View source for Number

{{Short description|Used to count, measure, and label}}
{{Other uses}}
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[[File:NumberSetinC.svg|thumb|[[Set inclusion]]s between the [[natural number]]s {{bug workaround|(ℕ), the [[integer]]s (ℤ), the [[rational number]]s (ℚ), the [[real number]]s (ℝ), and the [[complex number]]s (ℂ)}}]]
A '''number''' is a [[mathematical object]] used to count, measure, and label. The most basic examples are the [[natural number]]s 1, 2, 3, 4, and so forth.<ref>{{Cite journal |title=number, n. |url=http://www.oed.com/view/Entry/129082 |journal=OED Online |language=en-GB |publisher=Oxford University Press |access-date=2017-05-16 |archive-url=https://web.archive.org/web/20181004081907/http://www.oed.com/view/Entry/129082 |archive-date=2018-10-04 |url-status=live }}</ref> Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called ''numerals''; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly arranged in a [[numeral system]], which is an organized way to represent any number. The most common numeral system is the [[Hindu–Arabic numeral system]], which allows for the representation of any [[Integer|non-negative integer]] using a combination of ten fundamental numeric symbols, called [[numerical digit|digit]]s.<ref>{{Cite journal |title=numeral, adj. and n. |url=http://www.oed.com/view/Entry/129111 |journal=OED Online |publisher=Oxford University Press |access-date=2017-05-16 |archive-date=2022-07-30 |archive-url=https://web.archive.org/web/20220730095156/https://www.oed.com/start;jsessionid=B9929F0647C8EE5D4FDB3A3C1B2CA3C3?authRejection=true&url=%2Fview%2FEntry%2F129111 |url-status=live }}</ref>{{efn|In [[linguistics]], a [[numeral (linguistics)|numeral]] 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".}} In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with [[serial number]]s), and for codes (as with [[ISBN]]s). In common usage, a ''numeral'' is not clearly distinguished from the ''number'' that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0),<ref>{{Cite news |url=https://www.scientificamerican.com/article/history-of-zero/ |title=The Origin of Zero |last=Matson |first=John |work=Scientific American |access-date=2017-05-16 |language=en |archive-url=https://web.archive.org/web/20170826235655/https://www.scientificamerican.com/article/history-of-zero/ |archive-date=2017-08-26 |url-status=live }}</ref> [[negative number]]s,<ref name=":0">{{Cite book |url=https://books.google.com/books?id=f6HlhlBuQUgC&pg=PA88 |title=A History of Mathematics: From Mesopotamia to Modernity |last=Hodgkin |first=Luke |date=2005-06-02 |publisher=OUP Oxford |isbn=978-0-19-152383-0 |pages=85–88 |language=en |access-date=2017-05-16 |archive-url=https://web.archive.org/web/20190204012433/https://books.google.com/books?id=f6HlhlBuQUgC&pg=PA88 |archive-date=2019-02-04 |url-status=live }}</ref> [[rational number]]s such as [[one half]] <math>\left(\tfrac{1}{2}\right)</math>, [[real number]]s such as the [[square root of 2]] <math>\left(\sqrt{2}\right)</math> and [[pi|{{pi}}]],<ref>{{cite book |title=Mathematics across cultures : the history of non-western mathematics |date=2000 |publisher=Kluwer Academic |location=Dordrecht |isbn=1-4020-0260-2 |pages=410–411}}</ref> and [[complex number]]s<ref>{{Cite book |last=Descartes |first=René |title=La Géométrie: The Geometry of René Descartes with a facsimile of the first edition |url=https://archive.org/details/geometryofrenede00rend |year=1954 |author-link=René Descartes |orig-year=1637 |publisher=[[Dover Publications]] |isbn=0-486-60068-8 |access-date=20 April 2011 }}</ref> which extend the real numbers with a [[imaginary unit|square root of {{math|−1}}]] (and its combinations with real numbers by adding or subtracting its multiples).<ref name=":0" /> [[Calculation]]s with numbers are done with arithmetical operations, the most familiar being [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[exponentiation]]. Their study or usage is called [[arithmetic]], a term which may also refer to [[number theory]], the study of the properties of numbers.

