3586:, 1.0, 1.00, 1.000, ..., all represent the natural number 1. A given real number has only the following decimal representations: an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places or an exact value with only finitely many decimal places. In this last case, the last non-zero digit may be replaced by the digit one smaller followed by an unlimited number of 9s, or the last non-zero digit may be followed by an unlimited number of zeros. Thus the exact real number 3.74 can also be written 3.7399999999... and 3.74000000000.... Similarly, a decimal numeral with an unlimited number of 0s can be rewritten by dropping the 0s to the right of the rightmost nonzero digit, and a decimal numeral with an unlimited number of 9s can be rewritten by increasing by one the rightmost digit less than 9, and changing all the 9s to the right of that digit to 0s. Finally, an unlimited sequence of 0s to the right of a decimal place can be dropped. For example, 6.849999999999... = 6.85 and 6.850000000000... = 6.85. Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9s, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place. For example, 99.999... = 100.
4342:
6800:
954:. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news.
6760:
447:
6780:
6770:
6447:
6790:
40:
3971:, is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. The study of these questions belongs to
532:
2716:
3186:(supposed to be positive), then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator 1. For example â7 can be written
420:, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior
3323:
has a denominator whose prime factors are 2 or 5 or both, because these are the prime factors of 10, the base of the decimal system. Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02. Representing other real numbers as decimals would require an infinite
4123:
The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is
2973:
with an integer numerator and a positive integer denominator. Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between
844:
called them false roots as they cropped up in algebraic polynomials yet he found a way to swap true roots and false roots as well. At the same time, the
Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's
547:
is the first book that mentions zero as a number, hence
Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the
965:
numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. It had
2866:, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.
2862:, which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in
4266:. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.
552:
is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the
Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.
2325:
1265:, which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in
3283:. The following paragraph will focus primarily on positive real numbers. The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a
1750:
rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in
3324:
sequence of digits to the right of the decimal point. If this infinite sequence of digits follows a pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern. Such a decimal is called a
4124:
no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not.
2850:
is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10). In this base 10 system, the rightmost digit of a natural number has a
1663:
1393:
4160:-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what
1157:, an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the
3302:
is placed to the right of the digit with place value 1. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. For example, 123.456 represents
1512:
424:
around the late 14th century, and the HinduâArabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for
3319:, or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. A real number can be expressed by a finite number of decimal digits only if it is rational and its
4508:
1737:
4205:
Some number systems that are not included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes called
3982:
One answered question, as to whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes, was confirmed; this proven claim is called the
3998:
Many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. Example of such sets of integers are
1446:
338:
During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the
1915:, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the
1161:
mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. The symbol
3355:) denote exactly the rational numbers, i.e., all rational numbers are also real numbers, but it is not the case that every real number is rational. A real number that is not rational is called
4807:. "Today, it is no longer that easy to decide what counts as a 'number.' The objects from the original sequence of 'integer, rational, real, and complex' are certainly numbers, but so are the
3455:
3406:
218:
5728:
2727:(sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the
4058:
are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass, starting from a given segment of unit length, in a finite number of steps.
3169:
3078:
2884:
of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a
257:
3124:
5075:
2276:
1299:. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as
2833:
2804:
2075:
2046:
2528:
2490:
2452:
2414:
2376:
3475:
real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. They can only be approximated by decimal numerals, denoting
4456:
can refer to a symbol like 5, but also to a word or a phrase that names a number, like "five hundred"; numerals include also other words representing numbers, like "dozen".
3277:
6259:
6182:
6143:
6105:
6077:
6049:
6021:
5934:
5901:
5873:
5845:
3817:
3233:
2925:
2775:
2232:
2199:
2135:
2102:
2008:
1558:
819:
gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots".
5753:
1181:
3649:
1307:. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.
2947:
of the integers. As there is no common standard for the inclusion or not of zero in the natural numbers, the natural numbers without zero are commonly referred to as
3726:
3555:(truncation after the 3. decimal). Digits that suggest a greater accuracy than the measurement itself does, should be removed. The remaining digits are then called
4860:
2270:
of the next one. So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as
1067:. Other noteworthy contributions have been made by DruckenmĂŒller (1837), Kunze (1857), Lemke (1870), and GĂŒnther (1872). Ramus first connected the subject with
801:
During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician
4958:
Indian mathematicians invented the concept of zero and developed the "Arabic" numerals and system of place-value notation used in most parts of the world today
377:. These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.
1919:, which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the
4970:
856:
As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.
3411:
as it sometimes is, the ellipsis does not mean that the decimals repeat (they do not), but rather that there is no end to them. It has been proved that
990:
had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by
4811:-adics. The quaternions are rarely referred to as 'numbers,' on the other hand, though they can be used to coordinatize certain mathematical notions."
1873:
have been studied throughout recorded history. They are positive integers that are divisible only by 1 and themselves. Euclid devoted one book of the
1234:
showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of
384:
notation), which limits its representation of large numbers. Nonetheless, tallying systems are considered the first kind of abstract numeral system.
1581:
5711:
3582:
Just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation. For example,
4741:
1324:
5720:
1766:, showing that every polynomial over the complex numbers has a full set of solutions in that realm. Gauss studied complex numbers of the form
906:
is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math
5411:
5067:
4945:
4539:
3677:
polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a
4581:
4026:
are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called
6783:
772:
620:
seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting
388:
346:
system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as
1010:
into two groups having certain characteristic properties. The subject has received later contributions at the hands of
Weierstrass,
5795:
5310:
3464:, that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions
1473:
2747:) in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including 0 or not. The
6483:
5501:
2842:
numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten
3344:
can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0.
4794:
1116:
1682:
5626:
4939:
4804:
4717:
4682:
4575:
4051:
5541:
884:. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of
311:
Besides their practical uses, numbers have cultural significance throughout the world. For example, in
Western society, the
5959:
5208:, ed. Robert Fricke, Emmy Noether & Ăystein Ore (Braunschweig: Friedrich Vieweg & Sohn, 1932), vol. 3, pp. 315â334.
5196:
4131:
real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.
3789:
of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a
1283:
The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor
822:
European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although
5660:
4707:
1404:
998:(1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a
5954:
2896:
of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of
5688:
5596:
368:
4856:
3424:
5578:
5560:
5451:
5142:
5109:
5051:
5023:
4832:
4769:
4645:
4611:
4108:. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a
3983:
1878:
486:
1310:
This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When
994:(1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by
3601:
5939:
4479:
3379:
186:
6332:
1037:(formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of
565:
421:
130:
6410:
4974:
2855:
of 1, and every other digit has a place value ten times that of the place value of the digit to its right.
1936:
939:
composed between 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to
6514:
3823:
1763:
807:
505:
468:
3897:
is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "
3129:
3040:
1900:
to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the
826:
allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of
230:
6839:
6293:
5651:
4326:
extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form
3089:
1246:
numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of
1026:
927:
609:
There are other uses of zero before
Brahmagupta, though the documentation is not as complete as it is in the
5743:
5641:
2320:{\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }
125:. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a
6773:
6560:
5788:
1318:
for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation
636:
depend in part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether
1300:
6555:
6540:
6476:
5944:
5646:
5234:
Leonhard Euler, "Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis",
3567:
are measured as 1.23 m and 4.56 m, then multiplication gives an area for the rectangle between
3412:
765:
3559:. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001
3483:
real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only
354:, and the application of the term "number" is a matter of convention, without fundamental significance.
3827:
3605:
1278:
1164:
1059:, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of
877:
2809:
2780:
2051:
2022:
1072:
780:, black for negative. The first reference in a Western work was in the 3rd century AD in Greece.
6735:
6694:
6573:
6327:
6283:
5703:
3575:. Since not even the second digit after the decimal place is preserved, the following digits are not
2507:
2469:
2431:
2393:
2355:
335:, stimulating the investigation of many problems in number theory which are still of interest today.
4096:
digits of the computable number's decimal representation. Equivalent definitions can be given using
531:
6579:
6450:
6322:
4366:
4176:
3976:
3609:
3257:
1955:
1855:
1854:
took the key step of distinguishing between poles and branch points, and introduced the concept of
1851:
1063:, and at the opening of the 19th century were brought into prominence through the writings of
1022:
918:. Similarly, Babylonian math texts used sexagesimal (base 60) fractions with great frequency.
6242:
6165:
6126:
6088:
6060:
6032:
6004:
5917:
5884:
5856:
5828:
5333:
3800:
3216:
2908:
2758:
2215:
2182:
2118:
2085:
1991:
1536:
624:
and, by the
Medieval period, religious arguments about the nature and existence of 0 and the
141:. In addition to their use in counting and measuring, numerals are often used for labels (as with
6799:
6522:
6506:
5781:
5130:
4882:
Chrisomalis, Stephen (1 September 2003). "The
Egyptian origin of the Greek alphabetic numerals".
4428:
3785:; if the imaginary part is 0, then the number is a real number. Thus the real numbers are a
1886:
1266:
1112:
678:
and the
Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a
560:. Many ancient texts used 0. Babylonian and Egyptian texts used it. Egyptians used the word
464:
457:
31:
17:
4097:
687:
6763:
6583:
6532:
6469:
5601:
4217:
4184:
3615:
2970:
1859:
1572:
1247:
1096:
962:
293:
6803:
5443:
5437:
4929:
4531:
3669:. This set of numbers arose historically from trying to find closed formulas for the roots of
1467:
positive and the other negative. The incorrect use of this identity, and the related identity
373:
Bones and other artifacts have been discovered with marks cut into them that many believe are
6829:
6740:
6669:
6395:
6231:
5610:
5531:
Number, the language of science; a critical survey written for the cultured non-mathematician
5399:
5249:
Det
Kongelige Danske Videnskabernes Selskabs naturvidenskabelige og mathematiske Afhandlinger
5247:
Ramus, "Determinanternes Anvendelse til at bes temme Loven for de convergerende Bröker", in:
5218:
4928:
Bulliet, Richard; Crossley, Pamela; Headrick, Daniel; Hirsch, Steven; Johnson, Lyman (2010).
4565:
4453:
4371:
4278:
4031:
3002:
equal parts. Two different fractions may correspond to the same rational number; for example
2661:
1908:
1897:
1314:
coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See
1064:
4824:
The roots of civilization; the cognitive beginnings of man's first art, symbol, and notation
3979:
is an example of a still unanswered question: "Is every even number the sum of two primes?"
3701:
6834:
6730:
6565:
6148:
5906:
5158:
4422:
4055:
3987:
3880:
1928:
1912:
1747:
1224:
889:
510:
331:, permeated ancient and medieval thought. Numerology heavily influenced the development of
129:, which is an organized way to represent any number. The most common numeral system is the
6793:
4996:
4674:
Introduction to cultural mathematics : with case studies in the Otomies and the Incas
4172:-adic numbers contains the rational numbers, but is not contained in the complex numbers.
4127:
The set of computable numbers has the same cardinality as the natural numbers. Therefore,
1189:
defined the traditional Western notion of mathematical infinity. He distinguished between
664:
556:
The use of 0 as a number should be distinguished from its use as a placeholder numeral in
8:
6699:
6608:
6603:
6597:
6589:
6550:
6359:
6269:
6226:
6208:
5986:
5261:
5125:
Bernard Frischer (1984). "Horace and the Monuments: A New Interpretation of the Archytas
4410:
4387:
4382:
4327:
4237:
4206:
4200:
3839:
2835:
when it is necessary to indicate whether the set should start with 0 or 1, respectively.
