3593:, 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.
4408:
7002:
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.
6962:
447:
6982:
6972:
6649:
6992:
40:
3978:, 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:
2723:
3193:(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
3330:
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
4189:
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
2980:
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
2873:, 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.
2869:, 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
4332:. 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
3290:. 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
3331:
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
4190:
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.
2857:
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:
4226:-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
3309:
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
3326:, 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
4574:
1737:
4271:
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
3989:
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
4005:
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
3362:) 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
4873:. "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
3462:
3413:
218:
5905:
2734:(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
4124:
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.
3176:
3085:
2891:
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:
3131:
5141:
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
2840:
2811:
2075:
2046:
2528:
2490:
2452:
2414:
2376:
6146:
3482:
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
4522:
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".
3284:
6461:
6384:
6345:
6307:
6279:
6251:
6223:
6111:
6078:
6050:
6022:
3824:
3240:
2932:
2782:
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".
5930:
1181:
3656:
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.
2954:
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
3733:
3562:(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
4926:
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
5024:
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
5036:
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.
3418:
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
4877:-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:
5888:
3589:
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,
4807:
1324:
5897:
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
5477:
5133:
5011:
4605:
3684:
polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a
4647:
4033:
are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called
6985:
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,
5972:
5376:
3471:, that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions
1473:
2754:) in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including 0 or not. The
6685:
5567:
2849:
numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten
3351:
can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0.
4860:
1116:
1682:
5803:
5613:
5005:
4870:
4783:
4748:
4641:
4117:
5718:
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
6161:
5274:, ed. Robert Fricke, Emmy Noether & Ăystein Ore (Braunschweig: Friedrich Vieweg & Sohn, 1932), vol. 3, pp. 315â334.
5262:
4197:
real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.
3796:
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
5837:
4773:
1404:
998:(1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a
6156:
2903:
of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of
5865:
5773:
368:
4922:
3431:
5755:
5737:
5517:
5208:
5175:
5117:
5089:
4898:
4835:
4711:
4677:
4174:. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a
3990:
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
3608:
6116:
4545:
3386:
186:
6534:
1037:(formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of
565:
421:
130:
17:
6612:
5040:
2862:
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
6716:
3830:
1763:
807:
505:
468:
3904:
is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "
3136:
3047:
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:
7041:
6495:
5828:
4392:
extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form
3096:
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
5920:
5818:
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
6975:
6762:
5965:
4093:
The periods can be extended by permitting the integrand to be the product of an algebraic function and the
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:
6757:
6742:
6678:
6121:
5823:
5300:
Leonhard Euler, "Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis",
4058:
3574:
are measured as 1.23 m and 4.56 m, then multiplication gives an area for the rectangle between
3419:
2661:
765:
3566:. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001
3490:
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.
3834:
3612:
1278:
1164:
1059:, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of
877:
4075:. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known
2816:
2787:
2051:
2022:
1072:
780:, black for negative. The first reference in a Western work was in the 3rd century AD in Greece.
6937:
6896:
6775:
6529:
6485:
5880:
4098:
3582:. 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.
6127:
4162:
digits of the computable number's decimal representation. Equivalent definitions can be given using
531:
6781:
6652:
6524:
4432:
4242:
3983:
3616:
3264:
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.
6444:
6367:
6328:
6290:
6262:
6234:
6206:
6094:
6061:
6033:
6005:
5399:
3807:
3223:
2915:
2765:
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
7001:
6724:
6708:
5958:
5593:
5196:
4948:
Chrisomalis, Stephen (1 September 2003). "The
Egyptian origin of the Greek alphabetic numerals".
4494:
3792:; 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:
4163:
687:
6965:
6785:
6734:
6671:
5778:
4283:
4250:
3622:
2977:
1859:
1572:
1247:
1096:
962:
293:
7005:
5509:
5503:
4995:
4597:
3676:. 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
7031:
6942:
6871:
6597:
6433:
5787:
5708:
Number, the language of science; a critical survey written for the cultured non-mathematician
5465:
5315:
Det
Kongelige Danske Videnskabernes Selskabs naturvidenskabelige og mathematiske Afhandlinger
5313:
Ramus, "Determinanternes Anvendelse til at bes temme Loven for de convergerende Bröker", in:
5284:
4994:
Bulliet, Richard; Crossley, Pamela; Headrick, Daniel; Hirsch, Steven; Johnson, Lyman (2010).
4631:
4519:
4437:
4344:
4076:
4038:
3009:
equal parts. Two different fractions may correspond to the same rational number; for example
2668:
1908:
1897:
1314:
coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See
1064:
4890:
The roots of civilization; the cognitive beginnings of man's first art, symbol, and notation
3986:
is an example of a still unanswered question: "Is every even number the sum of two primes?"
3708:
7036:
6932:
6767:
6350:
6083:
5224:
4488:
4121:
4094:
4072:
3994:
3887:
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
6995:
5062:
4740:
Introduction to cultural mathematics : with case studies in the Otomies and the Incas
4238:-adic numbers contains the rational numbers, but is not contained in the complex numbers.
4193:
The set of computable numbers has the same cardinality as the natural numbers. Therefore,
4097:
of an algebraic function. This gives another countable ring: the exponential periods. The
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:
6901:
6810:
6805:
6799:
6791:
6752:
6561:
6471:
6428:
6410:
6188:
5327:
5191:
Bernard Frischer (1984). "Horace and the Monuments: A New Interpretation of the Archytas
4476:
4453:
4448:
4393:
4303:
4272:
4266:
4105:
3846:
2842:
when it is necessary to indicate whether the set should start with 0 or 1, respectively.
2755:
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:
6991:
5368:
3286:
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.
6947:
6891:
6795:
6712:
6466:
6178:
5422:
5105:
5037:"Historia Matematica Mailing List Archive: Re: [HM] The Zero Story: a question"
4973:
4801:
4703:
4413:
4367:
4315:
4087:
4068:
3897:
3842:
3681:
3563:
2944:
1920:
1669:
1220:
1194:
1056:
933:
557:
430:
347:
5559:
1042:
6861:
6624:
6587:
6551:
6490:
6476:
6171:
6151:
5799:
5769:
5751:
5733:
5682:
5636:
5609:
5534:
5513:
5441:
5204:
5171:
5113:
5085:
5001:
4977:
4965:
4904:
4894:
4866:
4841:
4831:
4789:
4779:
4754:
4744:
4707:
4697:
4673:
4637:
4464:
4458:
4407:
4371:
4321:
4179:
4133:
4046:
4034:
3597:
3498:. Thus 123.456 is considered an approximation of any real number greater or equal to
3363:
3332:
2900:
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:
5678:
5426:
4693:
1311:
841:
717:
7026:
6866:
6851:
6571:
6546:
6480:
6389:
6355:
6196:
6166:
6088:
5991:
5782:
5674:
5601:
5444:
5418:
5414:
4957:
4385:
4340:
4183:
4042:
4030:
4014:
4006:
3797:
3785:
3468:
2870:
2691:
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:
4699:
La Géométrie: The Geometry of René Descartes with a facsimile of the first edition
3672:
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
6880:
6856:
6771:
6519:
6423:
6055:
5884:
5722:
5715:
5605:
5592:
Kontsevich, Maxim; Zagier, Don (2001), Engquist, Björn; Schmid, Wilfried (eds.),
5473:
5266:
5259:
4888:
4825:
4738:
4491: â Elements of a field, e.g. real numbers, in the context of linear algebra
4427:
4422:
4356:
4329:
3596:
The real numbers also have an important but highly technical property called the
3495:
3327:
3220:
2971:
2936:
2912:
2888:
2850:
2751:
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:
6917:
6836:
6720:
6566:
6556:
6541:
6360:
6228:
5999:
5849:
5703:
5499:
5241:
4443:
4389:
4325:
4167:
4010:
3850:
3784:. If the real part of a complex number is 0, then the number is called an
3781:
3746:
are real numbers. Because of this, complex numbers correspond to points on the
3699:
3695:
3690:
3677:
3673:
3667:
3491:
3182:
2735:
2731:
2717:
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:
5270:(Braunschweig: Friedrich Vieweg & Sohn, 1872). Subsequently published in:
4961:
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
7020:
6876:
6728:
6694:
6629:
6602:
6511:
5686:
5537:
5163:
4969:
4793:
4758:
4352:
4214:
3979:
3870:
3747:
3604:
3373:
3359:
3306:
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:
5871:
4845:
4171:
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
6922:
6846:
6746:
6592:
6394:
5761:
5365:
4482:
4246:
4231:
3965:
3905:
3751:
3611:, is isomorphic to the real numbers. The real numbers are not, however, an
3494:. 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:
4908:
4253:). Therefore, they are often regarded as numbers by number theorists. The
3934:
the first non-negative odd numbers are {1, 3, 5, 7, ...}. Any even number
6418:
6200:
5845:
5743:
4670:
Mathematics across cultures : the history of non-western mathematics
4515:
4360:
4295:
4287:
4147:
3866:
3854:
3685:
3287:
3252:
2859:
2743:
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:
3956:. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}.