Besides their practical uses, numbers have cultural significance throughout the world.<ref name="Gilsdorf">{{Cite book |last=Gilsdorf |first=Thomas E. |url=https://books.google.com/books?id=IN8El-TTlSQC |title=Introduction to cultural mathematics : with case studies in the Otomies and the Incas |date=2012 |publisher=Wiley |isbn=978-1-118-19416-4 |location=Hoboken, N.J. |oclc=793103475}}</ref><ref name="Restivo">{{Cite book |last=Restivo |first=Sal P. |url=https://books.google.com/books?id=V0RuCQAAQBAJ&q=Mathematics+in+Society+and+History |title=Mathematics in society and history : sociological inquiries |date=1992 |isbn=978-94-011-2944-2 |location=Dordrecht |oclc=883391697}}</ref> For example, in Western society, the [[13 (number)|number 13]] is often regarded as [[unlucky]], and "[[One million|a million]]" may signify "a lot" rather than an exact quantity.<ref name="Gilsdorf" /> Though it is now regarded as [[pseudoscience]], belief in a mystical significance of numbers, known as [[numerology]], permeated ancient and medieval thought.<ref name="Ore">{{Cite book |last=Ore |first=Øystein |url=https://books.google.com/books?id=Sl_6BPp7S0AC |title=Number theory and its history |date=1988 |publisher=Dover |isbn=0-486-65620-9 |location=New York |oclc=17413345}}</ref> Numerology heavily influenced the development of [[Greek mathematics]], stimulating the investigation of many problems in number theory which are still of interest today.<ref name="Ore" />

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 [[hypercomplex number]]s, which consist of various extensions or modifications of the [[complex number]] system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as [[ring (mathematics)|rings]] and [[field (mathematics)|fields]], and the application of the term "number" is a matter of convention, without fundamental significance.<ref>Gouvêa, Fernando Q. ''[[The Princeton Companion to Mathematics]], Chapter II.1, "The Origins of Modern Mathematics"'', p. 82. Princeton University Press, September 28, 2008. {{isbn|978-0-691-11880-2}}. "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 ''p''-adics. The quaternions are rarely referred to as 'numbers,' on the other hand, though they can be used to coordinatize certain mathematical notions."</ref>

==History==

===First use of numbers===
{{main|History of ancient numeral systems}}
Bones and other artifacts have been discovered with marks cut into them that many believe are [[tally marks]].<ref>{{Cite book |last=Marshack |first=Alexander |url=https://books.google.com/books?id=vbQ9AAAAIAAJ |title=The roots of civilization; the cognitive beginnings of man's first art, symbol, and notation (1st ed.) |date=1971 |publisher=McGraw-Hill |isbn=0-07-040535-2 |location=New York |oclc=257105}}</ref> Some historians suggest that the [[Lebombo bone]] (dated about 43,000 years ago) and the [[Ishango bone]] (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed.<ref name="auto">{{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA2 2–3]}}</ref><ref>{{harvnb|Thiam|Rochon|2019|p=[https://books.google.com/books?id=EWSsDwAAQBAJ&pg=PA164 164]}}</ref> 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.<ref name="Ore" /> However, a [[Number sense|perceptual system for quantity]] thought to underlie numeracy, is shared with other species, a phylogenetic distribution suggesting it would have existed before the emergence of language.<ref>{{cite journal |last1=Coolidge |first1=Frederick L. |last2=Overmann |first2=Karenleigh A. |title=Numerosity, Abstraction, and the Emergence of Symbolic Thinking |journal=Current Anthropology |volume=53 |issue=2 |year=2012 |pages=204–225|doi=10.1086/664818 |s2cid=51918452 |url=https://osf.io/utn53/ }}</ref><ref name="auto"/>

A tallying system has no concept of place value (as in modern [[decimal]] notation), which limits its representation of large numbers. Nonetheless, tallying systems are considered the first kind of abstract numeral system.