2748:
1916:
1882:
1814:
1789:
1284:
1262:
1144:
991:
776:
contains methods for finding the areas of figures; red rods were used to denote positive
629:
351:
339:
75:
6789:
5302:
3279:
They include all the measuring numbers. Every real number corresponds to a point on the
770:
The abstract concept of negative numbers was recognized as early as 100â50 BC in China.
6745:
6689:
6593:
6510:
6264:
5976:
5356:
5039:
4971:"Historia Matematica Mailing List Archive: Re: [HM] The Zero Story: a question"
4907:
4735:
4637:
4347:
4301:
4249:
3890:
3835:
3674:
3556:
2937:
1920:
1669:
1220:
1194:
1056:
933:
557:
430:
347:
5493:
1042:
6659:
6422:
6385:
6349:
6288:
6274:
5969:
5949:
5622:
5592:
5574:
5556:
5468:
5447:
5375:
5138:
5105:
5047:
5019:
4935:
4911:
4899:
4838:
4828:
4800:
4775:
4765:
4723:
4713:
4688:
4678:
4641:
4631:
4607:
4571:
4398:
4392:
4341:
4305:
4255:
4113:
4067:
4039:
4027:
3590:
3491:. Thus 123.456 is considered an approximation of any real number greater or equal to
3356:
3325:
2893:
2645:
2616:
1961:
1564:
1255:
1235:
1038:
1030:
1011:
873:
869:
865:
812:
721:
588:. In mathematics texts this word often refers to the number zero. In a similar vein,
332:
5360:
4627:
1311:
841:
717:
6824:
6664:
6649:
6369:
6344:
6278:
6187:
6153:
5994:
5964:
5911:
5814:
5605:
5378:
5352:
5348:
4891:
4319:
4274:
4117:
4035:
4023:
4007:
3999:
3790:
3778:
3461:
2863:
2684:
2654:
2574:
2559:
1932:
1924:
1818:
1785:
1673:
1529:. This difficulty eventually led him to the convention of using the special symbol
1315:
1304:
1231:
1203:
1128:
1100:
1018:
983:
971:
951:
915:
903:
695:
517:
225:
142:
4633:
La Géométrie: The Geometry of René Descartes with a facsimile of the first edition
3665:
Moving to a greater level of abstraction, the real numbers can be extended to the
987:
520:. By this time (the 7th century) the concept had clearly reached Cambodia as
323:" may signify "a lot" rather than an exact quantity. Though it is now regarded as
6678:
6654:
6569:
6317:
6221:
5878:
5707:
5545:
5538:
5407:
5200:
5193:
4822:
4759:
4672:
4425: â Elements of a field, e.g. real numbers, in the context of linear algebra
4361:
4356:
4290:
4263:
3589:
The real numbers also have an important but highly technical property called the
3488:
3320:
3213:
2964:
2929:
2905:
2881:
2843:
2744:
2609:
2426:
2139:
1973:
1198:
1190:
1104:
1007:
958:
896:
846:
752:). An isolated use of their initial, N, was used in a table of Roman numerals by
516:. He treated 0 as a number and discussed operations involving it, including
276:(and its combinations with real numbers by adding or subtracting its multiples).
177:
173:
138:
55:
1090:
6715:
6634:
6518:
6364:
6354:
6339:
6158:
6026:
5822:
5672:
5526:
5433:
5175:
4377:
4323:
4259:
4101:
4003:
3843:
3777:. If the real part of a complex number is 0, then the number is called an
3774:
3739:
are real numbers. Because of this, complex numbers correspond to points on the
3692:
3688:
3683:
3670:
3666:
3660:
3484:
3175:
2728:
2724:
2710:
2602:
2502:
2350:
2236:
2012:
1948:
1822:
1568:
944:
895:, dating to roughly 300 BC. Of the Indian texts, the most relevant is the
707:
683:
617:
599:
536:
521:
411:
343:
297:
289:
270:
266:
126:
91:
87:
63:
47:
5204:(Braunschweig: Friedrich Vieweg & Sohn, 1872). Subsequently published in:
4895:
1877:
to the theory of primes; in it he proved the infinitude of the primes and the
816:
398: BC) and the earliest known base 10 system dates to 3100 BC in
6818:
6674:
6526:
6492:
6427:
6400:
6309:
5471:
5097:
4903:
4727:
4692:
4286:
4148:
3972:
3863:
3740:
3597:
3366:
3352:
3299:
2343:
1743:
1658:{\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta }
1243:
1050:
950:, who produced a (most likely geometrical) proof of the irrationality of the
899:, which also covers number theory as part of a general study of mathematics.
885:
881:
660:
656:
621:
525:
324:
305:
146:
43:
5694:
4779:
4105:
1211:
between infinite sets. But the next major advance in the theory was made by
663:. Maya arithmetic used base 4 and base 5 written as base 20.
589:
280:
with numbers are done with arithmetical operations, the most familiar being
6720:
6644:
6544:
6390:
6192:
5584:
5299:
4416:
4180:
4165:
3958:
3898:
3744:
3604:, is isomorphic to the real numbers. The real numbers are not, however, an
3487:. All measurements are, by their nature, approximations, and always have a
2552:
1977:
1939:
in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.
1893:
1870:
1840:
1836:
1762:
In the same year, Gauss provided the first generally accepted proof of the
1251:
1212:
1046:
999:
995:
979:
975:
936:
633:
110:
4842:
4187:). Therefore, they are often regarded as numbers by number theorists. The
3927:
the first non-negative odd numbers are {1, 3, 5, 7, ...}. Any even number
6216:
5998:
5668:
5566:
4604:
Mathematics across cultures : the history of non-western mathematics
4449:
4294:
4229:
4221:
4081:
3859:
3847:
3678:
3280:
3245:
2852:
2736:
2388:
2203:
1901:
1844:
1752:
1388:{\displaystyle \left({\sqrt {-1}}\right)^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}
1124:
1120:
1068:
1003:
829:
802:
794:
777:
679:
594:
539:, from an inscription from 683 AD. Early use of zero as a decimal figure.
513:
374:
320:
312:
285:
277:
221:
165:
83:
59:
3949:. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}.
3418:. Another well-known number, proven to be an irrational real number, is
1135:, so there is an uncountably infinite number of transcendental numbers.
6725:
6684:
6536:
6197:
6054:
5334:"Euler's 'mistake'? The radical product rule in historical perspective"
5283:
Darstellung der NĂ€herungswerthe von KettenbrĂŒchen in independenter Form
4857:"Egyptian Mathematical Papyri â Mathematicians of the African Diaspora"
4501:
4472:
4210:
4128:
4109:
3831:
3480:
3472:
3284:
2885:
2859:
2732:
1947:"Number system" redirects here. For systems which express numbers, see
1216:
1154:
940:
815:
that remains in use today. However, in the 12th century in India,
781:
675:
647:
people of south-central Mexico began to use a symbol for zero, a shell
637:
471: in this section. Unsourced material may be challenged and removed.
328:
301:
168:, the notion of number has been extended over the centuries to include
5285:(Erlangen: Eduard Besold, 1873); âââ, "Kettenbruchdeterminanten", in:
2331:
A more complete list of number sets appears in the following diagram.
5476:
5383:
4085:
3766:
3748:
3564:
2740:
1208:
1186:
1132:
845:
numeral. The first use of negative numbers in a European work was by
823:
3858:
is less than 1. In technical terms, the complex numbers lack a
3850:. That is, there is no consistent meaning assignable to saying that
3523:(rounding to 3 decimals), or of any real number greater or equal to
524:, and documentation shows the idea later spreading to China and the
446:
6305:
6236:
6082:
5102:
Mathematics across cultures: the history of non-Western mathematics
4233:
4225:
4018:
3838:
in the complex numbers. Like the reals, the complex numbers form a
3583:
3476:
3370:
3208:
2593:
1742:
The existence of complex numbers was not completely accepted until
1239:
1150:
1034:
947:
850:
745:
569:
281:
181:
79:
1197:âthe general consensus being that only the latter had true value.
744:, was used. These medieval zeros were used by all future medieval
5850:
5773:
3946:
3295:
2897:
2888:). As an example, the negative of 7 is written â7, and
2875:
2839:
2464:
2106:
1296:
1292:
734:, not as a symbol. When division produced 0 as a remainder,
710:
671:
603:
381:
316:
134:
51:
6461:
4709:
Mathematics in society and history : sociological inquiries
3854:
is greater than 1, nor is there any meaning in saying that
876:
notation for rational numbers in mathematical texts such as the
4168:
base provides the best mathematical properties. The set of the
3786:
2944:
2267:
1459:, and was also used in complex number calculations with one of
967:
749:
625:
137:
using a combination of ten fundamental numeric symbols, called
114:
39:
1021:
and higher degree equations was an important development, the
910:
to include calculations of decimal-fraction approximations to
811:
in 628, who used negative numbers to produce the general form
655:
but certainly by 40 BC, which became an integral part of
380:
A tallying system has no concept of place value (as in modern
342:, which consist of various extensions or modifications of the
4927:
4161:
3560:
2847:
1526:
1507:{\displaystyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}
1398:
seemed capriciously inconsistent with the algebraic identity
1060:
907:
864:
It is likely that the concept of fractional numbers dates to
686:. Because it was used alone, not as just a placeholder, this
667:
in 1961 reported a base 4, base 5 "finger" abacus.
648:
644:
399:
113:. More universally, individual numbers can be represented by
4281:. The hyperreals, or nonstandard reals (usually denoted as *
109:, and so forth. Numbers can be represented in language with
5176:"Ueber unendliche, lineare Punktmannichfaltigkeiten", pt. 5
2536:
1158:
753:
500:
426:
150:
121:; for example, "5" is a numeral that represents the number
4312:
to be reinterpreted as true first-order statements about *
3351:
It turns out that these repeating decimals (including the
2743:, i.e. 0 elements, where 0 is thus the smallest
2715:
5748:
4799:, p. 82. Princeton University Press, September 28, 2008.
3889:
is an integer that is "evenly divisible" by two, that is
3695:. The complex numbers consist of all numbers of the form
3608:, because they do not include a solution (often called a
2969:
A rational number is a number that can be expressed as a
2544:
1238:
represents a rigorous method of treating the ideas about
1071:, resulting, with the subsequent contributions of Heine,
327:, belief in a mystical significance of numbers, known as
4183:
and algebraic numbers have many similar properties (see
1732:{\displaystyle \cos \theta +i\sin \theta =e^{i\theta }.}
932:
The earliest known use of irrational numbers was in the
5373:
5013:
4567:
A History of Mathematics: From Mesopotamia to Modernity
4404:
3360:
2951:, and the natural numbers with zero are referred to as
2208:
The limit of a convergent sequence of rational numbers
911:
260:
4606:. Dordrecht: Kluwer Academic. 2000. pp. 410â411.
4191:-adic numbers play an important role in this analogy.
957:
The 16th century brought final European acceptance of
592:(5th century BC) used the null (zero) operator in the
195:
6639:
6629:
6245:
6168:
6129:
6091:
6063:
6035:
6007:
5920:
5887:
5859:
5831:
4931:
The Earth and Its Peoples: A Global History, Volume 1
4030:. Complex numbers which are not algebraic are called
3803:
3704:
3618:
3427:
3382:
3260:
3219:
3132:
3092:
3043:
2911:
2812:
2783:
2761:
2510:
2472:
2434:
2396:
2358:
2279:
2218:
2185:
2121:
2088:
2054:
2025:
1994:
1685:
1584:
1539:
1476:
1407:
1327:
1261:
A modern geometrical version of infinity is given by
1167:
233:
189:
169:
122:
106:
102:
98:
94:
5236:
Acta Academiae Scientiarum Imperialis Petropolitanae
4923:
4921:
4401: â Measurable property of a material or system
4337:
4164:
is used for the digits: any base is possible, but a
4013:
3579:. Therefore, the result is usually rounded to 5.61.
