
5859:Haar measure
5848:
5763:
5759:homeomorphic
5723:contractible
5707:
5703:
5694:
5690:
5687:metric space
5684:
5673:
5596:
5504:
5445:
5439:
5432:
5428:
5395:completeness
5384:
5377:
5375:is larger).
5370:
5365:
5355:
5352:
5342:
5311:
5307:
5305:
5237:
5235:
5153:
5142:
5131:
5124:
5119:
5115:
5109:
5102:
5096:
5088:
5084:
5081:
5076:
5072:
5071:A sequence (
5070:
5064:
5060:
5052:
5048:
5044:
5035:
5028:
5019:
5009:
5004:
5000:
4994:
4975:
4887:numeral base
4884:
4877:
4751:of the form
4696:
4687:
4651:
4476:one defines
4303:
4266:
3974:
3909:
3790:
3788:
3680:is zero and
3644:
3384:power of ten
3373:
3357:
3126:
3094:
2998:
2960:
2932:
2731:homomorphism
2728:
2716:Peano axioms
2709:
2620:
2545:Similarly a
2455:
2421:
2028:Greater than
1977:
1681:
1655:
1543:
900:which is an
896:and denoted
822:distributive
590:
498:
483:
442:
412:
387:
341:. A current
320:
302:Conversely,
301:
272:
215:
207:square roots
205:such as the
194:
159:
152:
133:
124:
93:
87:
72:
63:
44:
11103:Other types
10922:Bioctonions
10779:Quaternions
10482:Real number
9836:(1): 9–19.
8525:eigenvalues
8380:Lie algebra
8337:dimensional
8318:dimensional
7824:Baire space
7771:exponential
7604:fixed-point
7572:Computation
7264:isomorphism
7187:, and thus
7016:upper bound
6927:total order
6858:denote the
6814:isomorphism
6433:Middle Ages
6380:750–690 BC)
5932:are called
5880:alone: the
5871:Vitali sets
5766:square root
5586:and called
5526:cardinality
5507:uncountable
5501:Cardinality
5138:square root
5134:square root
5014:if for any
4937:and 9 with
3993:partial sum
3286:square root
3095:upper bound
2971:upper bound
1980:total order
1822:; that is,
1752:is denoted
1689:; that is,
1661:Subtraction
947:which is a
691:associative
606:commutative
435:total order
398:isomorphism
355:isomorphism
316:differences
279:number line
277:called the
264:= 3.1415...
148:derivatives
121:temperature
106:dimensional
94:real number
90:mathematics
58:introducing
11190:Categories
11057:Projective
11030:Infinities
10414:Vitali set
10344:Cantor set
9874:2015-11-15
9730:1611.09087
9510:2015-02-17
9477:2015-11-15
9107:References
8903:Irrational
8422:. It is a
8287:copies of
7816:set theory
7810:Set theory
7792:almost all
7783:computable
7775:polynomial
7648:algorithms
7565:Paul Cohen
7363:completion
7176:such that
6979:≥ 0, then
6903:The field
6826:isomorphic
6699:set theory
6528:polynomial
6508:("deaf").
6500:irrational
6492:cube roots
6445:fractional
6395:Pythagoras
6208:computable
5814:polynomial
5668:See also:
4383:such that
3981:truncating
3308:such that
3238:such that
3142:such that
3021:such that
1880:, denoted
409:Arithmetic
308:axis lines
251:= 1.414...
244:polynomial
157:by a bold
144:continuity
125:continuous
111:such as a
41:references
11141:solenoids
10961:Sedenions
10807:Octonions
10404:Real line
10285:EMS Press
10041:: 258–62.
10027:(1874). "
9755:118954904
9618:229023665
9574:123203471
9380:121416910
9112:Citations
9068:−
9062:∣
9056:∃
8942:Imaginary
8312:It is an
8198:∗
8193:−
8164:∗
8127:−
8047:∪
8010:≥
7981:−
7923:−
7788:countably
7725:∫
7698:
7640:stability
7581:computers
7545:ℵ
7514:ℵ
7367:converges
7192:+ 1 >
7088:∈
6812:up to an
6781:⋅
6685:and thus
6583:Lindemann
6551:Liouville
6524:Descartes
6465:magnitude
6350:Egyptians
6112:with the
5735:separable
5727:connected
5676:separable
5670:Real line
5606:ℵ
5592:power set
5568:ℵ
5252:∑
5197:∞
5182:∑
4948:−
4899:≥
4880:bijection
4705:≤
4602:−
4555:≤
4519:−
4418:induction
4401:≤
4334:⋯
4314:induction
4243:−
4208:∑
4164:∑
4123:⋯
4084:⋯
4073:−
4057:−
3957:≠
3892:…
3847:…
3836:−
3662:⋯
3653:π
3627:⋯
3561:⋯
3550:−
3534:−
3464:⋯
3428:⋯
3420:−
3073:; such a
3058:∈
3032:≤
2973:admits a
2910:⊂
2902:⊂
2595:≠
2507:−
2464:−
2383:not
2303:read as "
2285:≥
2238:not
2158:read as "
2140:≤
2059:read as "
2014:less than
1949:−
1851:−
1814:with the
1718:−
1700:−
1558:⋅
1156:−
1093:−
1041:−
1006:−
962:×
571:×
545:⋅
343:axiomatic
339:sequences
283:real line
66:July 2024
10575:Integers
10537:Sets of
10334:0.999...