3425:. Another well-known number, proven to be an irrational real number, is
1135:, so there is an uncountably infinite number of transcendental numbers.
6927:
6886:
6738:
6399:
6256:
5400:"Euler's 'mistake'? The radical product rule in historical perspective"
5349:
Darstellung der NĂ€herungswerthe von KettenbrĂŒchen in independenter Form
4923:"Egyptian Mathematical Papyri â Mathematicians of the African Diaspora"
4567:
4538:
4276:
4194:
4175:
3838:
3487:
3479:
3291:
2892:
2866:
2739:
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
5351:(Erlangen: Eduard Besold, 1873); âââ, "Kettenbruchdeterminanten", in:
2331:
A more complete list of number sets appears in the following diagram.
5542:
5449:
4151:
3773:
3755:
3571:
2747:
1208:
1186:
1132:
845:
numeral. The first use of negative numbers in a European work was by
823:
3865:
is less than 1. In technical terms, the complex numbers lack a
3857:. That is, there is no consistent meaning assignable to saying that
3530:(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:
6507:
6438:
6284:
5168:
Mathematics across cultures: the history of non-Western mathematics
4299:
4291:
4064:
4025:
3845:
in the complex numbers. Like the reals, the complex numbers form a
3590:
3483:
3377:
3215:
2593:
1742:
The existence of complex numbers was not completely accepted until
1239:
1150:
1034:
947:
850:
745:
569:
281:
181:
79:
5662:
4090:
and bridge the gap between algebraic and transcendental numbers.
1197:âthe general consensus being that only the latter had true value.
744:, was used. These medieval zeros were used by all future medieval
6027:
5950:
3953:
3302:
2904:
2895:). As an example, the negative of 7 is written â7, and
2882:
2846:
2464:
2106:
1296:
1292:
734:, not as a symbol. When division produced 0 as a remainder,
710:
671:
603:
381:
316:
134:
51:
6663:
4775:
Mathematics in society and history : sociological inquiries
3861:
is greater than 1, nor is there any meaning in saying that
876:
notation for rational numbers in mathematical texts such as the
4234:
base provides the best mathematical properties. The set of the
3793:
2951:
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
4993:
4227:
3567:
2854:
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
4347:. The hyperreals, or nonstandard reals (usually denoted as *
109:, and so forth. Numbers can be represented in language with
5242:"Ueber unendliche, lineare Punktmannichfaltigkeiten", pt. 5
2536:
1158:
753:
500:
426:
150:
121:; for example, "5" is a numeral that represents the number
4378:
to be reinterpreted as true first-order statements about *
3358:
It turns out that these repeating decimals (including the
2750:, i.e. 0 elements, where 0 is thus the smallest
2722:
5925:
4865:, p. 82. Princeton University Press, September 28, 2008.
4063:
A period is a complex number that can be expressed as an
3896:
is an integer that is "evenly divisible" by two, that is
3702:. The complex numbers consist of all numbers of the form
3615:, because they do not include a solution (often called a
2976:
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
5663:"Euler's constant: Euler's work and modern developments"
4249:
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
5439:
5079:
4633:
A History of Mathematics: From Mesopotamia to Modernity
4470:
4080:
3367:
2958:, and the natural numbers with zero are referred to as
2208:
The limit of a convergent sequence of rational numbers
911:
260:
4672:. Dordrecht: Kluwer Academic. 2000. pp. 410â411.
4257:-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:
6841:
6831:
6447:
6370:
6331:
6293:
6265:
6237:
6209:
6130:
6097:
6064:
6036:
6008:
4997:
The Earth and Its Peoples: A Global History, Volume 1
4037:. Complex numbers which are not algebraic are called
3810:
3711:
3625:
3434:
3389:
3267:
3226:
3139:
3099:
3050:
2918:
2819:
2790:
2768:
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:
5302:
Acta Academiae Scientiarum Imperialis Petropolitanae
4989:
4987:
4467: â Measurable property of a material or system
4403:
4230:
is used for the digits: any base is possible, but a
4020:
3586:. Therefore, the result is usually rounded to 5.61.
1441:{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}},}
5355:(Erlangen: Eduard Besold, 1875), c. 6, pp. 156â186.
4863:, Chapter II.1, "The Origins of Modern Mathematics"
4479: â Method for representing or encoding numbers
4324:, the natural numbers have been generalized to the
2742:
and other mathematicians started including 0 (
1084:
6455:
6378:
6339:
6301:
6273:
6245:
6217:
6140:
6105:
6072:
6044:
6016:
5945:
5600:, Berlin, Heidelberg: Springer, pp. 771â808,
5353:Lehrbuch der Determinanten-Theorie: FĂŒr Studirende
4461: â Universal and unchanging physical quantity
4052:
3818:
3727:
3650:
3456:
3407:
3278:
3234:
3170:
3125:
3079:
2926:
2834:
2805:
2776:
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:
5921:"Cuddling With 9, Smooching With 8, Winking At 7"
5766:Mathematical Thought from Ancient to Modern Times
4984:
2947:with the operations addition and multiplication.
1291:, when he considered the volume of an impossible
1183:is often used to represent an infinite quantity.
7018:
5591:
5532:
5190:
5039:. Sunsite.utk.edu. 26 April 1999. Archived from
4041:. The algebraic numbers that are solutions of a
4026:Algebraic, irrational and transcendental numbers
3457:{\displaystyle {\sqrt {2}}=1.41421356237\dots ,}
387:The first known system with place value was the
5336:Journal fĂŒr die reine und angewandte Mathematik
5289:Journal fĂŒr die reine und angewandte Mathematik
5229: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
5373:Interactive Mathematics Miscellany and Puzzles
4298:in addition to not being commutative, and the
4045: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:
6679:
5966:
5667:Bulletin of the American Mathematical Society
4440: â Fixed number that has received a name
3876:
706:), the Hellenistic zero had morphed into the
133:, which allows for the representation of any
5878:
4485: â Number divisible only by 1 or itself
4200:
4000:
3408:{\displaystyle \pi =3.14159265358979\dots ,}
2943: '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)}
5870:. BBC Radio 4. 9 March 2006. Archived from
5170:. Kluwer Academic Publishers. p. 451.
4947:
4118:constructions with straightedge and compass
2017:0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ...
835:
827:
735:
725:
6981:
6971:
6686:
6672:
6648:
5973:
5959:
5080:Staszkow, Ronald; Robert Bradshaw (2004).
4806:: CS1 maint: location missing publisher (
4732:
4730:
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.
6449:
6372:
6333:
6295:
6267:
6239:
6211:
6099:
6066:
6038:
6010:
5791:to *56, Cambridge University Press, 1910.