The earliest unambiguous numbers in the archaeological record are the [[Ancient Mesopotamian units of measurement|Mesopotamian base&nbsp;60]] system ({{circa|3400}}&nbsp;BC);<ref>{{Cite book |last=Schmandt-Besserat |first=Denise |title=Before Writing: From Counting to Cuneiform (2 vols) |publisher=University of Texas Press |date=1992}}</ref> place value emerged in it in the 3rd millennium BCE.<ref>{{Cite book |last=Robson |first=Eleanor |title=Mathematics in Ancient Iraq: A Social History |publisher=Princeton University Press |date=2008}}</ref> The earliest known base&nbsp;10 system dates to 3100&nbsp;BC in [[Egypt]].<ref>{{cite web |url=http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin |title=Egyptian Mathematical Papyri – Mathematicians of the African Diaspora |publisher=Math.buffalo.edu |access-date=2012-01-30 |archive-url=https://web.archive.org/web/20150407231917/http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin |archive-date=2015-04-07 |url-status=live }}</ref>

===Numerals===
{{main|Numeral system}}
Numbers should be distinguished from '''numerals''', 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.<ref>{{Cite journal |last=Chrisomalis |first=Stephen |date=2003-09-01 |title=The Egyptian origin of the Greek alphabetic numerals |journal=Antiquity |volume=77 |issue=297 |pages=485–96 |doi=10.1017/S0003598X00092541 |s2cid=160523072 |issn=0003-598X }}</ref> Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior [[Hindu–Arabic numeral system]] around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today.<ref name="Cengage Learning2">{{cite book |url=https://books.google.com/books?id=dOxl71w-jHEC&pg=PA192 |title=The Earth and Its Peoples: A Global History, Volume 1 |last2=Crossley |first2=Pamela |last3=Headrick |first3=Daniel |last4=Hirsch |first4=Steven |last5=Johnson |first5=Lyman |publisher=Cengage Learning |year=2010 |isbn=978-1-4390-8474-8 |page=192 |quote=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 |first1=Richard |last1=Bulliet |access-date=2017-05-16 |archive-url=https://web.archive.org/web/20170128072424/https://books.google.com/books?id=dOxl71w-jHEC&pg=PA192 |archive-date=2017-01-28 |url-status=live }}</ref> The key to the effectiveness of the system was the symbol for [[zero]], which was developed by ancient [[Indian mathematics|Indian mathematicians]] around 500 AD.<ref name="Cengage Learning2" />

===Zero{{anchor|History of zero}}===
The first known recorded use of [[zero]] dates to AD 628, and appeared in the ''[[Brāhmasphuṭasiddhānta]]'', the main work of the [[Indian mathematician]] [[Brahmagupta]]. He treated&nbsp;0 as a number and discussed operations involving it, including [[division by zero]]. By this time (the 7th&nbsp;century), the concept had clearly reached Cambodia in the form of [[Khmer numerals]],<ref>{{Cite magazine |last=Aczel |first=Amir D. |date=2015-05-07 |title=My Quest to Find the First Zero |url=https://time.com/3845786/my-quest-to-find-the-first-zero/ |access-date=2025-02-15 |magazine=TIME |language=en}}</ref> and documentation shows the idea later spreading to China and the [[Islamic world]].

[[File:Khmer Numerals - 605 from the Sambor inscriptions.jpg|thumb|The number 605 in [[Khmer numerals]], from an inscription from 683 AD. Early use of zero as a decimal figure.]]

Brahmagupta's ''Brāhmasphuṭasiddhānta'' 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 negative number". The ''Brāhmasphuṭasiddhānta'' 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.

The use of 0 as a number should be distinguished from its use as a placeholder numeral in [[place-value system]]s. Many ancient texts used&nbsp;0. Babylonian and Egyptian texts used it. Egyptians used the word ''nfr'' to denote zero&nbsp;balance in [[double-entry bookkeeping system|double entry accounting]]. Indian texts used a [[Sanskrit]] word {{lang|sa-Latn|Shunye}} or {{lang|sa|shunya}} to refer to the concept of ''void''. In mathematics texts this word often refers to the number zero.<ref>{{cite web |url=http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html |title=Historia Matematica Mailing List Archive: Re: [HM] The Zero Story: a question |publisher=Sunsite.utk.edu |date=1999-04-26 |access-date=2012-01-30 |url-status=dead |archive-url=https://web.archive.org/web/20120112073735/http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html |archive-date=2012-01-12 }}</ref> In a similar vein, [[Pāṇini]] (5th century BC) used the null (zero) operator in the ''[[Ashtadhyayi]]'', an early example of an [[formal grammar|algebraic grammar]] for the Sanskrit language (also see [[Pingala]]).

There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the ''Brāhmasphuṭasiddhānta''.