1441:{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}},}
5289:(Erlangen: Eduard Besold, 1875), c. 6, pp. 156â186.
4797:, Chapter II.1, "The Origins of Modern Mathematics"
4413: â Method for representing or encoding numbers
4258:, the natural numbers have been generalized to the
2735:
and other mathematicians started including 0 (
1084:
6253:
6176:
6137:
6099:
6071:
6043:
6015:
5928:
5895:
5867:
5839:
5768:
5287:Lehrbuch der Determinanten-Theorie: FĂŒr Studirende
4395: â Universal and unchanging physical quantity
3811:
3720:
3643:
3449:
3400:
3271:
3227:
3163:
3118:
3072:
2919:
2827:
2798:
2769:
2522:
2484:
2446:
2408:
2370:
2319:
2226:
2193:
2129:
2096:
2069:
2040:
2002:
1843:, which were expressed as geometrical entities by
1746:described the geometrical interpretation in 1799.
1731:
1657:
1552:
1506:
1440:
1387:
1175:
1115:proved in 1882 that Ï is transcendental. Finally,
798:, saying that the equation gave an absurd result.
251:
212:
5744:"Cuddling With 9, Smooching With 8, Winking At 7"
5589:Mathematical Thought from Ancient to Modern Times
4918:
2940:with the operations addition and multiplication.
1291:, when he considered the volume of an impossible
1183:is often used to represent an infinite quantity.
6816:
5466:
5124:
4973:. Sunsite.utk.edu. 26 April 1999. Archived from
4034:. The algebraic numbers that are solutions of a
4019:Algebraic, irrational and transcendental numbers
3450:{\displaystyle {\sqrt {2}}=1.41421356237\dots ,}
387:The first known system with place value was the
5270:Journal fĂŒr die reine und angewandte Mathematik
5223:Journal fĂŒr die reine und angewandte Mathematik
5163:Journal fĂŒr die reine und angewandte Mathematik
2111:..., â5, â4, â3, â2, â1, 0, 1, 2, 3, 4, 5, ...
716:Another true zero was used in tables alongside
5307:Interactive Mathematics Miscellany and Puzzles
4232:in addition to not being commutative, and the
4038:equation with integer coefficients are called
1149:The earliest known conception of mathematical
1138:
1033:1824) showed that they could not be solved by
970:. In 1872, the publication of the theories of
849:during the 15th century. He used them as
756:or a colleague about 725, a true zero symbol.
694:use of a true zero in the Old World. In later
579:
573:
6477:
5789:
4374: â Fixed number that has received a name
3869:
706:), the Hellenistic zero had morphed into the
133:, which allows for the representation of any
5701:
4419: â Number divisible only by 1 or itself
4134:
3993:
3401:{\displaystyle \pi =3.14159265358979\dots ,}
2936: 'number'. The set of integers forms a
1858:. This eventually led to the concept of the
1215:; in 1895 he published a book about his new
1076:
853:, but referred to them as "absurd numbers".
213:{\displaystyle \left({\tfrac {1}{2}}\right)}
5693:. BBC Radio 4. 9 March 2006. Archived from
5104:. Kluwer Academic Publishers. p. 451.
4881:
4052:constructions with straightedge and compass
2017:0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ...
835:
827:
735:
725:
6779:
6769:
6484:
6470:
6446:
5796:
5782:
5014:Staszkow, Ronald; Robert Bradshaw (2004).
4740:: CS1 maint: location missing publisher (
4666:
4664:
2514:
2476:
2438:
2400:
2362:
1980:. The main number systems are as follows:
682:numeral system otherwise using alphabetic
308:, the study of the properties of numbers.
6247:
6170:
6131:
6093:
6065:
6037:
6009:
5922:
5889:
5861:
5833:
5614:to *56, Cambridge University Press, 1910.
4626:
3846:, but unlike the real numbers, it is not
3826:asserts that the complex numbers form an
3805:
3705:
3359:. A famous irrational real number is the
3262:
3221:
3202:. The symbol for the rational numbers is
2913:
2815:
2786:
2763:
2516:
2478:
2440:
2402:
2364:
2313:
2305:
2297:
2289:
2281:
2220:
2187:
2123:
2090:
2057:
2028:
1996:
1451:which is valid for positive real numbers
1041:(all solutions to polynomial equations).
773:The Nine Chapters on the Mathematical Art
487:Learn how and when to remove this message
6440:
5769:Online Encyclopedia of Integer Sequences
5718:
5533:, New York, The Macmillan Company, 1930.
5331:
5137:. Harvard University Press. p. 83.
4820:
4670:
4045:
3793:. The symbol for the complex numbers is
3164:{\displaystyle {a\times d}={c\times b}.}
3073:{\displaystyle {1 \over 2}={2 \over 4}.}
2846:: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The
2714:
1835:. This generalization is largely due to
530:
252:{\displaystyle \left({\sqrt {2}}\right)}
38:
5639:
5553:Introduction to Mathematical Structures
4995:
4705:
4661:
4563:
4240:, neither associative nor commutative.
4194:
4050:Motivated by the classical problems of
3874:
3466:( 1 trillion = 10 = 1,000,000,000,000 )
3119:{\displaystyle {a \over b}={c \over d}}
784:referred to the equation equivalent to
362:
14:
6817:
5442:. Courier Dover Publications. p.
5432:
5313:from the original on 23 September 2010
5135:Harvard Studies in Classical Philology
4795:The Princeton Companion to Mathematics
4753:
4751:
4699:
4269:
4243:
3471:Not only these prominent examples but
2751:for the set of all natural numbers is
2731:.) However, in the 19th century,
1942:
1045:(1832) linked polynomial equations to
157:is not clearly distinguished from the
6465:
5777:
5721:"What's the World's Favorite Number?"
5491:
5467:
5414:from the original on 13 December 2019
5374:
5298:
5096:
5038:
4061:
2958:
1821:) of complex numbers derive from the
921:
503:dates to AD 628, and appeared in the
416:Numbers should be distinguished from
269:which extend the real numbers with a
5756:from the original on 6 November 2018
4948:from the original on 28 January 2017
4827:( ed.). New York: McGraw-Hill.
4584:from the original on 4 February 2019
4559:
4557:
3654:
2719:The natural numbers, starting with 1
1788:) or rational numbers. His student,
651:, in the New World, possibly by the
602:for the Sanskrit language (also see
469:adding citations to reliable sources
440:
5960:Set-theoretically definable numbers
5658:
5555:, Harcourt Brace Javanovich, 1989,
5206:âââ, Gesammelte mathematische Werke
5194:Stetigkeit & irrationale Zahlen
5068:"Classical Greek culture (article)"
5046:. Dover Publications. p. 259.
4757:
4748:
4542:from the original on 26 August 2017
4482:from the original on 4 October 2018
3250:The symbol for the real numbers is
2928:. Here the letter Z comes from
2260:is a formal square root of â1
1219:, introducing, among other things,
859:
759:
24:
5803:
5714:from the original on 8 April 2022.
5504:from the original on 5 August 2020
5272:, No. 56 (Jan. 1859): 87â99 at 97.
5159:"Die Elemente der Functionenlehre"
5016:The Mathematical Palette (3rd ed.)
4529:
4407: â Number, approximately 3.14
3952:
3905:may be constructed by the formula
3891:divisible by two without remainder
2723:The most familiar numbers are the
2704:
2266:Each of these number systems is a
1272:
499:The first known documented use of
369:History of ancient numeral systems
90:. The most basic examples are the
25:
6851:
6491:
5719:Krulwich, Robert (22 July 2011).
5633:
5621:, Oxford University Press, 2015,
5591:, Oxford University Press, 1990.
5539:What's special about this number?
5341:The American Mathematical Monthly
4934:. Cengage Learning. p. 192.
4863:from the original on 7 April 2015
4554:
4511:from the original on 30 July 2022
4236:, in which multiplication is not
4228:, in which multiplication is not
4220:, in which multiplication is not
4139:
4014:Subclasses of the complex numbers
3984:fundamental theorem of arithmetic
1879:fundamental theorem of arithmetic
1563:The 18th century saw the work of
1085:Transcendental numbers and reals
429:, which was developed by ancient
304:, a term which may also refer to
300:. Their study or usage is called
27:Used to count, measure, and label
6798:
6788:
6778:
6768:
6759:
6758:
6445:
5731:from the original on 18 May 2021
5702:Robin Wilson (7 November 2007).
5003:. Austin, Texas: self published.
4340:
3864:compatible with field operations
3834:with complex coefficients has a
2828:{\displaystyle \mathbb {N} _{1}}
2799:{\displaystyle \mathbb {N} _{0}}
2070:{\displaystyle \mathbb {N} _{1}}
2041:{\displaystyle \mathbb {N} _{0}}
1865:
1075:, and GĂŒnther, in the theory of
834:, 1202) and later as losses (in
445:
5485:
5460:
5426:
5392:
5367:
5325:
5292:
5275:
5254:
5241:
5228:
5211:
5185:
5168:
5151:
5118:
5090:
5078:from the original on 4 May 2022
5060:
5032:
5007:
4989:
4963:
4875:
4849:
4814:
4786:
4088:which, given a positive number
3239:
2523:{\displaystyle :\;\mathbb {N} }
2485:{\displaystyle :\;\mathbb {Z} }
2447:{\displaystyle :\;\mathbb {Q} }
2409:{\displaystyle :\;\mathbb {R} }
2371:{\displaystyle :\;\mathbb {C} }
1960:Numbers can be classified into
1560:to guard against this mistake.
564:to denote zero balance in
456:needs additional citations for
6537:analytic theory of L-functions
6515:non-abelian class field theory
5353:10.1080/00029890.2007.11920416
4620:
4596:
4570:. OUP Oxford. pp. 85â88.
4523:
4494:
4465:
4442:
3824:fundamental theorem of algebra
3290:Most real numbers can only be
2998:parts of a whole divided into
1817:). Other such classes (called
1764:fundamental theorem of algebra
1613:
1585:
966:remained almost dormant since
792:(the solution is negative) in
13:
1:
6294:Plane-based geometric algebra
5690:In Our Time: Negative Numbers
5520:
5332:MartĂnez, Alberto A. (2007).
5044:History of Modern Mathematics
4761:Number theory and its history
4564:Hodgkin, Luke (2 June 2005).
4304:. This principle allows true
4092:as input, produces the first
3272:{\displaystyle \mathbb {R} .}
928:History of irrational numbers
888:. The best known of these is
713:(otherwise meaning 70).
392:
6561:Transcendental number theory
6254:{\displaystyle \mathbb {S} }
6177:{\displaystyle \mathbb {C} }
6138:{\displaystyle \mathbb {R} }
6100:{\displaystyle \mathbb {O} }
6072:{\displaystyle \mathbb {H} }
6044:{\displaystyle \mathbb {C} }
6016:{\displaystyle \mathbb {R} }
5929:{\displaystyle \mathbb {A} }
5896:{\displaystyle \mathbb {Q} }
5868:{\displaystyle \mathbb {Z} }
5840:{\displaystyle \mathbb {N} }
5251:(Kjoebenhavn: 1855), p. 106.
4821:Marshack, Alexander (1971).
4671:Gilsdorf, Thomas E. (2012).