4692:
3853:, but unlike the real numbers, it is not
3833:asserts that the complex numbers form an
3812:
3712:
3366:. A famous irrational real number is the
3269:
3228:
3209:. The symbol for the rational numbers is
2920:
2822:
2793:
2770:
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
6642:
5946:Online Encyclopedia of Integer Sequences
5895:
5710:, New York, The Macmillan Company, 1930.
5660:
5397:
5203:. Harvard University Press. p. 83.
4886:
4736:
4111:
3800:. The symbol for the complex numbers is
3171:{\displaystyle {a\times d}={c\times b}.}
3080:{\displaystyle {1 \over 2}={2 \over 4}.}
2853:: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The
2721:
1835:. This generalization is largely due to
530:
252:{\displaystyle \left({\sqrt {2}}\right)}
38:
5816:
5730:Introduction to Mathematical Structures
5598:Mathematics Unlimited â 2001 and Beyond
5061:
4771:
4727:
4629:
4306:, neither associative nor commutative.
4260:
4116:Motivated by the classical problems of
3881:
3473:( 1 trillion = 10 = 1,000,000,000,000 )
3126:{\displaystyle {a \over b}={c \over d}}
784:referred to the equation equivalent to
362:
14:
7019:
5508:. Courier Dover Publications. p.
5498:
5379:from the original on 23 September 2010
5201:Harvard Studies in Classical Philology
4861:The Princeton Companion to Mathematics
4819:
4817:
4765:
4335:
4309:
4086:. The set of periods form a countable
3478:Not only these prominent examples but
2758:for the set of all natural numbers is
2738:.) However, in the 19th century,
1942:
1045:(1832) linked polynomial equations to
157:is not clearly distinguished from the
6667:
5954:
5898:"What's the World's Favorite Number?"
5661:Lagarias, Jeffrey C. (19 July 2013).
5634:
5587:
5585:
5557:
5533:
5480:from the original on 13 December 2019
5440:
5364:
5162:
5104:
4127:
2965:
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
5933:from the original on 6 November 2018
5014:from the original on 28 January 2017
4893:( ed.). New York: McGraw-Hill.
4650:from the original on 4 February 2019
4625:
4623:
3661:
2726: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:
6162:Set-theoretically definable numbers
5835:
5732:, Harcourt Brace Javanovich, 1989,
5272:âââ, Gesammelte mathematische Werke
5260:Stetigkeit & irrationale Zahlen
5134:"Classical Greek culture (article)"
5112:. Dover Publications. p. 259.
4823:
4814:
4608:from the original on 26 August 2017
4548:from the original on 4 October 2018
3257:The symbol for the real numbers is
2935:. Here the letter Z comes from
2260:is a formal square root of â1
1219:, introducing, among other things,
859:
759:
24:
6133:
5980:
5891:from the original on 8 April 2022.
5582:
5570:from the original on 5 August 2020
5338:, No. 56 (Jan. 1859): 87â99 at 97.
5225:"Die Elemente der Functionenlehre"
5082:The Mathematical Palette (3rd ed.)
4595:
4473: â Number, approximately 3.14
3959:
3912:may be constructed by the formula
3898:divisible by two without remainder
2730:The most familiar numbers are the
2711:
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:
7053:
6693:
5896:Krulwich, Robert (22 July 2011).
5810:
5798:, Oxford University Press, 2015,
5768:, Oxford University Press, 1990.
5716:What's special about this number?
5407:The American Mathematical Monthly
5000:. Cengage Learning. p. 192.
4929:from the original on 7 April 2015
4620:
4577:from the original on 30 July 2022
4302:, in which multiplication is not
4294:, in which multiplication is not
4286:, in which multiplication is not
4205:
4021:Subclasses of the complex numbers
3991: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
7000:
6990:
6980:
6970:
6961:
6960:
6647:
5908:from the original on 18 May 2021
5879:Robin Wilson (7 November 2007).
5069:. Austin, Texas: self published.
4406:
3871:compatible with field operations
3841:with complex coefficients has a
2835:{\displaystyle \mathbb {N} _{1}}
2806:{\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:
5679:10.1090/S0273-0979-2013-01423-X
5654:
5628:
5551:
5526:
5492:
5458:
5433:
5391:
5358:
5341:
5320:
5307:
5294:
5277:
5251:
5234:
5217:
5184:
5156:
5144:from the original on 4 May 2022
5126:
5098:
5073:
5055:
5029:
4941:
4915:
4880:
4852:
4154:which, given a positive number
4053:Periods and exponential periods
3246:
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
6739:analytic theory of L-functions
6717:non-abelian class field theory
6141:{\displaystyle {\mathcal {P}}}
5419:10.1080/00029890.2007.11920416
4686:
4662:
4636:. OUP Oxford. pp. 85â88.
4589:
4560:
4531:
4508:
3831:fundamental theorem of algebra
3297:Most real numbers can only be
3005: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:
6496:Plane-based geometric algebra
5867:In Our Time: Negative Numbers
5697:
5398:MartĂnez, Alberto A. (2007).
5110:History of Modern Mathematics
4827:Number theory and its history
4630:Hodgkin, Luke (2 June 2005).
4370:. This principle allows true
4158:as input, produces the first
3279:{\displaystyle \mathbb {R} .}
928:History of irrational numbers
888:. The best known of these is
713:(otherwise meaning 70).
392:
6763:Transcendental number theory
6456:{\displaystyle \mathbb {S} }
6379:{\displaystyle \mathbb {C} }
6340:{\displaystyle \mathbb {R} }
6302:{\displaystyle \mathbb {O} }
6274:{\displaystyle \mathbb {H} }
6246:{\displaystyle \mathbb {C} }
6218:{\displaystyle \mathbb {R} }
6106:{\displaystyle \mathbb {A} }
6073:{\displaystyle \mathbb {Q} }
6045:{\displaystyle \mathbb {Z} }
6017:{\displaystyle \mathbb {N} }
5606:10.1007/978-3-642-56478-9_39
5317:(Kjoebenhavn: 1855), p. 106.
4887:Marshack, Alexander (1971).
4737:Gilsdorf, Thomas E. (2012).
3819:{\displaystyle \mathbb {C} }
3619:) to the algebraic equation
3235:{\displaystyle \mathbb {Q} }
2927:{\displaystyle \mathbb {Z} }
2777:{\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:
6986:List of recreational topics
6758:Computational number theory
6743:probabilistic number theory
5824:Encyclopedia of Mathematics
5248:, 21, 4 (1883â12): 545â591.
5084:. Brooks Cole. p. 41.
4573:. Oxford University Press.
4544:. Oxford University Press.
4399:
4059:Period (algebraic geometry)
2950:The natural numbers form a
2876:
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:
7058:
5796:A Brief History of Numbers
5328:"Einige Eigenschaften der
5304:, 1779, 1 (1779): 162â187.
5291:, No. 101 (1887): 337â355.
4320:For dealing with infinite
4313:
4264:
4212:
4150:such that there exists an
4131:
4108:are exponential periods.
4056:
3974:, often shortened to just
3963:
3885:
3877:Subclasses of the integers
3835:algebraically closed field
3665:
3613:algebraically closed field
3250:
2969:
2880:
2715:
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:
6956:
6938:Diophantine approximation
6910:
6897:Chinese remainder theorem
6819:
6701:
6638:
6580:
6506:
6486:Algebra of physical space
6408:
6316:
6187:
5989:
5231:, No. 74 (1872): 172â188.
4962:10.1017/S0003598X00092541
4201:Extensions of the concept
4013:. For more examples, see
4001:Other classes of integers
3651:{\displaystyle x^{2}+1=0}
3603: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
6782:Arithmetic combinatorics
6542:Extended complex numbers
6525:Extended natural numbers
4772:Restivo, Sal P. (1992).
4743:. Hoboken, N.J.: Wiley.