Records show that the Ancient Greeks seemed unsure about the status of&nbsp;0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting [[philosophical]] and, by the Medieval period, religious arguments about the nature and existence of&nbsp;0 and the vacuum. The [[Zeno's paradoxes|paradoxes]] of [[Zeno of Elea]] depend in part on the uncertain interpretation of&nbsp;0. (The ancient Greeks even questioned whether&nbsp;{{num|1}} was a number.)

The late [[Olmec]] people of south-central Mexico began to use a symbol for zero, a shell [[glyph]], in the New World, possibly by the {{nowrap|4th century BC}} but certainly by 40&nbsp;BC, which became an integral part of [[Maya numerals]] and the [[Maya calendar]]. Maya arithmetic used base&nbsp;4 and base&nbsp;5 written as base&nbsp;20. [[George I. Sánchez]] in 1961 reported a base&nbsp;4, base&nbsp;5 "finger" abacus.<ref>{{Cite book |last=Sánchez |first=George I. |author-link=George I. Sánchez |title=Arithmetic in Maya |publisher=self published |year=1961 |place=Austin, Texas}}</ref>{{Better source needed|reason=The only source is a self-published book, albeit one by a respected educator. According to the (favorable) review by David H. Kelley in 'American Anthropologist', Sánchez was neither a Mayanist nor a mathematician. The review does not mention the abacus.|date=September 2020}}

By 130 AD, [[Ptolemy]], influenced by [[Hipparchus]] and the Babylonians, was using a symbol for&nbsp;0 (a small circle with a long overbar) within a [[sexagesimal]] numeral system otherwise using alphabetic [[Greek numerals]]. Because it was used alone, not as just a placeholder, this [[Greek numerals#Hellenistic zero|Hellenistic zero]] was the first ''documented'' use of a true zero in the Old World. In later [[Byzantine Empire|Byzantine]] manuscripts of his ''Syntaxis Mathematica'' (''Almagest''), the Hellenistic zero had morphed into the Greek letter [[Omicron]] (otherwise meaning&nbsp;70).

Another true zero was used in tables alongside [[Roman numerals#Zero|Roman numerals]] by 525 (first known use by [[Dionysius Exiguus]]), but as a word, {{lang|la|nulla}} meaning ''nothing'', not as a symbol. When division produced&nbsp;0 as a remainder, {{lang|la|nihil}}, also meaning ''nothing'', was used. These medieval zeros were used by all future medieval [[computus|computists]] (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by [[Bede]] or a colleague about 725, a true zero symbol.

===Negative numbers {{anchor|History of negative numbers}}===
{{further|History of negative numbers}}
The abstract concept of negative numbers was recognized as early as 100–50 BC in China. ''[[The Nine Chapters on the Mathematical Art]]'' contains methods for finding the areas of figures; red rods were used to denote positive [[coefficient]]s, black for negative.<ref>{{Cite book |last=Staszkow |first=Ronald |author2=Robert Bradshaw |title=The Mathematical Palette (3rd ed.) |publisher=Brooks Cole |year=2004 |page=41 |isbn=0-534-40365-4}}</ref> The first reference in a Western work was in the 3rd&nbsp;century AD in Greece. [[Diophantus]] referred to the equation equivalent to {{nowrap|4''x'' + 20 {{=}} 0}} (the solution is negative) in ''[[Arithmetica]]'', saying that the equation gave an absurd result.

During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician [[Brahmagupta]], in ''[[Brāhmasphuṭasiddhānta]]'' in 628, who used negative numbers to produce the general form [[quadratic formula]] that remains in use today. However, in the 12th&nbsp;century in India, [[Bhāskara II|Bhaskara]] 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".

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th&nbsp;century, although [[Fibonacci]] allowed negative solutions in financial problems where they could be interpreted as debts (chapter&nbsp;13 of {{Lang|la|[[Liber Abaci]]}}, 1202) and later as losses (in {{lang|la|Flos}}). [[René Descartes]] 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 numeral.<ref>{{Cite book |last=Smith |first=David Eugene |author-link=David Eugene Smith |title=History of Modern Mathematics |publisher=Dover Publications |year=1958 |page=259 |isbn=0-486-20429-4}}</ref> The first use of negative numbers in a European work was by [[Nicolas Chuquet]] during the 15th&nbsp;century. He used them as [[exponent]]s, but referred to them as "absurd numbers".