3812:{\displaystyle \mathbb {C} }
3612:) to the algebraic equation
3228:{\displaystyle \mathbb {Q} }
2920:{\displaystyle \mathbb {Z} }
2770:{\displaystyle \mathbb {N} }
2227:{\displaystyle \mathbb {C} }
2194:{\displaystyle \mathbb {R} }
2130:{\displaystyle \mathbb {Q} }
2097:{\displaystyle \mathbb {Z} }
2003:{\displaystyle \mathbb {N} }
1937:Charles de la Vallée-Poussin
1553:{\displaystyle {\sqrt {-1}}}
1525:are negative even bedeviled
1139:Infinity and infinitesimals
1049:giving rise to the field of
986:was brought about. In 1869,
7:
6784:List of recreational topics
6556:Computational number theory
6541:probabilistic number theory
5647:Encyclopedia of Mathematics
5182:, 21, 4 (1883â12): 545â591.
5018:. Brooks Cole. p. 41.
4507:. Oxford University Press.
4478:. Oxford University Press.
4333:
2943:The natural numbers form a
2869:
1119:showed that the set of all
943:, more specifically to the
766:History of negative numbers
720:by 525 (first known use by
584:to refer to the concept of
574:
422:HinduâArabic numeral system
405:
131:HinduâArabic numeral system
10:
6856:
5619:A Brief History of Numbers
5262:"Einige Eigenschaften der
5238:, 1779, 1 (1779): 162â187.
5225:, No. 101 (1887): 337â355.
4254:For dealing with infinite
4247:
4198:
4146:
4084:such that there exists an
4065:
3967:, often shortened to just
3956:
3878:
3870:Subclasses of the integers
3828:algebraically closed field
3658:
3606:algebraically closed field
3243:
2962:
2873:
2708:
1953:
1946:
1279:History of complex numbers
1276:
1209:one-to-one correspondences
1176:{\displaystyle {\text{â}}}
1142:
1088:
974:(by his pupil E. Kossak),
925:
878:Rhind Mathematical Papyrus
763:
409:
366:
357:
149:), and for codes (as with
29:
6754:
6736:Diophantine approximation
6708:
6695:Chinese remainder theorem
6617:
6499:
6436:
6378:
6304:
6284:Algebra of physical space
6206:
6114:
5985:
5812:
5165:, No. 74 (1872): 172â188.
4896:10.1017/S0003598X00092541
4135:Extensions of the concept
4006:. For more examples, see
3994:Other classes of integers
3644:{\displaystyle x^{2}+1=0}
3596:It can be shown that any
1856:essential singular points
1784:are integers (now called
1301:NiccolĂČ Fontana Tartaglia
1099:was first established by
598:, an early example of an
389:Mesopotamian base 60
145:), for ordering (as with
6580:Arithmetic combinatorics
6340:Extended complex numbers
6323:Extended natural numbers
4706:Restivo, Sal P. (1992).
4677:. Hoboken, N.J.: Wiley.
4435:
4367:List of types of numbers
4293:of the ordered field of
4177:algebraic function field
3610:square root of minus one
3373:. When pi is written as
1956:List of types of numbers
1852:Victor Alexandre Puiseux
1078:Kettenbruchdeterminanten
1017:The search for roots of
6551:Geometric number theory
6507:Algebraic number theory
5704:"4000 Years of Numbers"
5640:Nechaev, V.I. (2001) .
5219:"Ueber den Zahlbegriff"
4429:Subitizing and counting
4116:that contains the real
3916:for a suitable integer
3687:, a symbol assigned by
3681:of â1, denoted by
3539:and strictly less than
3507:and strictly less than
1887:greatest common divisor
566:double entry accounting
509:, the main work of the
436:
32:Number (disambiguation)
6670:Transcendental numbers
6584:additive number theory
6533:Analytic number theory
6396:Transcendental numbers
6255:
6232:Hyperbolic quaternions
6178:
6139:
6101:
6073:
6045:
6017:
5930:
5897:
5869:
5841:
5602:Alfred North Whitehead
5131:D.R. Shackleton Bailey
4502:"numeral, adj. and n."
4218:William Rowan Hamilton
4185:Function field analogy
4032:transcendental numbers
3813:
3722:
3721:{\displaystyle \,a+bi}
3645:
3451:
3402:
3273:
3229:
3165:
3120:
3074:
2921:
2829:
2800:
2771:
2720:
2610:Dyadic (finite binary)
2524:
2486:
2448:
2410:
2372:
2321:
2228:
2195:
2131:
2098:
2071:
2042:
2004:
1931:was finally proved by
1860:extended complex plane
1733:
1659:
1554:
1517:in the case when both
1508:
1442:
1389:
1248:infinitesimal calculus
1207:discussed the idea of
1177:
1111:is transcendental and
1097:transcendental numbers
1077:
948:Hippasus of Metapontum
836:
828:
736:
726:
616:Records show that the
580:
568:. Indian texts used a
548:negative number". The
540:
253:
214:
153:). In common usage, a
67:
6741:Irrationality measure
6731:Diophantine equations
6574:HodgeâArakelov theory
6328:Extended real numbers
6256:
6179:
6149:Split-complex numbers
6140:
6102:
6074:
6046:
6018:
5931:
5907:Constructible numbers
5898:
5870:
5842:
5611:Principia Mathematica
5180:Mathematische Annalen
4758:Ore, Ăystein (1988).
4372:Mathematical constant
4279:non-standard analysis
4098:Ό-recursive functions
4056:constructible numbers
4046:Constructible numbers
3986:. A proof appears in
3977:Goldbach's conjecture
3830:, meaning that every
3814:
3781:or is referred to as
3723:
3646:
3452:
3403:
3369:of any circle to its
3298:numerals, in which a
3274:
3230:
3166:
3121:
3075:
2953:non-negative integers
2922:
2830:
2801:
2772:
2718:
2525:
2487:
2449:
2411:
2373:
2322:
2256:are real numbers and
2229:
2196:
2132:
2099:
2072:
2043:
2005:
1909:Adrien-Marie Legendre
1898:Sieve of Eratosthenes
1831:for higher values of
1806:is a complex root of
1734:
1660:
1555:
1509:
1443:
1390:
1277:Further information:
1178:
1143:Further information:
1089:Further information:
1065:Joseph Louis Lagrange
926:Further information:
808:BrÄhmasphuáčasiddhÄnta
764:Further information:
611:BrÄhmasphuáčasiddhÄnta
550:BrÄhmasphuáčasiddhÄnta
545:BrÄhmasphuáčasiddhÄnta
534:
506:BrÄhmasphuáčasiddhÄnta
431:Indian mathematicians
315:is often regarded as
254:
215:
42:
6840:Mathematical objects
6700:Arithmetic functions
6566:Diophantine geometry
6360:Supernatural numbers
6270:Multicomplex numbers
6243:
6227:Dual-complex numbers
6166:
6127:
6089:
6061:
6033:
6005:
5987:Composition algebras
5955:Arithmetical numbers
5918:
5885:
5857:
5829:
5439:Axiomatic Set Theory
4859:. Math.buffalo.edu.
4792:GouvĂȘa, Fernando Q.
4532:"The Origin of Zero"
4423:Scalar (mathematics)
4216:, introduced by Sir
4207:hypercomplex numbers
4195:Hypercomplex numbers
3881:Even and odd numbers
3875:Even and odd numbers
3801:
3751:. In the expression
3702:
3616:
3563:. If the sides of a
3425:
3380:
3353:repetition of zeroes
3258:
3217:
3130:
3090:
3041:
3034:are equal, that is:
2909:
2810:
2781:
2759:
2655:Algebraic irrational
2508:
2470:
2432:
2394:
2356:
2277:
2216:
2183:
2119:
2086:
2077:are sometimes used.
2052:
2023:
1992:
1984:Main number systems
1929:prime number theorem
1913:prime number theorem
1881:, and presented the
1839:, who also invented
1757:De algebra tractatus
1748:Carl Friedrich Gauss
1683:
1582:
1537:
1474:
1405:
1325:
1225:continuum hypothesis
1223:and formulating the
1165:
1125:uncountably infinite
1107:proved in 1873 that
1023:AbelâRuffini theorem
700:Syntaxis Mathematica
511:Indian mathematician
465:improve this article
363:First use of numbers
340:hypercomplex numbers
231:
187:
161:that it represents.
135:non-negative integer
30:For other uses, see
6746:Continued fractions
6609:Arithmetic dynamics
6604:Arithmetic topology
6598:P-adic Hodge theory
6590:Arithmetic geometry
6523:IwasawaâTate theory
6265:Split-biquaternions
5977:Eisenstein integers
5940:Closed-form numbers
5659:Tallant, Jonathan.
5494:"Repeating Decimal"
5492:Weisstein, Eric W.
5404:Merriam-Webster.com
5040:Smith, David Eugene
4764:. New York: Dover.
4536:Scientific American
4411:Positional notation
4388:Orders of magnitude
4383:Numerical cognition
4270:Nonstandard numbers
4244:Transfinite numbers
4209:. They include the
4201:hypercomplex number
4175:The elements of an
3901:".) Any odd number
3365:, the ratio of the
2974:them. The fraction
2749:mathematical symbol
2346:
1985:
1943:Main classification
1917:Goldbach conjecture
1883:Euclidean algorithm
1815:Eisenstein integers
1792:, studied the type
1790:Gotthold Eisenstein
1573:De Moivre's formula
1285:Heron of Alexandria
1263:projective geometry
1221:transfinite numbers
1145:History of infinity
1127:but the set of all
1057:Continued fractions
992:Salvatore Pincherle
922:Irrational numbers
698:manuscripts of his
558:place-value systems
76:mathematical object
6690:Modular arithmetic
6660:Irrational numbers
6594:anabelian geometry
6511:class field theory
6423:Profinite integers
6386:Irrational numbers
6251:
6174:
6135:
6097:
6069:
6041:
6013:
5970:Gaussian rationals
5950:Computable numbers
5926:
5893:
5865:
5837:
5752:. 21 August 2011.
5661:"Do Numbers Exist"
5573:, Springer, 1974,
5544:2018-02-23 at the
5469:Weisstein, Eric W.
5376:Weisstein, Eric W.
5303:"What's a number?"
5281:Siegmund GĂŒnther,
5199:2021-07-09 at the
5191:Richard Dedekind,
5001:Arithmetic in Maya
4997:SĂĄnchez, George I.
4977:on 12 January 2012
4638:Dover Publications
4348:Mathematics portal
4302:transfer principle
4300:and satisfies the
4250:transfinite number
4112:, and thus form a
4062:Computable numbers
4040:algebraic integers
4028:irrational numbers
3809:
3761:, the real number
3718:
3641:
3557:significant digits
3447:
3398:
3269:
3254:, also written as
3225:
3161:
3116:
3070:
2917:
2825:
2796:
2767:
2721:
2520:
2482:
2444:
2406:
2368:
2342:
2317:
2224:
2191:
2127:
2094:
2067:
2038:
2000:
1983:
1921:Riemann hypothesis
1729:
1655:
1550:
1504:
1438:
1385:
1195:potential infinity
1173:
1133:countably infinite
724:), but as a word,
541:
535:The number 605 in
249:
210:
204:
68:
6812:
6811:
6709:Advanced concepts
6665:Algebraic numbers
6650:Composite numbers
6459:
6458:
6370:Superreal numbers
6350:Levi-Civita field
6345:Hyperreal numbers
6289:Spacetime algebra
6275:Geometric algebra
6188:Bicomplex numbers
6154:Split-quaternions
5995:Division algebras
5965:Gaussian integers
5912:Algebraic numbers
5815:definable numbers
5627:978-0-19-870259-7
5551:Steven Galovich,
5498:Wolfram MathWorld
5266:schen Funktionen"
4941:978-1-4390-8474-8
4805:978-0-691-11880-2
4719:978-94-011-2944-2
4684:978-1-118-19416-4
4577:978-0-19-152383-0
4399:Physical quantity
4393:Physical constant
4308:statements about
4289:that is a proper
4275:Hyperreal numbers
4118:algebraic numbers
4114:real closed field
4074:computable number
4068:Computable number
4024:Algebraic numbers
4000:Fibonacci numbers
3988:Euclid's Elements
3691:, and called the
3591:least upper bound
3433:
3326:repeating decimal
3287:, e.g. â123.456.