10240:(2013).
10184:(1966).
10156:(1986).
10129:(1971).
10112:. Heath.
10081:(2001).
10065:(1964).
10049:(1960).
10001:(1986).
9971:(2001).
9948:Archived
9893:(1996).
9707:27854250
9399:Springer
9324:(1999),
9297:Springer
9249:(1980),
9223:(2013).
8993:0.999...
8851:Fraction
8684:Rational
8544:See also
8521:matrices
8497:and the
7644:accuracy
7409:and the
7168:; then,
7014:with an
6975:≥ 0 and
6894:addition
6866:The set
6642:In 1854
6631:(1821),
6619:calculus
6593:(1893),
6543:Legendre
6488:equation
6482:, or as
6476:850–930)
6371:such as
6218:or not.
5749:but not
5739:complete
5718:topology
5145:calculus
5093:ε > 0
5024:distance
5016:ε > 0
4997:sequence
4982:complete
4873:0.999...
4871:9. (see
3047:for all
2951:compiler
2645:subfield
1742:Division
1675:and the
1415:implies
1080:denoted
998:denoted
443:addition
433:, and a
427:addition
392:that is
323:theorems
229:fraction
227:and the
136:calculus
123:. Here,
117:duration
113:distance
109:quantity
11131:numbers
10963: (
10809: (
10781: (
10753: (
10725: (
10669:Periods
10638: (
10605: (
10577: (
10549: (
10531:systems
10287:, 2001
10210:, 1930.
9962:Sources
9903:114–115
9735:Bibcode
9695:Bibcode
9360:Bibcode
9231:(ed.).
9158:, B.4.
8760:Natural
8722:Integer
8608:Complex
8251:-tuples
7798:. Some
7473:Leibniz
7444:and in
7388:Physics
6952:, then
6595:Hurwitz
6591:Hilbert
6573:Hermite
6535:Lambert
6517:decimal
6463:" and "
6393:led by
6222:History
5851:measure
5751:compact
5725:(hence
5470:. Thus
5446:largest
5427:is the
5391:uniform
5140:of 2).
3659:3.14159
3168:(take,
2576:(where
2479:(where
994:has an
591:product
222:integer
191:
169:
155:denoted
102:measure
54:improve
10931:Other
10529:Number
10256:
10226:
10196:
10170:
10141:
10097:
10013:
9987:
9954:, p. 6
9909:
9865:
9805:
9780:
9753:
9705:
9616:
9606:
9572:
9548:und π"
9501:
9468:
9441:
9405:
9378:
9303:
9267:
9165:plural
9145:"real"
9019:unique
8490:group.
8369:matrix
7822:, the
7695:arctan
7630:, but
7616:binary
7536:after
7481:Cauchy
7349:, see
7040:has a
6822:Tarski
6665:, and
6637:Cauchy
6633:Cauchy
6623:limits
6599:Gordan
6597:, and
6569:Cantor
6494:, and
6486:in an
6461:number
6449:Indian
6373:Manava
6365:600 BC
6191:has a
6167:subset
5853:, the
5716:; the
5057:Cauchy
4978:limits
3346:Every
3203:where
2712:axioms
2311:" or "
2166:" or "
1457:, and
1076:has a
347:unique
140:limits
98:number
43:, but
11139:-adic
11129:-adic
10886:Over
10847:Over
10841:types
10839:Split
9935:Paris
9826:(PDF)
9751:S2CID
9725:arXiv
9703:JSTOR
9663:(PDF)
9570:S2CID
9376:S2CID
9227:. In
8962:Notes
8515:on a
8462:* + ℵ
8378:real
8367:real
8348:point
7870:field
7864:) or
7614:uses
7477:Euler
7432:Logic
7266:from
7181:>
7129:least
6890:field
6888:is a
6707:logic
6354:Vedic
6118:basis
6086:over
6011:field
5680:dense
4747:is a
4720:with
3786:etc.