4501:
4433:List of types of numbers
4359:of the ordered field of
4243:algebraic function field
3617:square root of minus one
3380:. When pi is written as
1956:List of types of numbers
1852:Victor Alexandre Puiseux
1078:Kettenbruchdeterminanten
1017:The search for roots of
6753:Geometric number theory
6709:Algebraic number theory
5881:"4000 Years of Numbers"
5817:Nechaev, V.I. (2001) .
5285:"Ueber den Zahlbegriff"
4495:Subitizing and counting
4182:that contains the real
3923:for a suitable integer
3694:, a symbol assigned by
3688:of â1, denoted by
3546:and strictly less than
3514:and strictly less than
1887:greatest common divisor
566:double entry accounting
509:, the main work of the
436:
32:Number (disambiguation)
6872:Transcendental numbers
6786:additive number theory
6735:Analytic number theory
6598:Transcendental numbers
6457:
6434:Hyperbolic quaternions
6380:
6341:
6303:
6275:
6247:
6219:
6142:
6107:
6074:
6046:
6018:
5779:Alfred North Whitehead
5197:D.R. Shackleton Bailey
4568:"numeral, adj. and n."
4284:William Rowan Hamilton
4251:Function field analogy
4077:mathematical constants
4039:transcendental numbers
3820:
3729:
3728:{\displaystyle \,a+bi}
3652:
3458:
3409:
3280:
3236:
3172:
3127:
3081:
2928:
2836:
2807:
2778:
2727:
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:
6943:Irrationality measure
6933:Diophantine equations
6776:HodgeâArakelov theory
6530:Extended real numbers
6458:
6381:
6351:Split-complex numbers
6342:
6304:
6276:
6248:
6220:
6143:
6108:
6084:Constructible numbers
6075:
6047:
6019:
5788:Principia Mathematica
5641:mathworld.wolfram.com
5246:Mathematische Annalen
4824:Ore, Ăystein (1988).
4438:Mathematical constant
4345:non-standard analysis
4164:Ό-recursive functions
4122:constructible numbers
4112:Constructible numbers
3993:. A proof appears in
3984:Goldbach's conjecture
3837:, meaning that every
3821:
3788:or is referred to as
3730:
3653:
3459:
3410:
3376:of any circle to its
3305:numerals, in which a
3281:
3237:
3173:
3128:
3082:
2960:non-negative integers
2929:
2837:
2808:
2779:
2725:
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:
7042:Mathematical objects
6902:Arithmetic functions
6768:Diophantine geometry
6562:Supernatural numbers
6472:Multicomplex numbers
6445:
6429:Dual-complex numbers
6368:
6329:
6291:
6263:
6235:
6207:
6189:Composition algebras
6157:Arithmetical numbers
6128:
6095:
6062:
6034:
6006:
5505:Axiomatic Set Theory
4925:. Math.buffalo.edu.
4858:GouvĂȘa, Fernando Q.
4598:"The Origin of Zero"
4489:Scalar (mathematics)
4282:, introduced by Sir
4273:hypercomplex numbers
4261:Hypercomplex numbers
3888:Even and odd numbers
3882:Even and odd numbers
3808:
3758:. In the expression
3709:
3623:
3570:. If the sides of a
3432:
3387:
3360:repetition of zeroes
3265:
3224:
3137:
3097:
3048:
3041:are equal, that is:
2916:
2817:
2788:
2766:
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
6948:Continued fractions
6811:Arithmetic dynamics
6806:Arithmetic topology
6800:P-adic Hodge theory
6792:Arithmetic geometry
6725:IwasawaâTate theory
6467:Split-biquaternions
6179:Eisenstein integers
6117:Closed-form numbers
5836:Tallant, Jonathan.
5635:Weisstein, Eric W.
5560:"Repeating Decimal"
5558:Weisstein, Eric W.
5470:Merriam-Webster.com
5106:Smith, David Eugene
4830:. New York: Dover.
4602:Scientific American
4477:Positional notation
4454:Orders of magnitude
4449:Numerical cognition
4336:Nonstandard numbers
4310:Transfinite numbers
4275:. They include the
4267:hypercomplex number
4241:The elements of an
3908:".) Any odd number
3372:, the ratio of the
2981:them. The fraction
2756: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
6892:Modular arithmetic
6862:Irrational numbers
6796:anabelian geometry
6713:class field theory
6625:Profinite integers
6588:Irrational numbers
6453:
6376:
6337:
6299:
6271:
6243:
6215:
6172:Gaussian rationals
6152:Computable numbers
6138:
6103:
6070:
6042:
6014:
5929:. 21 August 2011.
5838:"Do Numbers Exist"
5750:, Springer, 1974,
5721:2018-02-23 at the
5637:"Algebraic Period"
5535:Weisstein, Eric W.
5442:Weisstein, Eric W.
5369:"What's a number?"
5347:Siegmund GĂŒnther,
5265:2021-07-09 at the
5257:Richard Dedekind,
5067:Arithmetic in Maya
5063:SĂĄnchez, George I.
5043:on 12 January 2012
4704:Dover Publications
4414:Mathematics portal
4368:transfer principle
4366:and satisfies the
4316:transfinite number
4178:, and thus form a
4128:Computable numbers
4071:over an algebraic
4069:algebraic function
4047:algebraic integers
4035:irrational numbers
3816:
3768:, the real number
3725:
3648:
3564:significant digits
3454:
3405:
3276:
3261:, also written as
3232:
3168:
3123:
3077:
2924:
2832:
2803:
2774:
2728:
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:
7014:
7013:
6911:Advanced concepts
6867:Algebraic numbers
6852:Composite numbers
6661:
6660:
6572:Superreal numbers
6552:Levi-Civita field
6547:Hyperreal numbers
6491:Spacetime algebra
6477:Geometric algebra
6390:Bicomplex numbers
6356:Split-quaternions
6197:Division algebras
6167:Gaussian integers
6089:Algebraic numbers
5992:definable numbers
5804:978-0-19-870259-7
5728:Steven Galovich,
5615:978-3-642-56478-9
5564:Wolfram MathWorld
5332:schen Funktionen"
5007:978-1-4390-8474-8
4871:978-0-691-11880-2
4785:978-94-011-2944-2
4750:978-1-118-19416-4
4643:978-0-19-152383-0
4465:Physical quantity
4459:Physical constant
4374:statements about
4355:that is a proper
4341:Hyperreal numbers
4184:algebraic numbers
4180:real closed field
4140:computable number
4134:Computable number
4031:Algebraic numbers
4007:Fibonacci numbers
3995:Euclid's Elements
3698:, and called the
3598:least upper bound
3440:
3333:repeating decimal
3294:, e.g. â123.456.