3114:
3101:
3065:
3052:
2949:positive integers
2702:
2701:
2698:
2697:
2694:
2693:
2690:
2689:
2679:
2678:
2675:
2674:
2671:
2670:
2667:
2666:
2636:
2635:
2632:
2631:
2628:
2627:
2624:
2623:
2617:Repeating decimal
2584:
2583:
2580:
2579:
2575:Negative integers
2569:
2568:
2565:
2564:
2560:Composite numbers
2264:
2263:
2171:are integers and
1819:cyclotomic fields
1786:Gaussian integers
1565:Abraham de Moivre
1548:
1502:
1501:
1487:
1486:
1433:
1420:
1413:
1374:
1364:
1341:
1236:hyperreal numbers
1171:
1129:algebraic numbers
1095:The existence of
1039:algebraic numbers
1006:, separating all
1002:in the system of
904:decimal fractions
874:Egyptian fraction
870:Ancient Egyptians
866:prehistoric times
860:Rational numbers
813:quadratic formula
760:Negative numbers
722:Dionysius Exiguus
665:George I. SĂĄnchez
600:algebraic grammar
497:
496:
489:
333:Greek mathematics
243:
203:
143:telephone numbers
16:(Redirected from
6847:
6802:
6792:
6782:
6781:
6772:
6771:
6762:
6761:
6655:Rational numbers
6486:
6479:
6472:
6463:
6462:
6449:
6448:
6416:
6406:
6318:Cardinal numbers
6279:Clifford algebra
6260:
6258:
6257:
6252:
6250:
6222:Dual quaternions
6183:
6181:
6180:
6175:
6173:
6144:
6142:
6141:
6136:
6134:
6106:
6104:
6103:
6098:
6096:
6078:
6076:
6075:
6070:
6068:
6050:
6048:
6047:
6042:
6040:
6022:
6020:
6019:
6014:
6012:
5935:
5933:
5932:
5927:
5925:
5902:
5900:
5899:
5894:
5892:
5879:Rational numbers
5874:
5872:
5871:
5866:
5864:
5846:
5844:
5843:
5838:
5836:
5798:
5791:
5784:
5775:
5774:
5765:
5763:
5761:
5740:
5738:
5736:
5715:
5698:
5684:
5682:
5680:
5671:. Archived from
5655:
5606:Bertrand Russell
5571:Naive Set Theory
5536:Erich Friedman,
5514:
5513:
5511:
5509:
5489:
5483:
5482:
5481:
5464:
5458:
5457:
5430:
5424:
5423:
5421:
5419:
5400:"natural number"
5396:
5390:
5389:
5388:
5379:"Natural Number"
5371:
5365:
5364:
5338:
5329:
5323:
5322:
5320:
5318:
5296:
5290:
5279:
5273:
5258:
5252:
5245:
5239:
5232:
5226:
5215:
5209:
5189:
5183:
5172:
5166:
5155:
5149:
5148:
5122:
5116:
5115:
5094:
5088:
5087:
5085:
5083:
5064:
5058:
5057:
5036:
5030:
5029:
5011:
5005:
5004:
4993:
4987:
4986:
4984:
4982:
4967:
4961:
4960:
4955:
4953:
4925:
4916:
4915:
4879:
4873:
4872:
4870:
4868:
4853:
4847:
4846:
4818:
4812:
4790:
4784:
4783:
4755:
4746:
4745:
4739:
4731:
4703:
4697:
4696:
4668:
4659:
4658:
4656:
4654:
4624:
4618:
4617:
4600:
4594:
4593:
4591:
4589:
4561:
4552:
4551:
4549:
4547:
4527:
4521:
4520:
4518:
4516:
4498:
4492:
4491:
4489:
4487:
4469:
4457:
4446:
4350:
4345:
4344:
4264:cardinal numbers
4078:recursive number
4076:, also known as
4036:monic polynomial
4008:Integer sequence
3940:
3926:
3920:. Starting with
3915:
3818:
3816:
3815:
3810:
3808:
3791:Gaussian integer
3783:purely imaginary
3779:imaginary number
3760:
3727:
3725:
3724:
3719:
3650:
3648:
3647:
3642:
3628:
3627:
3600:, which is also
3574:
3570:
3554:
3552:
3551:
3548:
3545:
3538:
3536:
3535:
3532:
3529:
3522:
3520:
3519:
3516:
3513:
3506:
3504:
3503:
3500:
3497:
3467:
3462:square root of 2
3456:
3454:
3453:
3448:
3434:
3429:
3415:
3407:
3405:
3404:
3399:
3390:3.14159265358979
3363:
3347:
3343:
3341:
3340:
3337:
3334:
3318:
3316:
3315:
3312:
3309:
3278:
3276:
3275:
3270:
3265:
3234:
3232:
3231:
3226:
3224:
3212:), also written
3201:
3199:
3198:
3195:
3192:
3182:is greater than
3170:
3168:
3167:
3162:
3157:
3143:
3125:
3123:
3122:
3117:
3115:
3107:
3102:
3094:
3079:
3077:
3076:
3071:
3066:
3058:
3053:
3045:
3033:
3031:
3030:
3027:
3024:
3017:
3015:
3014:
3011:
3008:
2993:
2991:
2990:
2985:
2982:
2959:Rational numbers
2926:
2924:
2923:
2918:
2916:
2891:
2864:Peano Arithmetic
2834:
2832:
2831:
2826:
2824:
2823:
2818:
2805:
2803:
2802:
2797:
2795:
2794:
2789:
2777:, and sometimes
2776:
2774:
2773:
2768:
2766:
2651:
2650:
2642:
2641:
2599:
2598:
2590:
2589:
2533:
2532:
2529:
2527:
2526:
2521:
2519:
2499:
2498:
2495:
2494:
2491:
2489:
2488:
2483:
2481:
2461:
2460:
2457:
2456:
2453:
2451:
2450:
2445:
2443:
2423:
2422:
2419:
2418:
2415:
2413:
2412:
2407:
2405:
2385:
2384:
2381:
2380:
2377:
2375:
2374:
2369:
2367:
2347:
2341:
2338:
2337:
2334:
2333:
2326:
2324:
2323:
2318:
2316:
2308:
2300:
2292:
2284:
2233:
2231:
2230:
2225:
2223:
2200:
2198:
2197:
2192:
2190:
2162:
2160:
2159:
2154:
2151:
2140:Rational numbers
2136:
2134:
2133:
2128:
2126:
2103:
2101:
2100:
2095:
2093:
2076:
2074:
2073:
2068:
2066:
2065:
2060:
2047:
2045:
2044:
2039:
2037:
2036:
2031:
2009:
2007:
2006:
2001:
1999:
1986:
1982:
1933:Jacques Hadamard
1925:Bernhard Riemann
1923:, formulated by
1911:conjectured the
1904:and later eras.
1889:of two numbers.
1885:for finding the
1830:
1812:
1801:
1775:
1738:
1736:
1735:
1730:
1725:
1724:
1676:(1748) gave us:
1674:complex analysis
1664:
1662:
1661:
1656:
1621:
1620:
1559:
1557:
1556:
1551:
1549:
1541:
1513:
1511:
1510:
1505:
1503:
1494:
1493:
1488:
1482:
1478:
1447:
1445:
1444:
1439:
1434:
1426:
1421:
1416:
1414:
1409:
1394:
1392:
1391:
1386:
1375:
1367:
1365:
1357:
1352:
1351:
1346:
1342:
1334:
1316:imaginary number
1305:Gerolamo Cardano
1290:
1273:Complex numbers
1232:Abraham Robinson
1204:Two New Sciences
1182:
1180:
1179:
1174:
1172:
1169:
1080:
1008:rational numbers
984:Richard Dedekind
972:Karl Weierstrass
952:square root of 2
916:square root of 2
839:
833:
791:
748:(calculators of
739:
729:
688:Hellenistic zero
674:, influenced by
654:
583:
577:
492:
485:
481:
478:
472:
449:
441:
397:
394:
274:
263:
258:
256:
255:
250:
248:
244:
239:
226:square root of 2
219:
217:
216:
211:
209:
205:
196:
178:rational numbers
174:negative numbers
56:rational numbers
21:
6855:
6854:
6850:
6849:
6848:
6846:
6845:
6844:
6815:
6814:
6813:
6808:
6750:
6716:Quadratic forms
6704:
6679:P-adic analysis
6635:Natural numbers
6613:
6570:Arakelov theory
6495:
6490:
6460:
6455:
6432:
6411:
6401:
6374:
6365:Surreal numbers
6355:Ordinal numbers
6300:
6246:
6244:
6241:
6240:
6202:
6169:
6167:
6164:
6163:
6161:
6159:Split-octonions
6130:
6128:
6125:
6124:
6116:
6110:
6092:
6090:
6087:
6086:
6064:
6062:
6059:
6058:
6036:
6034:
6031:
6030:
6027:Complex numbers
6008:
6006:
6003:
6002:
5981:
5921:
5919:
5916:
5915:
5888:
5886:
5883:
5882:
5860:
5858:
5855:
5854:
5832:
5830:
5827:
5826:
5823:Natural numbers
5808:
5802:
5759:
5757:
5742:
5734:
5732:
5708:Gresham College
5697:on 31 May 2022.
5687:
5678:
5676:
5675:on 8 March 2016
5636:
5546:Wayback Machine
5523:
5518:
5517:
5507:
5505:
5490:
5486:
5465:
5461:
5454:
5434:Suppes, Patrick
5431:
5427:
5417:
5415:
5408:Merriam-Webster
5398:
5397:
5393:
5372:
5368:
5336:
5330:
5326:
5316:
5314:
5297:
5293:
5280:
5276:
5259:
5255:
5246:
5242:
5233:
5229:
5216:
5212:
5201:Wayback Machine
5190:
5186:
5173:
5169:
5156:
5152:
5145:
5123:
5119:
5112:
5095:
5091:
5081:
5079:
5066:
5065:
5061:
5054:
5037:
5033:
5026:
5012:
5008:
4994:
4990:
4980:
4978:
4969:
4968:
4964:
4951:
4949:
4942:
4926:
4919:
4890:(297): 485â96.