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.

===Rational numbers {{anchor|History of rational numbers}}===
It is likely that the concept of fractional numbers dates to [[prehistoric times]]. The [[Ancient Egyptians]] used their [[Egyptian fraction]] notation for rational numbers in mathematical texts such as the [[Rhind Mathematical Papyrus]] and the [[Kahun Papyrus]]. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of [[number theory]].<ref>{{Cite web |title=Classical Greek culture (article) |url=https://www.khanacademy.org/humanities/world-history/ancient-medieval/classical-greece/a/greek-culture |access-date=2022-05-04 |website=Khan Academy |language=en |archive-date=2022-05-04 |archive-url=https://web.archive.org/web/20220504133917/https://www.khanacademy.org/humanities/world-history/ancient-medieval/classical-greece/a/greek-culture |url-status=live }}</ref> The best known of these is [[Euclid's Elements|Euclid's ''Elements'']], dating to roughly 300&nbsp;BC. Of the Indian texts, the most relevant is the [[Sthananga Sutra]], which also covers number theory as part of a general study of mathematics.

The concept of [[decimal fraction]]s 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 [[sutra]] to include calculations of decimal-fraction approximations to [[pi]] or the [[square root of 2]].{{Citation needed|date=September 2020}} Similarly, Babylonian math texts used sexagesimal (base&nbsp;60) fractions with great frequency.

===Irrational numbers {{anchor|History of irrational numbers}}===
{{further|History of irrational numbers}}
The Babylonians, as early as 1800 BCE, demonstrated remarkably accurate numerical approximations of irrational quantities such as √2 on clay tablets, using these values primarily for practical calculations in geometry and land measurement.<ref>{{cite book
 | last = Neugebauer | first = Otto | title = The Exact Sciences in Antiquity | publisher = Dover Publications | location = New York | year = 1969 | isbn = 978-0486233567 | pages = 36–38}}</ref> There were practical approximations of irrational numbers in the [[Indian mathematics|Indian]] [[Shulba Sutras]] composed between 800 and 500&nbsp;BC.<ref>{{Cite book |editor-last=Selin |editor-first=Helaine |editor-link=Helaine Selin |title=Mathematics across cultures: the history of non-Western mathematics |publisher=Kluwer Academic Publishers |year=2000 |page=412 |isbn=0-7923-6481-3|url=https://ia902901.us.archive.org/14/items/mathematicsacrossculturesthehistoryofnonwesternmathematicshelaineselin1946/Mathematics%20across%20cultures%3B%20the%20history%20of%20non-western%20mathematics%20-%20Helaine%20Selin%2C%201946.pdf}}</ref> The first existence proofs of irrational numbers is usually attributed to [[Pythagoras]], more specifically to the [[Pythagoreanism|Pythagorean]] [[Hippasus]], who produced a (most likely geometrical) proof of the irrationality of the [[square root of 2]]. 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. He could not disprove the existence of irrational numbers, or accept them, so according to legend, he sentenced Hippasus to death by drowning, to impede the spread of this unsettling news.<ref>{{cite book |title=Harvard Studies in Classical Philology |chapter=Horace and the Monuments: A New Interpretation of the Archytas ''Ode'' |author=Bernard Frischer |editor=D.R. Shackleton Bailey |editor-link=D. R. Shackleton Bailey |page=83 |publisher=Harvard University Press |year=1984 |isbn=0-674-37935-7}}</ref>