3121:
3108:
3072:
3059:
2956:positive integers
2709:
2708:
2705:
2704:
2701:
2700:
2697:
2696:
2686:
2685:
2682:
2681:
2678:
2677:
2674:
2673:
2662:Irrational period
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
7049:
7004:
6994:
6984:
6983:
6974:
6973:
6964:
6963:
6857:Rational numbers
6688:
6681:
6674:
6665:
6664:
6651:
6650:
6618:
6608:
6520:Cardinal numbers
6481:Clifford algebra
6462:
6460:
6459:
6454:
6452:
6424:Dual quaternions
6385:
6383:
6382:
6377:
6375:
6346:
6344:
6343:
6338:
6336:
6308:
6306:
6305:
6300:
6298:
6280:
6278:
6277:
6272:
6270:
6252:
6250:
6249:
6244:
6242:
6224:
6222:
6221:
6216:
6214:
6147:
6145:
6144:
6139:
6137:
6136:
6112:
6110:
6109:
6104:
6102:
6079:
6077:
6076:
6071:
6069:
6056:Rational numbers
6051:
6049:
6048:
6043:
6041:
6023:
6021:
6020:
6015:
6013:
5975:
5968:
5961:
5952:
5951:
5942:
5940:
5938:
5917:
5915:
5913:
5892:
5875:
5861:
5859:
5857:
5848:. Archived from
5832:
5783:Bertrand Russell
5748:Naive Set Theory
5713:Erich Friedman,
5691:
5690:
5658:
5652:
5651:
5649:
5647:
5632:
5626:
5625:
5624:
5622:
5589:
5580:
5579:
5577:
5575:
5555:
5549:
5548:
5547:
5530:
5524:
5523:
5496:
5490:
5489:
5487:
5485:
5466:"natural number"
5462:
5456:
5455:
5454:
5445:"Natural Number"
5437:
5431:
5430:
5404:
5395:
5389:
5388:
5386:
5384:
5362:
5356:
5345:
5339:
5324:
5318:
5311:
5305:
5298:
5292:
5281:
5275:
5255:
5249:
5238:
5232:
5221:
5215:
5214:
5188:
5182:
5181:
5160:
5154:
5153:
5151:
5149:
5130:
5124:
5123:
5102:
5096:
5095:
5077:
5071:
5070:
5059:
5053:
5052:
5050:
5048:
5033:
5027:
5026:
5021:
5019:
4991:
4982:
4981:
4945:
4939:
4938:
4936:
4934:
4919:
4913:
4912:
4884:
4878:
4856:
4850:
4849:
4821:
4812:
4811:
4805:
4797:
4769:
4763:
4762:
4734:
4725:
4724:
4722:
4720:
4690:
4684:
4683:
4666:
4660:
4659:
4657:
4655:
4627:
4618:
4617:
4615:
4613:
4593:
4587:
4586:
4584:
4582:
4564:
4558:
4557:
4555:
4553:
4535:
4523:
4512:
4416:
4411:
4410:
4330:cardinal numbers
4144:recursive number
4142:, also known as
4106:Euler's constant
4043:monic polynomial
4015:Integer sequence
3947:
3933:
3927:. Starting with
3922:
3825:
3823:
3822:
3817:
3815:
3798:Gaussian integer
3790:purely imaginary
3786:imaginary number
3767:
3734:
3732:
3731:
3726:
3657:
3655:
3654:
3649:
3635:
3634:
3607:, which is also
3581:
3577:
3561:
3559:
3558:
3555:
3552:
3545:
3543:
3542:
3539:
3536:
3529:
3527:
3526:
3523:
3520:
3513:
3511:
3510:
3507:
3504:
3474:
3469:square root of 2
3463:
3461:
3460:
3455:
3441:
3436:
3422:
3414:
3412:
3411:
3406:
3397:3.14159265358979
3370:
3354:
3350:
3348:
3347:
3344:
3341:
3325:
3323:
3322:
3319:
3316:
3285:
3283:
3282:
3277:
3272:
3241:
3239:
3238:
3233:
3231:
3219:), also written
3208:
3206:
3205:
3202:
3199:
3189:is greater than
3177:
3175:
3174:
3169:
3164:
3150:
3132:
3130:
3129:
3124:
3122:
3114:
3109:
3101:
3086:
3084:
3083:
3078:
3073:
3065:
3060:
3052:
3040:
3038:
3037:
3034:
3031:
3024:
3022:
3021:
3018:
3015:
3000:
2998:
2997:
2992:
2989:
2966:Rational numbers
2933:
2931:
2930:
2925:
2923:
2898:
2871:Peano Arithmetic
2841:
2839:
2838:
2833:
2831:
2830:
2825:
2812:
2810:
2809:
2804:
2802:
2801:
2796:
2784:, and sometimes
2783:
2781:
2780:
2775:
2773:
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:
7057:
7056:
7052:
7051:
7050:
7048:
7047:
7046:
7017:
7016:
7015:
7010:
6952:
6918:Quadratic forms
6906:
6881:P-adic analysis
6837:Natural numbers
6815:
6772:Arakelov theory
6697:
6692:
6662:
6657:
6634:
6613:
6603:
6576:
6567:Surreal numbers
6557:Ordinal numbers
6502:
6448:
6446:
6443:
6442:
6404:
6371:
6369:
6366:
6365:
6363:
6361:Split-octonions
6332:
6330:
6327:
6326:
6318:
6312:
6294:
6292:
6289:
6288:
6266:
6264:
6261:
6260:
6238:
6236:
6233:
6232:
6229:Complex numbers
6210:
6208:
6205:
6204:
6183:
6132:
6131:
6129:
6126:
6125:
6098:
6096:
6093:
6092:
6065:
6063:
6060:
6059:
6037:
6035:
6032:
6031:
6009:
6007:
6004:
6003:
6000:Natural numbers
5985:
5979:
5936:
5934:
5919:
5911:
5909:
5885:Gresham College
5874:on 31 May 2022.
5864:
5855:
5853:
5852:on 8 March 2016
5813:
5723:Wayback Machine
5700:
5695:
5694:
5659:
5655:
5645:
5643:
5633:
5629:
5620:
5618:
5616:
5590:
5583:
5573:
5571:
5556:
5552:
5531:
5527:
5520:
5500:Suppes, Patrick
5497:
5493:
5483:
5481:
5474:Merriam-Webster
5464:
5463:
5459:
5438:
5434:
5402:
5396:
5392:
5382:
5380:
5363:
5359:
5346:
5342:
5325:
5321:
5312:
5308:
5299:
5295:
5282:
5278:
5267:Wayback Machine
5256:
5252:
5239:
5235:
5222:
5218:
5211:
5189:
5185:
5178:
5161:
5157:
5147:
5145:
5132:
5131:
5127:
5120:
5103:
5099:
5092:
5078:
5074:
5060:
5056:
5046:
5044:
5035:
5034:
5030:
5017:
5015:
5008:
4992:
4985:
4956:(297): 485â96.