4880:
4876:
4866:
4864:
4855:
4854:
4850:
4835:
4819:
4815:
4791:
4787:
4772:
4756:
4749:
4733:
4732:
4720:
4704:
4700:
4685:
4669:
4662:
4652:
4650:
4648:
4628:Descartes, René
4625:
4621:
4614:
4602:
4601:
4597:
4587:
4585:
4578:
4562:
4555:
4545:
4543:
4528:
4524:
4514:
4512:
4500:
4499:
4495:
4485:
4483:
4471:
4470:
4466:
4461:
4460:
4447:
4443:
4438:
4433:
4378:Complex numbers
4362:List of numbers
4357:Concrete number
4346:
4339:
4336:
4324:surreal numbers
4272:
4260:ordinal numbers
4252:
4246:
4203:
4197:
4154:
4145:
4137:
4102:Turing machines
4070:
4064:
4048:
4021:
4016:
4004:perfect numbers
3996:
3961:
3955:
3932:
3921:
3906:
3883:
3877:
3872:
3804:
3802:
3799:
3798:
3752:
3703:
3700:
3699:
3667:complex numbers
3663:
3657:
3655:Complex numbers
3623:
3619:
3617:
3614:
3613:
3572:
3568:
3549:
3546:
3543:
3542:
3540:
3533:
3530:
3527:
3526:
3524:
3517:
3514:
3511:
3510:
3508:
3501:
3498:
3495:
3494:
3492:
3489:margin of error
3465:
3428:
3426:
3423:
3422:
3413:
3381:
3378:
3377:
3361:
3345:
3338:
3335:
3332:
3331:
3329:
3321:fractional part
3313:
3310:
3307:
3306:
3304:
3261:
3259:
3256:
3255:
3248:
3242:
3220:
3218:
3215:
3214:
3196:
3193:
3190:
3189:
3187:
3147:
3133:
3131:
3128:
3127:
3126:if and only if
3106:
3093:
3091:
3088:
3087:
3057:
3044:
3042:
3039:
3038:
3028:
3025:
3022:
3021:
3019:
3012:
3009:
3006:
3005:
3003:
2986:
2983:
2978:
2977:
2975:
2967:
2965:Rational number
2961:
2912:
2910:
2907:
2906:
2889:
2878:
2872:
2819:
2814:
2813:
2811:
2808:
2807:
2790:
2785:
2784:
2782:
2779:
2778:
2762:
2760:
2757:
2756:
2755:, also written
2745:cardinal number
2725:natural numbers
2713:
2707:
2705:Natural numbers
2515:
2509:
2506:
2505:
2477:
2471:
2468:
2467:
2439:
2433:
2430:
2429:
2401:
2395:
2392:
2391:
2363:
2357:
2354:
2353:
2312:
2304:
2296:
2288:
2280:
2278:
2275:
2274:
2237:Complex numbers
2219:
2217:
2214:
2213:
2186:
2184:
2181:
2180:
2155:
2152:
2147:
2146:
2144:
2122:
2120:
2117:
2116:
2089:
2087:
2084:
2083:
2061:
2056:
2055:
2053:
2050:
2049:
2032:
2027:
2026:
2024:
2021:
2020:
2018:
2013:Natural numbers
1995:
1993:
1990:
1989:
1974:natural numbers
1958:
1952:
1945:
1868:
1825:
1807:
1793:
1767:
1717:
1713:
1684:
1681:
1680:
1670:Euler's formula
1616:
1612:
1583:
1580:
1579:
1575:(1730) states:
1540:
1538:
1535:
1534:
1492:
1477:
1475:
1472:
1471:
1425:
1415:
1408:
1406:
1403:
1402:
1366:
1356:
1347:
1333:
1329:
1328:
1326:
1323:
1322:
1288:
1281:
1275:
1199:Galileo Galilei
1191:actual infinity
1168:
1166:
1163:
1162:
1153:appears in the
1147:
1141:
1093:
1087:
930:
924:
902:The concept of
897:Sthananga Sutra
862:
847:Nicolas Chuquet
785:
768:
762:
740:, also meaning
652:
640:was a number.)
493:
482:
476:
473:
462:
450:
439:
433:around 500 AD.
414:
408:
395:
371:
365:
360:
272:
271:square root of
267:complex numbers
261:
238:
234:
232:
229:
228:
194:
190:
188:
185:
184:
92:natural numbers
64:complex numbers
48:natural numbers
35:
28:
23:
22:
15:
12:
11:
5:
6853:
6843:
6842:
6837:
6832:
6827:
6810:
6809:
6807:
6806:
6796:
6786:
6776:
6774:List of topics
6766:
6755:
6752:
6751:
6749:
6748:
6743:
6738:
6733:
6728:
6723:
6718:
6712:
6710:
6706:
6705:
6703:
6702:
6697:
6692:
6687:
6682:
6675:P-adic numbers
6672:
6667:
6662:
6657:
6652:
6647:
6642:
6637:
6632:
6627:
6621:
6619:
6615:
6614:
6612:
6611:
6606:
6601:
6587:
6577:
6563:
6558:
6553:
6548:
6530:
6519:Iwasawa theory
6503:
6501:
6497:
6496:
6489:
6488:
6481:
6474:
6466:
6457:
6456:
6454:
6453:
6443:
6441:Classification
6437:
6434:
6433:
6431:
6430:
6428:Normal numbers
6425:
6420:
6398:
6393:
6388:
6382:
6380:
6376:
6375:
6373:
6372:
6367:
6362:
6357:
6352:
6347:
6342:
6337:
6336:
6335:
6325:
6320:
6314:
6312:
6310:infinitesimals
6302:
6301:
6299:
6298:
6297:
6296:
6291:
6286:
6272:
6267:
6262:
6249:
6234:
6229:
6224:
6219:
6213:
6211:
6204:
6203:
6201:
6200:
6195:
6190:
6185:
6172:
6156:
6151:
6146:
6133:
6120:
6118:
6112:
6111:
6109:
6108:
6095:
6080:
6067:
6052:
6039:
6024:
6011:
5991:
5989:
5983:
5982:
5980:
5979:
5974:
5973:
5972:
5962:
5957:
5952:
5947:
5942:
5937:
5924:
5909:
5904:
5891:
5876:
5863:
5848:
5835:
5819:
5817:
5810:
5809:
5801:
5800:
5793:
5786:
5778:
5772:
5771:
5766:
5716:
5699:
5685:
5656:
5635:
5634:External links
5632:
5631:
5630:
5615:
5599:
5597:978-0195061352
5582:
5564:
5549:
5534:
5527:Tobias Dantzig
5522:
5519:
5516:
5515:
5484:
5459:
5452:
5425:
5391:
5366:
5347:(4): 273â285.
5324:
5291:
5274:
5260:Eduard Heine,
5253:
5240:
5227:
5217:L. Kronecker,
5210:
5184:
5174:Georg Cantor,
5167:
5157:Eduard Heine,
5150:
5143:
5117:
5110:
5100:, ed. (2000).
5098:Selin, Helaine
5089:
5059:
5052:
5031:
5024:
5006:
4988:
4962:
4940:
4917:
4874:
4848:
4833:
4813:
4785:
4770:
4747:
4718:
4698:
4683:
4660:
4646:
4619:
4612:
4595:
4576:
4553:
4530:Matson, John.
4522:
4493:
4463:
4462:
4459:
4458:
4440:
4439:
4437:
4434:
4432:
4431:
4426:
4420:
4414:
4408:
4402:
4396:
4390:
4385:
4380:
4375:
4369:
4364:
4359:
4353:
4352:
4351:
4335:
4332:
4271:
4268:
4248:Main article:
4245:
4242:
4199:Main article:
4196:
4193:
4147:Main article:
4144:
4138:
4136:
4133:
4066:Main article:
4063:
4060:
4047:
4044:
4020:
4017:
4015:
4012:
3995:
3992:
3957:Main article:
3954:
3951:
3879:Main article:
3876:
3873:
3871:
3868:
3807:
3775:imaginary part
3773:is called the
3765:is called the
3729:
3728:
3717:
3714:
3711:
3708:
3693:imaginary unit
3689:Leonhard Euler
3661:Complex number
3659:Main article:
3656:
3653:
3640:
3637:
3634:
3631:
3626:
3622:
3485:countably many
3458:
3457:
3446:
3443:
3440:
3437:
3432:
3409:
3408:
3397:
3394:
3391:
3388:
3385:
3268:
3264:
3244:Main article:
3241:
3238:
3223:
3176:absolute value
3172:
3171:
3160:
3156:
3153:
3150:
3146:
3142:
3139:
3136:
3113:
3110:
3105:
3100:
3097:
3081:
3080:
3069:
3064:
3061:
3056:
3051:
3048:
2963:Main article:
2960:
2957:
2915:
2874:Main article:
2871:
2868:
2822:
2817:
2793:
2788:
2765:
2729:Ancient Greeks
2711:Natural number
2709:Main article:
2706:
2703:
2700:
2699:
2696:
2695:
2692:
2691:
2688:
2687:
2681:
2680:
2677:
2676:
2673:
2672:
2669:
2668:
2665:
2664:
2662:Transcendental
2658:
2657:
2648:
2638:
2637:
2634:
2633:
2630:
2629:
2626:
2625:
2622:
2621:
2619:
2613:
2612:
2606:
2605:
2603:Finite decimal
2596:
2586:
2585:
2582:
2581:
2578:
2577:
2571:
2570:
2567:
2566:
2563:
2562:
2556:
2555:
2549:
2548:
2541:
2540:
2530:
2518:
2513:
2492:
2480:
2475:
2454:
2442:
2437:
2416:
2404:
2399:
2378:
2366:
2361:
2344:Number systems
2329:
2328:
2315:
2311:
2307:
2303:
2299:
2295:
2291:
2287:
2283:
2262:
2261:
2239:
2234:
2222:
2210:
2209:
2206:
2201:
2189:
2177:
2176:
2142:
2137:
2125:
2113:
2112:
2109:
2104:
2092:
2080:
2079:
2064:
2059:
2035:
2030:
2015:
2010:
1998:
1972:, such as the
1970:number systems
1949:Numeral system
1944:
1941:
1867:
1866:Prime numbers
1864:
1823:roots of unity
1740:
1739:
1728:
1723:
1720:
1716:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1666:
1665:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1619:
1615:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1569:Leonhard Euler
1547:
1544:
1515:
1514:
1500:
1497:
1491:
1485:
1481:
1449:
1448:
1437:
1432:
1429:
1424:
1419:
1412:
1396:
1395:
1384:
1381:
1378:
1373:
1370:
1363:
1360:
1355:
1350:
1345:
1340:
1337:
1332:
1312:René Descartes
1289:1st century AD
1274:
1271:
1230:In the 1960s,
1140:
1137:
1103:(1844, 1851).