The 16th century brought final European acceptance of negative integral and fractional 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 remained almost dormant since [[Euclid]]. In 1872, the publication of the theories of [[Karl Weierstrass]] (by his pupil E. Kossak), [[Eduard Heine]],<ref>Eduard Heine, [[doi:10.1515/crll.1872.74.172|"Die Elemente der Functionenlehre"]], ''[Crelle's] Journal für die reine und angewandte Mathematik'', No. 74 (1872): 172–188.</ref> [[Georg Cantor]],<ref>Georg Cantor, [[doi:10.1007/BF01446819|"Ueber unendliche, lineare Punktmannichfaltigkeiten", pt. 5]], ''Mathematische Annalen'', 21, 4 (1883‑12): 545–591.</ref> and [[Richard Dedekind]]<ref>Richard Dedekind, ''[https://books.google.com/books?id=n-43AAAAMAAJ Stetigkeit & irrationale Zahlen] {{Webarchive|url=https://web.archive.org/web/20210709184745/https://books.google.ca/books?id=n-43AAAAMAAJ |date=2021-07-09 }}'' (Braunschweig: Friedrich Vieweg & Sohn, 1872). Subsequently published in: ''———, Gesammelte mathematische Werke'', ed. Robert Fricke, Emmy Noether & Öystein Ore (Braunschweig: Friedrich Vieweg & Sohn, 1932), vol. 3, pp. 315–334.</ref> was brought about. In 1869, [[Charles Méray]] 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 [[Salvatore Pincherle]] (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by [[Paul Tannery]] (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a [[Dedekind cut|cut (Schnitt)]] in the system of [[real number]]s, separating all [[rational number]]s into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, [[Leopold Kronecker|Kronecker]],<ref>L. Kronecker, [[doi:10.1515/crll.1887.101.337|"Ueber den Zahlbegriff"]], ''[Crelle's] Journal für die reine und angewandte Mathematik'', No. 101 (1887): 337–355.</ref> and Méray.

The search for roots of [[Quintic equation|quintic]] and higher degree equations was an important development, the [[Abel–Ruffini theorem]] ([[Paolo Ruffini (mathematician)|Ruffini]] 1799, [[Niels Henrik Abel|Abel]] 1824) showed that they could not be solved by [[nth root|radicals]] (formulas involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of [[algebraic numbers]] (all solutions to polynomial equations). [[Évariste Galois|Galois]] (1832) linked polynomial equations to [[group theory]] giving rise to the field of [[Galois theory]].

[[Simple continued fraction]]s, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of [[Euler]],<ref>Leonhard Euler, "Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis", ''Acta Academiae Scientiarum Imperialis Petropolitanae'', 1779, 1 (1779): 162–187.</ref> and at the opening of the 19th&nbsp;century were brought into prominence through the writings of [[Joseph Louis Lagrange]]. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus<ref>Ramus, "Determinanternes Anvendelse til at bes temme Loven for de convergerende Bröker", in: ''Det Kongelige Danske Videnskabernes Selskabs naturvidenskabelige og mathematiske Afhandlinger'' (Kjoebenhavn: 1855), p. 106.</ref> first connected the subject with [[determinant]]s, resulting, with the subsequent contributions of Heine,<ref>Eduard Heine, [[doi:10.1515/crll.1859.56.87|"Einige Eigenschaften der ''Lamé''schen Funktionen"]], ''[Crelle's] Journal für die reine und angewandte Mathematik'', No. 56 (Jan. 1859): 87–99 at 97.</ref> [[August Ferdinand Möbius|Möbius]], and Günther,<ref>Siegmund Günther, ''Darstellung der Näherungswerthe von Kettenbrüchen in independenter Form'' (Erlangen: Eduard Besold, 1873); ———, "Kettenbruchdeterminanten", in: ''Lehrbuch der Determinanten-Theorie: Für Studirende'' (Erlangen: Eduard Besold, 1875), c. 6, pp. 156–186.</ref> in the theory of {{Lang|de|Kettenbruchdeterminanten}}.

===Transcendental numbers and reals {{anchor|History of transcendental numbers and reals}}===
{{further|History of π}}

The existence of [[transcendental numbers]]<ref>{{cite web |last=Bogomolny |first=A. |author-link=Cut-the-Knot |title=What's a number? |work=Interactive Mathematics Miscellany and Puzzles |url=http://www.cut-the-knot.org/do_you_know/numbers.shtml |access-date=11 July 2010 |archive-url=https://web.archive.org/web/20100923231547/http://www.cut-the-knot.org/do_you_know/numbers.shtml |archive-date=23 September 2010 |url-status=live }}</ref> was first established by [[Joseph Liouville|Liouville]] (1844, 1851). [[Charles Hermite|Hermite]] proved in 1873 that ''e'' is transcendental and [[Ferdinand von Lindemann|Lindemann]] proved in 1882 that π is transcendental. Finally, [[Cantor's first uncountability proof|Cantor]] showed that the set of all [[real number]]s is [[uncountable|uncountably infinite]] but the set of all [[algebraic number]]s is [[countable|countably infinite]], so there is an uncountably infinite number of transcendental numbers.