4946:
4942:
4932:
4930:
4921:
4920:
4916:
4901:
4885:
4881:
4857:
4853:
4838:
4822:
4815:
4799:
4798:
4786:
4770:
4766:
4751:
4735:
4728:
4718:
4716:
4714:
4694:Descartes, René
4691:
4687:
4680:
4668:
4667:
4663:
4653:
4651:
4644:
4628:
4621:
4611:
4609:
4594:
4590:
4580:
4578:
4566:
4565:
4561:
4551:
4549:
4537:
4536:
4532:
4527:
4526:
4513:
4509:
4504:
4499:
4444:Complex numbers
4428:List of numbers
4423:Concrete number
4412:
4405:
4402:
4390:surreal numbers
4338:
4326:ordinal numbers
4318:
4312:
4269:
4263:
4220:
4211:
4203:
4168:Turing machines
4136:
4130:
4114:
4061:
4055:
4028:
4023:
4011:perfect numbers
4003:
3968:
3962:
3939:
3928:
3913:
3890:
3884:
3879:
3811:
3809:
3806:
3805:
3759:
3710:
3707:
3706:
3674:complex numbers
3670:
3664:
3662:Complex numbers
3630:
3626:
3624:
3621:
3620:
3579:
3575:
3556:
3553:
3550:
3549:
3547:
3540:
3537:
3534:
3533:
3531:
3524:
3521:
3518:
3517:
3515:
3508:
3505:
3502:
3501:
3499:
3496:margin of error
3472:
3435:
3433:
3430:
3429:
3420:
3388:
3385:
3384:
3368:
3352:
3345:
3342:
3339:
3338:
3336:
3328:fractional part
3320:
3317:
3314:
3313:
3311:
3268:
3266:
3263:
3262:
3255:
3249:
3227:
3225:
3222:
3221:
3203:
3200:
3197:
3196:
3194:
3154:
3140:
3138:
3135:
3134:
3133:if and only if
3113:
3100:
3098:
3095:
3094:
3064:
3051:
3049:
3046:
3045:
3035:
3032:
3029:
3028:
3026:
3019:
3016:
3013:
3012:
3010:
2993:
2990:
2985:
2984:
2982:
2974:
2972:Rational number
2968:
2919:
2917:
2914:
2913:
2896:
2885:
2879:
2826:
2821:
2820:
2818:
2815:
2814:
2797:
2792:
2791:
2789:
2786:
2785:
2769:
2767:
2764:
2763:
2762:, also written
2752:cardinal number
2732:natural numbers
2720:
2714:
2712: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:
18:Numerical value
15:
12:
11:
5:
7055:
7045:
7044:
7039:
7034:
7029:
7012:
7011:
7009:
7008:
6998:
6988:
6978:
6976:List of topics
6968:
6957:
6954:
6953:
6951:
6950:
6945:
6940:
6935:
6930:
6925:
6920:
6914:
6912:
6908:
6907:
6905:
6904:
6899:
6894:
6889:
6884:
6877:P-adic numbers
6874:
6869:
6864:
6859:
6854:
6849:
6844:
6839:
6834:
6829:
6823:
6821:
6817:
6816:
6814:
6813:
6808:
6803:
6789:
6779:
6765:
6760:
6755:
6750:
6732:
6721:Iwasawa theory
6705:
6703:
6699:
6698:
6691:
6690:
6683:
6676:
6668:
6659:
6658:
6656:
6655:
6645:
6643:Classification
6639:
6636:
6635:
6633:
6632:
6630:Normal numbers
6627:
6622:
6600:
6595:
6590:
6584:
6582:
6578:
6577:
6575:
6574:
6569:
6564:
6559:
6554:
6549:
6544:
6539:
6538:
6537:
6527:
6522:
6516:
6514:
6512:infinitesimals
6504:
6503:
6501:
6500:
6499:
6498:
6493:
6488:
6474:
6469:
6464:
6451:
6436:
6431:
6426:
6421:
6415:
6413:
6406:
6405:
6403:
6402:
6397:
6392:
6387:
6374:
6358:
6353:
6348:
6335:
6322:
6320:
6314:
6313:
6311:
6310:
6297:
6282:
6269:
6254:
6241:
6226:
6213:
6193:
6191:
6185:
6184:
6182:
6181:
6176:
6175:
6174:
6164:
6159:
6154:
6149:
6135:
6119:
6114:
6101:
6086:
6081:
6068:
6053:
6040:
6025:
6012:
5996:
5994:
5987:
5986:
5978:
5977:
5970:
5963:
5955:
5949:
5948:
5943:
5893:
5876:
5862:
5833:
5812:
5811:External links
5809:
5808:
5807:
5792:
5776:
5774:978-0195061352
5759:
5741:
5726:
5711:
5704:Tobias Dantzig
5699:
5696:
5693:
5692:
5673:(4): 527â628.
5653:
5627:
5614:
5581:
5550:
5525:
5518:
5491:
5457:
5432:
5413:(4): 273â285.
5390:
5357:
5340:
5326:Eduard Heine,
5319:
5306:
5293:
5283:L. Kronecker,
5276:
5250:
5240:Georg Cantor,
5233:
5223:Eduard Heine,
5216:
5209:
5183:
5176:
5166:, ed. (2000).
5164:Selin, Helaine
5155:
5125:
5118:
5097:
5090:
5072:
5054:
5028:
5006:
4983:
4940:
4914:
4899:
4879:
4851:
4836:
4813:
4784:
4764:
4749:
4726:
4712:
4685:
4678:
4661:
4642:
4619:
4596:Matson, John.
4588:
4559:
4529:
4528:
4525:
4524:
4506:
4505:
4503:
4500:
4498:
4497:
4492:
4486:
4480:
4474:
4468:
4462:
4456:
4451:
4446:
4441:
4435:
4430:
4425:
4419:
4418:
4417:
4401:
4398:
4337:
4334:
4314:Main article:
4311:
4308:
4265:Main article:
4262:
4259:
4213:Main article:
4210:
4204:
4202:
4199:
4132:Main article:
4129:
4126:
4113:
4110:
4057:Main article:
4054:
4051:
4027:
4024:
4022:
4019:
4002:
3999:
3964:Main article:
3961:
3958:
3886:Main article:
3883:
3880:
3878:
3875:
3814:
3782:imaginary part
3780:is called the
3772:is called the
3736:
3735:
3724:
3721:
3718:
3715:
3700:imaginary unit
3696:Leonhard Euler
3668:Complex number
3666:Main article:
3663:
3660:
3647:
3644:
3641:
3638:
3633:
3629:
3492:countably many
3465:
3464:
3453:
3450:
3447:
3444:
3439:
3416:
3415:
3404:
3401:
3398:
3395:
3392:
3275:
3271:
3251:Main article:
3248:
3245:
3230:
3183:absolute value
3179:
3178:
3167:
3163:
3160:
3157:
3153:
3149:
3146:
3143:
3120:
3117:
3112:
3107:
3104:
3088:
3087:
3076:
3071:
3068:
3063:
3058:
3055:
2970:Main article:
2967:
2964:
2922:
2881:Main article:
2878:
2875:
2829:
2824:
2800:
2795:
2772:
2736:Ancient Greeks
2718:Natural number
2716:Main article:
2713:
2710:
2707:
2706:
2703:
2702:
2699:
2698:
2695:
2694:
2688:
2687:
2684:
2683:
2680:
2679:
2676:
2675:
2672:
2671:
2669:Transcendental
2665:
2664:
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:
7054:
7043:
7040:
7038:
7035:
7033:
7030:
7028:
7025:
7024:
7022:
7007:
7003:
6999:
6997:
6993:
6989:
6987:
6979:
6977:
6969:
6967:
6959:
6958:
6955:
6949:
6946:
6944:
6941:
6939:
6936:
6934:
6931:
6929:
6926:
6924:
6923:Modular forms
6921:
6919:
6916:
6915:
6913:
6909:
6903:
6900:
6898:
6895:
6893:
6890:
6888:
6885:
6882:
6878:
6875:
6873:
6870:
6868:
6865:
6863:
6860:
6858:
6855:
6853:
6850:
6848:
6847:Prime numbers
6845:
6843:
6840:
6838:
6835:
6833:
6830:
6828:
6825:
6824:
6822:
6818:
6812:
6809:
6807:
6804:
6801:
6797:
6793:
6790:
6787:
6783:
6780:
6777:
6773:
6769:
6766:
6764:
6761:
6759:
6756:
6754:
6751:
6748:
6744:
6740:
6736:
6733:
6730:
6729:Kummer theory
6726:
6722:
6718:
6714:
6710:
6707:
6706:
6704:
6700:
6696:
6695:Number theory
6689:
6684:
6682:
6677:
6675:
6670:
6669:
6666:
6654:
6646:
6644:
6641:
6640:
6637:
6631:
6628:
6626:
6623:
6620:
6616:
6610:
6606:
6601:
6599:
6596:
6594:
6593:Fuzzy numbers
6591:
6589:
6586:
6585:
6583:
6579:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6536:
6533:
6532:
6531:
6528:
6526:
6523:
6521:
6518:
6517:
6515:
6513:
6509:
6505:
6497:
6494:
6492:
6489:
6487:
6484:
6483:
6482:
6478:
6475:
6473:
6470:
6468:
6465:
6440:
6437:
6435:
6432:
6430:
6427:
6425:
6422:
6420:
6417:
6416:
6414:
6412:
6407:
6401:
6398:
6396:
6395:Biquaternions
6393:
6391:
6388:
6362:
6359:
6357:
6354:
6352:
6349:
6324:
6323:
6321:
6315:
6286:
6283:
6258:
6255:
6230:
6227:
6202:
6198:
6195:
6194:
6192:
6190:
6186:
6180:
6177:
6173:
6170:
6169:
6168:
6165:
6163:
6160:
6158:
6155:
6153:
6150:
6123:
6120:
6118:
6115:
6090:
6087:
6085:
6082:
6057:
6054:
6029:
6026:
6001:
5998:
5997:
5995:
5993:
5988:
5983:
5976:
5971:
5969:
5964:
5962:
5957:
5956:
5953:
5947:
5944:
5932:
5928:
5927:
5922:
5907:
5903:
5899:
5894:
5890:
5886:
5882:
5877:
5873:
5869:
5868:
5863:
5851:
5847:
5843:
5839:
5834:
5830:
5826:
5825:
5820:
5815:
5814:
5805:
5801:
5797:
5793:
5790:
5789:
5784:
5780:
5777:
5775:
5771:
5767:
5763:
5760:
5757:
5756:0-387-90092-6
5753:
5749:
5745:
5742:
5739:
5738:0-15-543468-3
5735:
5731:
5727:
5725:
5724:
5720:
5717:
5712:
5709:
5705:
5702:
5701:
5688:
5684:
5680:
5676:
5672:
5668:
5664:
5657:
5642:
5638:
5631:
5617:
5611:
5607:
5603:
5599:
5595:
5588:
5586:
5569:
5565:
5561:
5554:
5545:
5544:
5539:
5536:
5529:
5521:
5519:0-486-61630-4
5515:
5511:
5507:
5506:
5501:
5495:
5479:
5475:
5471:
5467:
5461:
5452:
5451:
5446:
5443:
5436:
5428:
5424:
5420:
5416:
5412:
5408:
5401:
5394:
5378:
5374:
5370:
5367:
5366:Bogomolny, A.