1086:
1083:
923:
920:
861:
858:
842:René Descartes
761:
758:
718:Roman numerals
690:was the first
684:Greek numerals
653:4th century BC
618:Ancient Greeks
543:Brahmagupta's
537:Khmer numerals
522:Khmer numerals
495:
494:
453:
451:
444:
438:
435:
412:Numeral system
410:Main article:
407:
404:
367:Main article:
364:
361:
359:
356:
344:complex number
298:exponentiation
290:multiplication
247:
242:
237:
208:
202:
199:
193:
147:serial numbers
127:numeral system
44:Set inclusions
26:
9:
6:
4:
3:
2:
6852:
6841:
6838:
6836:
6833:
6831:
6828:
6826:
6823:
6822:
6820:
6805:
6801:
6797:
6795:
6791:
6787:
6785:
6777:
6775:
6767:
6765:
6757:
6756:
6753:
6747:
6744:
6742:
6739:
6737:
6734:
6732:
6729:
6727:
6724:
6722:
6721:Modular forms
6719:
6717:
6714:
6713:
6711:
6707:
6701:
6698:
6696:
6693:
6691:
6688:
6686:
6683:
6680:
6676:
6673:
6671:
6668:
6666:
6663:
6661:
6658:
6656:
6653:
6651:
6648:
6646:
6645:Prime numbers
6643:
6641:
6638:
6636:
6633:
6631:
6628:
6626:
6623:
6622:
6620:
6616:
6610:
6607:
6605:
6602:
6599:
6595:
6591:
6588:
6585:
6581:
6578:
6575:
6571:
6567:
6564:
6562:
6559:
6557:
6554:
6552:
6549:
6546:
6542:
6538:
6534:
6531:
6528:
6527:Kummer theory
6524:
6520:
6516:
6512:
6508:
6505:
6504:
6502:
6498:
6494:
6493:Number theory
6487:
6482:
6480:
6475:
6473:
6468:
6467:
6464:
6452:
6444:
6442:
6439:
6438:
6435:
6429:
6426:
6424:
6421:
6418:
6414:
6408:
6404:
6399:
6397:
6394:
6392:
6391:Fuzzy numbers
6389:
6387:
6384:
6383:
6381:
6377:
6371:
6368:
6366:
6363:
6361:
6358:
6356:
6353:
6351:
6348:
6346:
6343:
6341:
6338:
6334:
6331:
6330:
6329:
6326:
6324:
6321:
6319:
6316:
6315:
6313:
6311:
6307:
6303:
6295:
6292:
6290:
6287:
6285:
6282:
6281:
6280:
6276:
6273:
6271:
6268:
6266:
6263:
6238:
6235:
6233:
6230:
6228:
6225:
6223:
6220:
6218:
6215:
6214:
6212:
6210:
6205:
6199:
6196:
6194:
6193:Biquaternions
6191:
6189:
6186:
6160:
6157:
6155:
6152:
6150:
6147:
6122:
6121:
6119:
6113:
6084:
6081:
6056:
6053:
6028:
6025:
6000:
5996:
5993:
5992:
5990:
5988:
5984:
5978:
5975:
5971:
5968:
5967:
5966:
5963:
5961:
5958:
5956:
5953:
5951:
5948:
5946:
5943:
5941:
5938:
5913:
5910:
5908:
5905:
5880:
5877:
5852:
5849:
5824:
5821:
5820:
5818:
5816:
5811:
5806:
5799:
5794:
5792:
5787:
5785:
5780:
5779:
5776:
5770:
5767:
5755:
5751:
5750:
5745:
5730:
5726:
5722:
5717:
5713:
5709:
5705:
5700:
5696:
5692:
5691:
5686:
5674:
5670:
5666:
5662:
5657:
5653:
5649:
5648:
5643:
5638:
5637:
5628:
5624:
5620:
5616:
5613:
5612:
5607:
5603:
5600:
5598:
5594:
5590:
5586:
5583:
5580:
5579:0-387-90092-6
5576:
5572:
5568:
5565:
5562:
5561:0-15-543468-3
5558:
5554:
5550:
5548:
5547:
5543:
5540:
5535:
5532:
5528:
5525:
5524:
5503:
5499:
5495:
5488:
5479:
5478:
5473:
5470:
5463:
5455:
5453:0-486-61630-4
5449:
5445:
5441:
5440:
5435:
5429:
5413:
5409:
5405:
5401:
5395:
5386:
5385:
5380:
5377:
5370:
5362:
5358:
5354:
5350:
5346:
5342:
5335:
5328:
5312:
5308:
5304:
5301:
5300:Bogomolny, A.
5295:
5288:
5284:
5278:
5271:
5267:
5265:
5257:
5250:
5244:
5237:
5231:
5224:
5220:
5214:
5207:
5203:
5202:
5198:
5195:
5188:
5181:
5177:
5171:
5164:
5160:
5154:
5146:
5144:0-674-37935-7
5140:
5136:
5132:
5128:
5121:
5113:
5111:0-7923-6481-3
5107:
5103:
5099:
5093:
5077:
5073:
5069:
5063:
5055:
5053:0-486-20429-4
5049:
5045:
5041:
5035:
5027:
5025:0-534-40365-4
5021:
5017:
5010:
5002:
4998:
4992:
4976:
4972:
4966:
4959:
4947:
4943:
4937:
4933:
4932:
4924:
4922:
4913:
4909:
4905:
4901:
4897:
4893:
4889:
4885:
4878:
4862:
4858:
4852:
4844:
4840:
4836:
4834:0-07-040535-2
4830:
4826:
4825:
4817:
4810:
4806:
4802:
4798:
4796:
4789:
4781:
4777:
4773:
4771:0-486-65620-9
4767:
4763:
4762:
4754:
4752:
4743:
4737:
4729:
4725:
4721:
4715:
4712:. Dordrecht.
4711:
4710:
4702:
4694:
4690:
4686:
4680:
4676:
4675:
4667:
4665:
4649:
4647:0-486-60068-8
4643:
4639:
4635:
4634:
4629:
4623:
4615:
4613:1-4020-0260-2
4609:
4605:
4599:
4583:
4579:
4573:
4569:
4568:
4560:
4558:
4541:
4537:
4533:
4526:
4510:
4506:
4503:
4497:
4481:
4477:
4474:
4468:
4464:
4455:
4451:
4445:
4441:
4430:
4427:
4424:
4421:
4418:
4415:
4412:
4409:
4406:
4403:
4400:
4397:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4373:
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4354:
4349:
4343:
4338:
4331:
4329:
4325:
4321:
4317:
4315:
4311:
4307:
4303:
4299:
4296:
4292:
4288:
4287:ordered field
4285:), denote an
4284:
4280:
4276:
4267:
4265:
4261:
4257:
4251:
4241:
4239:
4235:
4231:
4227:
4223:
4219:
4215:
4212:
4208:
4202:
4192:
4190:
4186:
4182:
4178:
4173:
4171:
4167:
4163:
4159:
4153:
4151:
4143:-adic numbers
4142:
4132:
4130:
4125:
4121:
4119:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4083:
4079:
4075:
4069:
4059:
4057:
4053:
4043:
4041:
4037:
4033:
4029:
4025:
4011:
4009:
4005:
4001:
3991:
3989:
3985:
3980:
3978:
3974:
3973:number theory
3970:
3966:
3960:
3953:Prime numbers
3950:
3948:
3944:
3939:
3935:
3931:has the form
3930:
3924:
3919:
3913:
3909:
3904:
3900:
3896:
3892:
3888:
3882:
3867:
3865:
3861:
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3829:
3825:
3820:
3796:
3792:
3788:
3784:
3780:
3776:
3772:
3768:
3764:
3759:
3755:
3750:
3746:
3742:
3741:complex plane
3738:
3734:
3715:
3712:
3709:
3706:
3698:
3697:
3696:
3694:
3690:
3686:
3685:
3680:
3676:
3672:
3668:
3662:
3652:
3638:
3635:
3632:
3629:
3624:
3620:
3611:
3607:
3603:
3599:
3598:ordered field
3594:
3592:
3587:
3585:
3580:
3578:
3566:
3562:
3558:
3490:
3486:
3482:
3478:
3474:
3469:
3463:
3444:
3441:
3439:1.41421356237
3438:
3435:
3430:
3421:
3420:
3419:
3417:
3416:is irrational
3395:
3392:
3389:
3386:
3383:
3376:
3375:
3374:
3372:
3368:
3367:circumference
3364:
3358:
3354:
3349:
3327:
3322:
3301:
3300:decimal point
3297:
3293:
3288:
3286:
3282:
3266:
3253:
3247:
3237:
3236:
3211:
3210:
3205:
3185:
3181:
3177:
3158:
3154:
3151:
3148:
3144:
3140:
3137:
3134:
3111:
3108:
3103:
3098:
3095:
3086:
3085:
3084:
3067:
3062:
3059:
3054:
3049:
3046:
3037:
3036:
3035:
3001:
2997:
2989:
2981:
2972:
2966:
2956:
2954:
2950:
2946:
2941:
2939:
2935:
2931:
2927:
2904:also written
2903:
2899:
2895:
2887:
2883:
2877:
2867:
2865:
2861:
2856:
2854:
2849:
2848:radix or base
2845:
2841:
2836:
2820:
2791:
2754:
2750:
2746:
2742:
2738:
2734:
2733:set theorists
2730:
2726:
2717:
2712:
2686:
2683:
2682:
2663:
2660:
2659:
2656:
2653:
2652:
2649:
2647:
2644:
2643:
2640:
2639:
2620:
2618:
2615:
2614:
2611:
2608:
2607:
2604:
2601:
2600:
2597:
2595:
2592:
2591:
2588:
2587:
2576:
2573:
2572:
2561:
2558:
2557:
2554:
2553:Prime numbers
2551:
2550:
2546:
2543:
2542:
2538:
2535:
2534:
2531:
2511:
2504:
2501:
2500:
2497:
2496:
2493:
2473:
2466:
2463:
2462:
2459:
2458:
2455:
2435:
2428:
2425:
2424:
2421:
2420:
2417:
2397:
2390:
2387:
2386:
2383:
2382:
2379:
2359:
2352:
2349:
2348:
2345:
2340:
2339:
2336:
2335:
2332:
2309:
2301:
2293:
2285:
2273:
2272:
2271:
2269:
2259:
2255:
2251:
2247:
2243:
2240:
2238:
2235:
2212:
2211:
2207:
2205:
2202:
2179:
2178:
2174:
2170:
2166:
2158:
2150:
2143:
2141:
2138:
2115:
2114:
2110:
2108:
2105:
2082:
2081:
2078:
2062:
2033:
2016:
2014:
2011:
1988:
1987:
1981:
1979:
1975:
1971:
1967:
1963:
1957:
1950:
1940:
1938:
1934:
1930:
1927:in 1859. The
1926:
1922:
1918:
1914:
1910:
1905:
1903:
1899:
1895:
1890:
1888:
1884:
1880:
1876:
1872:
1871:Prime numbers
1863:
1861:
1857:
1853:
1848:
1846:
1842:
1841:ideal numbers
1838:
1834:
1828:
1824:
1820:
1816:
1810:
1805:
1800:
1796:
1791:
1787:
1783:
1779:
1774:
1770:
1765:
1760:
1758:
1754:
1749:
1745:
1744:Caspar Wessel
1726:
1721:
1718:
1714:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1679:
1678:
1677:
1675:
1671:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1617:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1578:
1577:
1576:
1574:
1570:
1566:
1561:
1545:
1542:
1532:
1528:
1524:
1520:
1498:
1495:
1489:
1483:
1479:
1470:
1469:
1468:
1466:
1462:
1458:
1454:
1435:
1430:
1427:
1422:
1417:
1410:
1401:
1400:
1399:
1382:
1379:
1376:
1371:
1368:
1361:
1358:
1353:
1348:
1343:
1338:
1335:
1330:
1321:
1320:
1319:
1317:
1313:
1308:
1306:
1302:
1298:
1294:
1286:
1280:
1270:
1268:
1264:
1259:
1257:
1253:
1249:
1245:
1244:infinitesimal
1241:
1237:
1233:
1228:
1226:
1222:
1218:
1214:
1210:
1206:
1205:
1200:
1196:
1192:
1188:
1184:
1160:
1156:
1152:
1146:
1136:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1092:
1082:
1079:
1074:
1070:
1066:
1062:
1058:
1054:
1052:
1051:Galois theory
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1015:
1014:, and MĂ©ray.