2839:from
2788:from
351:up to
242:of a
96:is a
11175:List
11032:and
10254:ISBN
10224:ISBN
10194:ISBN
10168:ISBN
10139:ISBN
10095:ISBN
10011:ISBN
9985:ISBN
9907:ISBN
9863:ISBN
9803:ISBN
9778:ISBN
9614:OCLC
9604:ISBN
9499:ISBN
9466:ISBN
9439:ISBN
9429:(PI)
9403:ISBN
9301:ISBN
9265:ISBN
9033:>
8804:: 1
8796:: 0
8794:Zero
8646:Real
8452:The
8433:The
8418:and
8410:The
8399:The
8385:noun
8376:and
8359:real
8178:and
8112:and
8081:zero
8075:and
7966:and
7908:and
7834:The
7642:and
7579:and
7501:The
7451:The
7233:and
7109:<
6983:≥ 0.
6937:and
6896:and
6836:Let
6791:<
6557:nor
6505:surd
6451:and
6437:zero
6431:The
6135:The
6009:The
5737:and
5729:and
5693:and
5429:only
5047:and
4826:>
4728:<
4458:<
3921:>
3910:(If
3352:root
3262:<
3249:<
3153:<
2941:and
2672:The
2614:and
2580:and
2437:and
2426:and
2393:<
2333:<
2248:<
2188:<
2106:<
2078:>
2041:>
1993:<
1978:The
1667:and
1622:<
1595:<
1523:<
1512:and
1497:<
1468:<
1432:<
1400:<
1369:<
1340:<
1329:and
1314:<
1288:<
1234:<
1219:and
1197:<
894:zero
885:and
813:and
755:and
682:and
646:and
597:and
505:and
497:The
490:and
449:and
441:The
429:and
275:line
240:root
195:real
146:and
92:, a
10246:doi
10087:doi
9977:doi
9933:of
9838:doi
9743:doi
9562:doi
9435:170
9368:doi
9356:500
9156:n.2
8802:One
8283:of
7836:set
7814:In
7716:or
7489:'s
7417:or
7295:to
7185:– 1
7018:in
6992:of
6971:if
6944:if
6860:set
6709:of
6502:or
6201:V=L
6169:of
5936:of
5768:in
5733:),
5373:+ 1
4988:or
4690:of
4312:by
3097:of
2997:is
2733:of
2012:is
1937:max
1818:of
1782:or
1685:of
1277:or
1110:or
593:of
560:or
486:of
484:sum
368:of
353:an
281:or
209:of
119:or
88:In
11192::
10721::
10283:,
10277:,
10252:.
10162:.
10125:;
10121:;
10093:.
10039:77
9983:.
9937:,
9921:^
9905:.
9882:^
9832:.
9828:.
9776:,
9749:.
9741:.
9733:.
9721:47
9719:.
9701:.
9691:74
9689:.
9665:.
9645:,
9639:,
9612:.
9568:.
9558:43
9556:.
9550:.
9513:,
9437:.
9401:,
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9374:,
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9299:,
9295:,
9287:;
9255:,
9205:.
9194:^
9184:.
9148:.
9128:.
8531:.
8420:−∞
8416:+∞
8372:,
8354:.
7479:,
7475:,
7459:,
7448:.
7425:.
7405:,
7401:,
7397:,
7379:.
7353:.
7201:.
6981:xy
6964:+
6960:≥
6956:+
6948:≥
6941::
6933:,
6828:.
6728:.
6661:,
6601:.
6474:c.
6439:,
6412:.
6378:c.
6363:c.
6108:.
6006:.
5873:.
5846:.
5706:−
5368:,
5345:.
5122:.
5108:−
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5034:−
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4951:1.
4765:10
4694:.
4629:10
4546:10
4239:10
4195:10
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2953:.
2726:.
2707:.
2618:.
2453:.
2274::
2129::
2030::
881:,
809:,
425:,
357:)
270:.
249:√2
225:−5
213:.
211:−1
167:,
150:.
142:,
131:.
115:,
11143:)
11137:p
11133:(
11127:p
11001:/
10985:)
10972:S
10908::
10895:C
10869::
10856:R
10831:)
10818:O
10803:)
10790:H
10775:)
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10734:R
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10627:)
10614:Q
10599:)
10586:Z
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10558:N
10521:e
10514:t
10507:v
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10248::
10232:.