5361:
5354:
5350:
5344:
5337:
5333:
5331:
5323:
5316:
5310:
5303:
5297:
5290:
5286:
5280:
5273:
5269:
5268:
5264:
5261:
5254:
5247:
5243:
5237:
5230:
5226:
5220:
5212:
5210:0-674-37935-7
5206:
5202:
5198:
5194:
5187:
5179:
5177:0-7923-6481-3
5173:
5169:
5165:
5159:
5143:
5139:
5135:
5129:
5121:
5119:0-486-20429-4
5115:
5111:
5107:
5101:
5093:
5091:0-534-40365-4
5087:
5083:
5076:
5068:
5064:
5058:
5042:
5038:
5032:
5025:
5013:
5009:
5003:
4999:
4998:
4990:
4988:
4979:
4975:
4971:
4967:
4963:
4959:
4955:
4951:
4944:
4928:
4924:
4918:
4910:
4906:
4902:
4900:0-07-040535-2
4896:
4892:
4891:
4883:
4876:
4872:
4868:
4864:
4862:
4855:
4847:
4843:
4839:
4837:0-486-65620-9
4833:
4829:
4828:
4820:
4818:
4809:
4803:
4795:
4791:
4787:
4781:
4778:. Dordrecht.
4777:
4776:
4768:
4760:
4756:
4752:
4746:
4742:
4741:
4733:
4731:
4715:
4713:0-486-60068-8
4709:
4705:
4701:
4700:
4695:
4689:
4681:
4679:1-4020-0260-2
4675:
4671:
4665:
4649:
4645:
4639:
4635:
4634:
4626:
4624:
4607:
4603:
4599:
4592:
4576:
4572:
4569:
4563:
4547:
4543:
4540:
4534:
4530:
4521:
4517:
4511:
4507:
4496:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4420:
4415:
4409:
4404:
4397:
4395:
4391:
4387:
4383:
4381:
4377:
4373:
4369:
4365:
4362:
4358:
4354:
4353:ordered field
4351:), denote an
4350:
4346:
4342:
4333:
4331:
4327:
4323:
4317:
4307:
4305:
4301:
4297:
4293:
4289:
4285:
4281:
4278:
4274:
4268:
4258:
4256:
4252:
4248:
4244:
4239:
4237:
4233:
4229:
4225:
4219:
4217:
4209:-adic numbers
4208:
4198:
4196:
4191:
4187:
4185:
4181:
4177:
4173:
4169:
4165:
4161:
4157:
4153:
4149:
4145:
4141:
4135:
4125:
4123:
4119:
4109:
4107:
4103:
4102:
4096:
4091:
4089:
4085:
4084:
4078:
4074:
4070:
4066:
4060:
4050:
4048:
4044:
4040:
4036:
4032:
4018:
4016:
4012:
4008:
3998:
3996:
3992:
3987:
3985:
3981:
3980:number theory
3977:
3973:
3967:
3960:Prime numbers
3957:
3955:
3951:
3946:
3942:
3938:has the form
3937:
3931:
3926:
3920:
3916:
3911:
3907:
3903:
3899:
3895:
3889:
3874:
3872:
3868:
3864:
3860:
3856:
3852:
3848:
3844:
3840:
3836:
3832:
3827:
3803:
3799:
3795:
3791:
3787:
3783:
3779:
3775:
3771:
3766:
3762:
3757:
3753:
3749:
3748:complex plane
3745:
3741:
3722:
3719:
3716:
3713:
3705:
3704:
3703:
3701:
3697:
3693:
3692:
3687:
3683:
3679:
3675:
3669:
3659:
3645:
3642:
3639:
3636:
3631:
3627:
3618:
3614:
3610:
3606:
3605:ordered field
3601:
3599:
3594:
3592:
3587:
3585:
3573:
3569:
3565:
3497:
3493:
3489:
3485:
3481:
3476:
3470:
3451:
3448:
3446:1.41421356237
3445:
3442:
3437:
3428:
3427:
3426:
3424:
3423:is irrational
3402:
3399:
3396:
3393:
3390:
3383:
3382:
3381:
3379:
3375:
3374:circumference
3371:
3365:
3361:
3356:
3334:
3329:
3308:
3307:decimal point
3304:
3300:
3295:
3293:
3289:
3273:
3260:
3254:
3244:
3243:
3218:
3217:
3212:
3192:
3188:
3184:
3165:
3161:
3158:
3155:
3151:
3147:
3144:
3141:
3118:
3115:
3110:
3105:
3102:
3093:
3092:
3091:
3074:
3069:
3066:
3061:
3056:
3053:
3044:
3043:
3042:
3008:
3004:
2996:
2988:
2979:
2973:
2963:
2961:
2957:
2953:
2948:
2946:
2942:
2938:
2934:
2911:also written
2910:
2906:
2902:
2894:
2890:
2884:
2874:
2872:
2868:
2863:
2861:
2856:
2855:radix or base
2852:
2848:
2843:
2827:
2798:
2761:
2757:
2753:
2749:
2745:
2741:
2740:set theorists
2737:
2733:
2724:
2719:
2693:
2690:
2689:
2670:
2667:
2666:
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:
7032:Group theory
6826:
6820:Key concepts
6747:sieve theory
6614:
6604:
6419:Dual numbers
6411:hypercomplex
6201:Real numbers
5981:
5937:17 September
5935:. Retrieved
5924:
5912:17 September
5910:. Retrieved
5901:
5872:the original
5866:
5854:. Retrieved
5850:the original
5841:
5822:
5795:
5786:
5765:
5762:Morris Kline
5747:
5729:
5714:
5707:
5670:
5666:
5656:
5646:22 September
5644:. Retrieved
5640:
5630:
5621:22 September
5619:, retrieved
5597:
5572:. Retrieved
5563:
5553:
5541:
5528:
5504:
5494:
5482:. Retrieved
5469:
5460:
5448:
5435:
5410:
5406:
5393:
5381:. Retrieved
5372:
5360:
5352:
5348:
5343:
5335:
5329:
5322:
5314:
5309:
5301:
5296:
5288:
5279:
5271:
5258:
5253:
5245:
5236:
5228:
5219:
5200:
5192:
5186:
5167:
5158:
5146:. Retrieved
5138:Khan Academy
5137:
5128:
5109:
5100:
5081:
5075:
5066:
5057:
5045:. Retrieved
5041:the original
5031:
5023:
5016:. Retrieved
4996:
4953:
4949:
4943:
4931:. Retrieved
4917:
4889:
4882:
4874:
4859:
4854:
4826:
4774:
4767:
4739:
4717:. Retrieved
4698:
4688:
4669:
4664:
4652:. Retrieved
4632:
4610:. Retrieved
4601:
4591:
4579:. Retrieved
4570:
4562:
4550:. Retrieved
4541:
4539:"number, n."