===Infinity and infinitesimals {{anchor|History of infinity and infinitesimals}}===
{{further|History of infinity}}
The earliest known conception of mathematical [[infinity]] appears in the [[Yajur Veda]], 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 [[Jain]] mathematicians c. 400&nbsp;BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. The symbol <math>\text{∞}</math> is often used to represent an infinite quantity.

[[Aristotle]] defined the traditional Western notion of mathematical infinity. He distinguished between [[actual infinity]] and [[potential infinity]]—the general consensus being that only the latter had true value. [[Galileo Galilei]]'s ''[[Two New Sciences]]'' discussed the idea of [[bijection|one-to-one correspondences]] between infinite sets. But the next major advance in the theory was made by [[Georg Cantor]]; in 1895 he published a book about his new [[set theory]], introducing, among other things, [[transfinite number]]s and formulating the [[continuum hypothesis]].

In the 1960s, [[Abraham Robinson]] showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of [[hyperreal numbers]] represents a rigorous method of treating the ideas about [[infinity|infinite]] and [[infinitesimal]] numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of [[infinitesimal calculus]] by [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]].

A modern geometrical version of infinity is given by [[projective geometry]], 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 [[perspective (graphical)|perspective]] drawing.

===Complex numbers {{anchor|History of complex numbers}}===
{{further|History of complex numbers}}
The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor [[Heron of Alexandria]] in the {{nowrap|1st century AD}}, when he considered the volume of an impossible [[frustum]] of a [[pyramid]]. They became more prominent when in the 16th&nbsp;century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as [[Niccolò Fontana Tartaglia]] and [[Gerolamo Cardano]]. 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.

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When [[René Descartes]] coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See [[imaginary number]] for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation
:<math>\left ( \sqrt{-1}\right )^2 =\sqrt{-1}\sqrt{-1}=-1</math>
seemed capriciously inconsistent with the algebraic identity
:<math>\sqrt{a}\sqrt{b}=\sqrt{ab},</math>
which is valid for positive real numbers ''a'' and ''b'', and was also used in complex number calculations with one of ''a'', ''b'' positive and the other negative. The incorrect use of this identity, and the related identity
:<math>\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}</math>
in the case when both ''a'' and ''b'' are negative even bedeviled [[Euler]].<ref>{{cite journal |last=Martínez |first=Alberto A. |year=2007 |title=Euler's 'mistake'? The radical product rule in historical perspective |journal=The American Mathematical Monthly |volume=114 |issue=4 |pages=273–285 |doi=10.1080/00029890.2007.11920416 |s2cid=43778192 |url = https://www.martinezwritings.com/m/Euler_files/EulerMonthly.pdf }}</ref> This difficulty eventually led him to the convention of using the special symbol ''i'' in place of <math>\sqrt{-1}</math> to guard against this mistake.

The 18th century saw the work of [[Abraham de Moivre]] and [[Leonhard Euler]]. [[De Moivre's formula]] (1730) states:
:<math>(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta </math>
while [[Euler's formula]] of [[complex analysis]] (1748) gave us:
:<math>\cos \theta + i\sin \theta = e ^{i\theta }. </math>

The existence of complex numbers was not completely accepted until [[Caspar Wessel]] described the geometrical interpretation in 1799. [[Carl Friedrich Gauss]] 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 [[John Wallis|Wallis]]'s ''De algebra tractatus''.

In the same year, Gauss provided the first generally accepted proof of the [[fundamental theorem of algebra]], showing that every polynomial over the complex numbers has a full set of solutions in that realm. Gauss studied complex numbers of the form {{nowrap|''a'' + ''bi''}}, where ''a'' and ''b'' are integers (now called [[Gaussian integer]]s) or rational numbers. His student, [[Gotthold Eisenstein]], studied the type {{nowrap|''a'' + ''bω''}}, where ''ω'' is a complex root of {{nowrap|''x''<sup>3</sup> − 1 {{=}} 0}} (now called [[Eisenstein integers]]). Other such classes (called [[cyclotomic field]]s) of complex numbers derive from the [[roots of unity]] {{nowrap|''x''<sup>''k''</sup> − 1 {{=}} 0}} for higher values of ''k''. This generalization is largely due to [[Ernst Kummer]], who also invented [[ideal number]]s, which were expressed as geometrical entities by [[Felix Klein]] in 1893.