1013:
1009:
1005:
1001:
1000:cut (Schnitt)
997:
993:
989:
988:Charles MĂ©ray
985:
981:
977:
973:
969:
964:
961:integral and
960:
955:
953:
949:
946:
942:
938:
935:
929:
919:
917:
913:
909:
905:
900:
898:
894:
893:
887:
886:number theory
883:
882:Kahun Papyrus
879:
875:
871:
867:
857:
854:
852:
848:
843:
838:
832:
831:
825:
820:
818:
814:
810:
809:
804:
799:
797:
796:
789:
783:
779:
775:
774:
767:
757:
755:
751:
747:
743:
738:
733:
728:
723:
719:
714:
712:
709:
705:
701:
697:
693:
689:
685:
681:
677:
673:
668:
666:
662:
661:Maya calendar
658:
657:Maya numerals
650:
646:
641:
639:
635:
631:
627:
623:
622:philosophical
619:
614:
612:
607:
605:
601:
597:
596:
591:
587:
582:
576:
571:
567:
563:
559:
554:
551:
546:
538:
533:
529:
527:
526:Islamic world
523:
519:
515:
512:
508:
507:
502:
491:
488:
480:
477:November 2022
470:
466:
460:
459:
454:This section
452:
448:
443:
442:
434:
432:
428:
423:
419:
413:
403:
401:
390:
385:
383:
378:
376:
370:
355:
353:
349:
345:
341:
336:
334:
330:
326:
325:pseudoscience
322:
318:
314:
309:
307:
306:number theory
303:
299:
295:
291:
287:
283:
279:
275:
268:
264:
245:
240:
235:
227:
223:
206:
200:
197:
191:
183:
179:
175:
171:
167:
162:
160:
156:
152:
148:
144:
140:
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
96:
93:
89:
85:
81:
77:
73:
65:
62:(â), and the
61:
57:
53:
49:
45:
41:
37:
33:
19:
6830:Group theory
6624:
6618:Key concepts
6545:sieve theory
6412:
6402:
6217:Dual numbers
6209:hypercomplex
5999:Real numbers
5804:
5760:17 September
5758:. Retrieved
5747:
5735:17 September
5733:. Retrieved
5724:
5695:the original
5689:
5677:. Retrieved
5673:the original
5664:
5645:
5618:
5609:
5588:
5585:Morris Kline
5570:
5552:
5537:
5530:
5506:. Retrieved
5497:
5487:
5475:
5462:
5438:
5428:
5416:. Retrieved
5403:
5394:
5382:
5369:
5344:
5340:
5327:
5315:. Retrieved
5306:
5294:
5286:
5282:
5277:
5269:
5263:
5256:
5248:
5243:
5235:
5230:
5222:
5213:
5205:
5192:
5187:
5179:
5170:
5162:
5153:
5134:
5126:
5120:
5101:
5092:
5080:. Retrieved
5072:Khan Academy
5071:
5062:
5043:
5034:
5015:
5009:
5000:
4991:
4979:. Retrieved
4975:the original
4965:
4957:
4950:. Retrieved
4930:
4887:
4883:
4877:
4865:. Retrieved
4851:
4823:
4816:
4808:
4793:
4788:
4760:
4708:
4701:
4673:
4651:. Retrieved
4632:
4622:
4603:
4598:
4586:. Retrieved
4566:
4544:. Retrieved
4535:
4525:
4513:. Retrieved
4504:
4496:
4484:. Retrieved
4475:
4473:"number, n."
4467:
4444:
4417:Prime number
4318:
4313:
4309:
4297:
4295:real numbers
4282:
4277:are used in
4273:
4253:
4213:
4204:
4188:
4181:finite field
4174:
4169:
4166:prime number
4157:
4155:
4152:-adic number
4149:
4140:
4126:
4122:
4093:
4089:
4077:
4073:
4071:
4049:
4022:
3997:
3981:
3968:
3965:prime number
3964:
3962:
3959:Prime number
3945:is again an
3942:
3937:
3933:
3928:
3922:
3917:
3911:
3907:
3902:
3894:
3886:
3884:
3855:
3851:
3821:
3794:
3782:
3770:
3762:
3757:
3753:
3747:of two real
3745:vector space
3736:
3732:
3730:
3682:
3664:
3595:
3588:
3581:
3576:
3470:
3459:
3410:
3350:
3292:approximated
3291:
3289:
3251:
3249:
3240:Real numbers
3207:
3203:
3183:
3179:
3173:
3083:In general,
3082:
2999:
2995:
2987:
2979:
2968:
2952:
2948:
2942:
2933:
2901:
2890:7 + (â7) = 0
2879:
2857:
2837:
2752:
2722:
2330:
2265:
2257:
2253:
2249:
2245:
2241:
2204:Real numbers
2172:
2168:
2164:
2156:
2148:
2019:
1978:real numbers
1969:
1965:
1959:
1906:
1894:Eratosthenes
1891:
1874:
1869:
1849:
1837:Ernst Kummer
1832:
1826:
1813:(now called
1808:
1803:
1798:
1794:
1781:
1777:
1772:
1768:
1761:
1756:
1741:
1667:
1562:
1533:in place of
1530:
1522:
1518:
1516:
1464:
1460:
1456:
1452:
1450:
1397:
1309:
1282:
1260:
1229:
1213:Georg Cantor
1202:
1185:
1148:
1121:real numbers
1108:
1094:
1091:History of Ï
1069:determinants
1055:
1047:group theory
1016:
1004:real numbers
996:Paul Tannery
980:Georg Cantor
976:Eduard Heine
956:
937:Sulba Sutras
931:
901:
891:
863:
855:
821:
806:
800:
793:
787:
778:coefficients
771:
769:
741:
731:
715:
708:Greek letter
703:
699:
691:
669:
642:
634:Zeno of Elea
615:
610:
608:
593:
585:
561:
555:
549:
544:
542:
504:
498:
483:
474:
463:Please help
458:verification
455:
417:
415:
386:
379:
372:
337:
310:
278:Calculations
224:such as the
222:real numbers
163:
158:
154:
118:
111:number words
71:
69:
60:real numbers
46:between the
36:
6835:Abstraction
6804:Wikiversity
6726:L-functions
6379:Other types
6198:Bioctonions
6055:Quaternions
5669:Brady Haran
5665:Numberphile
5567:Paul Halmos
4450:linguistics
4306:first-order
4262:and to the
4238:alternative
4230:associative
4222:commutative
4211:quaternions
4082:real number
3887:even number
3860:total order
3842:, which is
3679:square root
3577:significant
3468:of digits.
3281:number line
3246:Real number
2994:represents
2892:. When the
2853:place value
2737:cardinality
1966:number sets
1902:Renaissance
1892:In 240 BC,
1845:Felix Klein
1267:perspective
945:Pythagorean
872:used their
830:Liber Abaci
803:Brahmagupta
795:Arithmetica
680:sexagesimal
670:By 130 AD,
595:Ashtadhyayi
514:Brahmagupta
396: 3400
375:tally marks
286:subtraction
166:mathematics
6819:Categories
6685:Arithmetic
6333:Projective
6306:Infinities
5617:Leo Cory,
5521:References
4981:30 January
4867:30 January
4505:OED Online
4476:OED Online
4129:almost all
4110:polynomial
4106:λ-calculus
3895:odd number
3832:polynomial
3749:dimensions
3593:property.
3573:5.603011 m
3569:5.614591 m
3473:almost all
3357:irrational
3285:minus sign
2886:minus sign
2860:set theory
2646:Irrational
1954:See also:
1217:set theory
1155:Yajur Veda
963:fractional
941:Pythagoras
782:Diophantus
746:computists
692:documented
676:Hipparchus
329:numerology
302:arithmetic
6417:solenoids
6237:Sedenions
6083:Octonions
5652:EMS Press
5477:MathWorld
5472:"Integer"
5418:4 October
5384:MathWorld
4912:160523072
4904:0003-598X
4884:Antiquity
4736:cite book
4728:883391697
4693:793103475
4630:(1954) .
4320:Superreal
4291:extension
4234:sedenions
4226:octonions
4086:algorithm
3899:divisible
3767:real part
3675:quadratic
3565:rectangle
3481:truncated
3442:…
3393:…
3384:π
3152:×
3138:×
2741:empty set
2685:Imaginary
2310:⊂
2302:⊂
2294:⊂
2286:⊂
2175:is not 0
1964:, called
1907:In 1796,
1896:used the
1847:in 1893.
1722:θ
1708:θ
1705:
1693:θ
1690:
1653:θ
1647:
1635:θ
1629:
1610:θ
1607:
1595:θ
1592:
1543:−
1380:−
1369:−
1359:−
1336:−
1269:drawing.
1187:Aristotle
1113:Lindemann
1101:Liouville
1012:Kronecker
890:Euclid's
851:exponents
824:Fibonacci
696:Byzantine
643:The late
630:paradoxes
321:a million
313:number 13
117:, called
58:(â), the
54:(â€), the
50:(â), the
6794:Wikibook
6764:Category
5851:Integers
5813:Sets of
5754:Archived
5729:Archived
5712:Archived
5642:"Number"
5542:Archived
5502:Archived
5436:(1972).
5412:Archived
5361:43778192
5311:Archived
5197:Archived
5076:Archived
5042:(1958).
4999:(1961).
4946:Archived
4861:Archived
4780:17413345
4653:20 April
4582:Archived
4540:Archived
4509:Archived
4480:Archived
4334:See also
3862:that is
3844:complete
3602:complete
3584:0.999...
3371:diameter
3209:quotient
2971:fraction
2898:integers
2882:negative
2870:Integers
2594:Fraction
2427:Rational
2107:Integers
1976:and the
1875:Elements
1850:In 1850
1802:, where
1776:, where
1240:infinite
1170:∞
1151:infinity
1035:radicals
959:negative
892:Elements
880:and the
817:Bhaskara
790:+ 20 = 0
730:meaning
704:Almagest
659:and the
570:Sanskrit
518:division
418:numerals
406:Numerals
391:system (
294:division
282:addition
182:one half
180:such as
119:numerals
78:used to
52:integers
6825:Numbers
6625:Numbers
6407:numbers
6239: (
6085: (
6057: (
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5945:Periods
5914: (
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5807:systems
5679:6 April
5508:23 July
5317:11 July
5133:(ed.).
4454:numeral
4179:over a
4080:, is a
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3848:ordered
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3512:1234565
3509:
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3328:. Thus
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2840:base 10
2838:In the
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2465:Integer
2351:Complex
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1829:â 1 = 0
1811:â 1 = 0
1297:pyramid
1293:frustum
1287:in the
1256:Leibniz
1105:Hermite
1027:Ruffini
1019:quintic
914:or the
742:nothing
732:nothing
711:Omicron
672:Ptolemy
604:Pingala
382:decimal
358:History
319:, and "
317:unlucky
155:numeral
115:symbols
84:measure
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6500:Fields
6207:Other
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3731:where
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3308:123456
2945:subset
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2844:digits
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2163:where
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1668:while
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1117:Cantor
1073:Möbius
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982:, and
968:Euclid
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868:. The
750:Easter
628:. The
626:vacuum
590:PÄáčini
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575:Shunye
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296:, and
265:, and
159:number
139:digits
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6640:Unity
6415:-adic
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6162:Over
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5337:(PDF)
5082:4 May
4908:S2CID
4436:Notes
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3893:; an
3840:field
3671:cubic
3518:10000
3502:10000
3206:(for
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1295:of a
1061:Euler
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727:nulla
649:glyph
645:Olmec
572:word
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348:rings
172:(0),
151:ISBNs
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5762:2011
5737:2011
5681:2013
5623:ISBN
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5448:ISBN
5420:2014
5319:2010
5264:Lamé
5139:ISBN
5106:ISBN
5084:2022
5048:ISBN
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4983:2012
4954:2017
4936:ISBN
4900:ISSN
4869:2012
4839:OCLC
4829:ISBN
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4776:OCLC
4766:ISBN
4742:link
4724:OCLC
4714:ISBN
4689:OCLC
4679:ISBN
4655:2011
4642:ISBN
4608:ISBN
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4572:ISBN
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