4533:
4510:
4483:Prime number
4384:
4379:
4375:
4363:
4361:real numbers
4348:
4343:are used in
4339:
4319:
4279:
4270:
4254:
4247:finite field
4240:
4235:
4232:prime number
4223:
4221:
4218:-adic number
4215:
4206:
4192:
4188:
4159:
4155:
4143:
4139:
4137:
4115:
4100:
4092:
4082:
4079:such as the
4062:
4029:
4004:
3988:
3975:
3972:prime number
3971:
3969:
3966:Prime number
3952:is again an
3949:
3944:
3940:
3935:
3929:
3924:
3918:
3914:
3909:
3901:
3893:
3891:
3862:
3858:
3828:
3801:
3789:
3777:
3769:
3764:
3760:
3754:of two real
3752:vector space
3743:
3739:
3737:
3689:
3671:
3602:
3595:
3588:
3583:
3477:
3466:
3417:
3357:
3299:approximated
3298:
3296:
3258:
3256:
3247:Real numbers
3214:
3210:
3190:
3186:
3180:
3090:In general,
3089:
3006:
3002:
2994:
2986:
2975:
2959:
2955:
2949:
2940:
2908:
2897:7 + (â7) = 0
2886:
2864:
2844:
2759:
2729:
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:
7037:Abstraction
7006:Wikiversity
6928:L-functions
6581:Other types
6400:Bioctonions
6257:Quaternions
5846:Brady Haran
5842:Numberphile
5744:Paul Halmos
4516:linguistics
4372:first-order
4328:and to the
4304:alternative
4296:associative
4288:commutative
4277:quaternions
4148:real number
4104:as well as
4095:exponential
3894:even number
3867:total order
3849:, which is
3686:square root
3584:significant
3475:of digits.
3288:number line
3253:Real number
3001:represents
2899:. When the
2860:place value
2744: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
7021:Categories
6887:Arithmetic
6535:Projective
6508:Infinities
5794:Leo Cory,
5698:References
5047:30 January
4933:30 January
4571:OED Online
4542:OED Online
4195:almost all
4176:polynomial
4172:λ-calculus
3902:odd number
3839:polynomial
3756:dimensions
3600:property.
3580:5.603011 m
3576:5.614591 m
3480:almost all
3364:irrational
3292:minus sign
2893:minus sign
2867: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
6619:solenoids
6439:Sedenions
6285:Octonions
5829:EMS Press
5687:0273-0979
5594:"Periods"
5543:MathWorld
5538:"Integer"
5484:4 October
5450:MathWorld
4978:160523072
4970:0003-598X
4950:Antiquity
4802:cite book
4794:883391697
4759:793103475
4696:(1954) .
4386:Superreal
4357:extension
4300:sedenions
4292:octonions
4152:algorithm
3906:divisible
3774:real part
3682:quadratic
3572:rectangle
3488:truncated
3449:…
3400:…
3391:π
3159:×
3145:×
2748:empty set
2692: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
6996:Wikibook
6966:Category
6028:Integers
5990:Sets of
5931:Archived
5906:Archived
5889:Archived
5819:"Number"
5719:Archived
5568:Archived
5502:(1972).
5478:Archived
5427:43778192
5377:Archived
5263:Archived
5142:Archived
5108:(1958).
5065:(1961).
5012:Archived
4927:Archived
4846:17413345
4719:20 April
4648:Archived
4606:Archived
4575:Archived
4546:Archived
4400:See also
4065:integral
3869:that is
3851:complete
3609:complete
3591:0.999...
3378:diameter
3216:quotient
2978:fraction
2905:integers
2889:negative
2877: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
7027:Numbers
6827:Numbers
6609:numbers
6441: (
6287: (
6259: (
6231: (
6203: (
6124: (
6122:Periods
6091: (
6058: (
6030: (
6002: (
5984:systems
5856:6 April
5574:23 July
5383:11 July
5199:(ed.).
4520:numeral
4245:over a
4146:, is a
4099:number
4081:number
3954:integer
3855:ordered
3560:
3548:
3544:
3532:
3528:
3519:1234565
3516:
3512:
3503:1234555
3500:
3484:rounded
3349:
3337:
3335:. Thus
3324:
3312:
3303:decimal
3207:
3195:
3181:If the
3039:
3027:
3023:
3011:
2999:
2983:
2883:Integer
2847:base 10
2845:In the
2746:of the
2503:Natural
2465:Integer
2351:Complex
2161:
2145:
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
6702:Fields
6409:Other
5982:Number
5802:
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5174:
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4654:16 May
4640:
4612:16 May
4581:16 May
4552:16 May
4394:fields
4290:, the
4120:, the
4073:domain
4067:of an
3948:where
3794:subset
3738:where
3551:123457
3535:123456
3315:123456
2952:subset
2937:German
2851:digits
2268:subset
2248:where
2163:where
1753:Wallis
1668:while
1252:Newton
1117:Cantor
1073:Möbius
1043:Galois
1029:1799,
982:, and
968:Euclid
934:Indian
868:. The
750:Easter
628:. The
626:vacuum
590:PÄáčini
581:shunya
575:Shunye
352:fields
296:, and
265:, and
159:number
139:digits
86:, and
72:number
6842:Unity
6617:-adic
6607:-adic
6364:Over
6325:Over
6319:types
6317:Split
5423:S2CID
5403:(PDF)
5148:4 May
4974:S2CID
4502:Notes
3976:prime
3900:; an
3847:field
3678:cubic
3525:10000
3509:10000
3213:(for
2939:
1527:Euler
1295:of a
1061:Euler
908:sutra
805:, in
737:nihil
727:nulla
649:glyph
645:Olmec
572:word
400:Egypt
348:rings
172:(0),
151:ISBNs
88:label
80:count
74:is a
6653:List
6510:and
5939:2011
5914:2011
5858:2013
5800:ISBN
5781:and
5770:ISBN
5752:ISBN
5734:ISBN
5683:ISSN
5648:2024
5623:2024
5610:ISBN
5576:2020
5514:ISBN
5486:2014
5385:2010
5330:Lamé
5205:ISBN
5172:ISBN
5150:2022
5114:ISBN
5086:ISBN
5049:2012
5020:2017
5002:ISBN
4966:ISSN
4935:2012
4905:OCLC
4895:ISBN
4867:ISBN
4842:OCLC
4832:ISBN
4808:link
4790:OCLC
4780:ISBN
4755:OCLC
4745:ISBN
4721:2011
4708:ISBN
4674:ISBN
4656:2017
4638:ISBN
4614:2017
4583:2017
4554:2017
4518:, a
4388:and
4322:sets
4228:base
4222:The
4088:ring
4009:and
3932:= 0,
3921:+ 1,
3843:root
3829:The
3776:and
3750:, a
3742:and
3680:and
3578:and
3557:1000
3541:1000
3467:the
3321:1000
3025:and
2945:ring
2941:Zahl
2887:The
2547:: 1
2539:: 0
2537:Zero
2389:Real
2252:and
2167:and
1962:sets
1935:and
1780:and
1567:and
1521:and
1455:and
1303:and
1254:and
1242:and
1193:and
1159:Jain
1031:Abel
837:Flos
754:Bede
586:void
501:zero
437:Zero
427:zero
350:and
259:and
170:zero
123:five
5926:NPR
5902:NPR
5675:doi
5602:doi
5415:doi
5411:114
5193:Ode
4958:doi
4514:In
4170:or
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3892:An
3804:or
3486:or
3301:by
3185:of
2901:set
2865:In
2813:or
2545:One
2048:or
1968:or
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1687:cos
1672:of
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1626:cos
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632:of
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578:or
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