10202:.
10176:.
10147:.
10103:.
10089::
10019:.
9993:.
9979::
9939:"
9915:.
9844:.
9840::
9834:1
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9745::
9737::
9727::
9709:.
9697::
9620:.
9576:.
9564::
9546:e
9481:.
9447:.
9427:π
9370::
9362::
9310:.
9209:.
9087:,
9082:2
9078:z
9074:=
9071:y
9065:x
9059:z
9036:y
9030:x
8774:N
8769::
8736:Z
8731::
8698:Q
8693::
8660:R
8655::
8622:C
8617::
8509:.
8486:1
8484:ℵ
8479:1
8477:ℵ
8473:*
8471:1
8469:ℵ
8464:1
8460:1
8458:ℵ
8449:.
8439:∞
8430:.
8335:-
8333:n
8329:n
8316:-
8314:n
8300:.
8296:R
8285:n
8275:(
8262:R
8249:n
8232:n
8227:R
8203:.
8188:R
8159:+
8154:R
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8097:+
8093:R
8059:.
8056:}
8053:0
8050:{
8042:+
8037:R
8013:0
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7976:R
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7947:R
7918:R
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7866:R
7860:(
7847:R
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7750:d
7744:x
7740:x
7734:1
7729:0
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7701:5
7676:,
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7549:0
7518:1
7371:π
7336:R
7309:2
7305:R
7281:1
7276:R
7248:2
7243:R
7219:1
7214:R
7199:N
7194:N
7190:n
7183:N
7179:n
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7170:N
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7142:2
7115:}
7112:2
7104:2
7100:x
7096::
7092:Q
7085:x
7082:{
7066:.
7053:R
7027:R
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6931:x
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6875:R
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6796:)
6786:;
6776:;
6771:+
6766:;
6762:R
6758:(
6587:π
6578:e
6560:e
6555:e
6547:π
6539:π
6472:(
6376:(
6356:"
6330:)
6326:N
6322:(
6302:)
6298:Z
6294:(
6274:)
6270:Q
6266:(
6246:)
6242:R
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6178:R
6152:R
6130:B
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5971:R
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5710:|
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5704:x
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5691:x
5634:c
5610:0
5572:0
5543:.
5538:c
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5414:R
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5366:z
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5327:x
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5312:N
5308:M
5288:!
5285:n
5279:n
5275:x
5267:M
5262:N
5259:=
5256:n
5238:x
5218:!
5215:n
5209:n
5205:x
5192:0
5189:=
5186:n
5178:=
5173:x
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5112:|
5110:x
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5089:x
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4669:n
4665:D
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4638:.
4633:n
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4586:n
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4160:=
4146:n
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4126:+
4120:+
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3889:,
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3768:=
3763:2
3759:a
3739:,
3736:1
3733:=
3728:1
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3701:3
3698:=
3693:0
3689:b
3678:k
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3656:=
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3459:2
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3441:.
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3407:k
3403:b
3392:k
3388:x
3332:.
3329:x
3326:=
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3317:r
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3282:x
3277:.
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3232:x
3225:x
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3188:1
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3179:=
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3150:x
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3122:S
3108:.
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3081:u
3061:S
3055:s
3035:u
3029:s
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2985:S
2918:.
2914:R
2906:Q
2898:N
2874:.
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2772:,
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2720:1
2691:,
2686:2
2660:.
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2612:p
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2582:q
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2560:/
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2533:.
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2510:n
2487:n
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2439:1
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2428:1
2424:0
2402:.
2399:)
2396:b
2390:a
2387:(
2362:,
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2353:=
2350:a
2347:(
2339:)
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2327:(
2317:b
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2254:)
2251:a
2245:b
2242:(
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2208:=
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2202:(
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2172:b
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2164:b
2160:a
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2112:.
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2103:b
2081:b
2075:a
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2038:a
2017:b
2010:a
1996:b
1990:a
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1840:=
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1712:+
1709:a
1706:=
1703:b
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1626:a
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1598:1
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1567:0
1564:=
1561:a
1555:0
1529:.
1526:b
1520:0
1500:a
1494:0
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1471:a
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1426:+
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1372:c
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1337:b
1317:b
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1200:b
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1164:=
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1120:1
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987:.
985:a
971:a
968:=
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921:=
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778:=
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743:)
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737:+
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731:(
728:+
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722:=
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716:+
713:)
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686:.
684:b
680:a
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660:=
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625:=
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601:.
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