In 1850 [[Victor Alexandre Puiseux]] took the key step of distinguishing between poles and branch points, and introduced the concept of [[mathematical singularity|essential singular points]].{{clarify|reason=Why is this a key step in the history of complex numbers?|date=September 2020}} This eventually led to the concept of the [[extended complex plane]].

===Prime numbers {{anchor|History of prime numbers}}===
[[Prime number]]s have been studied throughout recorded history.{{Citation needed|reason=Wikipedia's prime number article says the Greeks were the first to explicitly study prime numbers and mentions only the Rhind Papyrus as implicitly recognizing a distinction between prime and composite numbers.|date=September 2020}} They are positive integers that are divisible only by 1 and themselves. Euclid devoted one book of the ''Elements'' to the theory of primes; in it he proved the infinitude of the primes and the [[fundamental theorem of arithmetic]], and presented the [[Euclidean algorithm]] for finding the [[greatest common divisor]] of two numbers.

In 240 BC, [[Eratosthenes]] used the [[Sieve of Eratosthenes]] to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the [[Renaissance]] and later eras.{{Citation needed|reason=Need citation for activity (or lack thereof) during era between Eratosthenes and Legendre.|date=September 2020}}

In 1796, [[Adrien-Marie Legendre]] conjectured the [[prime number theorem]], 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 [[Goldbach conjecture]], 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 [[Riemann hypothesis]], formulated by [[Bernhard Riemann]] in 1859. The [[prime number theorem]] was finally proved by [[Jacques Hadamard]] and [[Charles de la Vallée-Poussin]] in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.

==Main classification{{anchor|Classification|Classification of numbers}}==
{{Redirect|Number system|systems which express numbers|Numeral system}}
{{See also|List of types of numbers}}
Numbers can be classified into [[set (mathematics)|sets]], called '''number sets''' or '''number systems''', such as the [[natural numbers]] and the [[real numbers]]. The main number systems are as follows:
{|class="wikitable" style="margin: 1em auto; max-width: 600px; overflow-x: auto"
|+ Main number systems
!Symbol
!Name
!Examples/Explanation
|-
!<math>\mathbb{N}</math>
![[Natural number]]s
| 0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ...<br />
<math>\mathbb{N}_0</math> or <math>\mathbb{N}_1</math> are sometimes used.
|-
!<math>\mathbb{Z}</math>
![[Integer]]s
|..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
|-
!<math>\mathbb{Q}</math>
![[Rational number]]s
|{{sfrac|''a''|''b''}} where ''a'' and ''b'' are integers and ''b'' is not 0
|-
!<math>\mathbb{R}</math>
![[Real number]]s
|The limit of a convergent sequence of rational numbers
|-
!<math>\mathbb{C}</math>
![[Complex number]]s
|''a'' + ''bi'' where ''a'' and ''b'' are real numbers and ''i'' is a formal square root of&nbsp;−1
|}

Each of these number systems is a [[subset]] 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
:<math>\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}</math>.

A more complete list of number sets appears in the following diagram.
{{Classification_of_numbers}}

===Natural numbers===
{{Main|Natural number}}
[[File:Nat num.svg|thumb|The natural numbers, starting with 1]]
The most familiar numbers are the [[natural number]]s (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with&nbsp;1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th&nbsp;century, [[set theory|set theorists]] and other mathematicians started including&nbsp;0 ([[cardinality]] of the [[empty set]], i.e. 0&nbsp;elements, where&nbsp;0 is thus the smallest [[cardinal number]]) in the set of natural numbers.<ref>
{{MathWorld|title=Natural Number|id=NaturalNumber}}</ref><ref>{{Cite web |url=http://www.merriam-webster.com/dictionary/natural%20number |title=natural number |work=Merriam-Webster.com |publisher=[[Merriam-Webster]] |access-date=4 October 2014 |archive-url=https://web.archive.org/web/20191213133201/https://www.merriam-webster.com/dictionary/natural%20number |archive-date=13 December 2019 |url-status=live }}</ref> Today, different mathematicians use the term to describe both sets, including&nbsp;0 or not. The [[mathematical symbol]] for the set of all natural numbers is '''N''', also written <math>\mathbb{N}</math>, and sometimes <math>\mathbb{N}_0</math> or <math>\mathbb{N}_1</math> when it is necessary to indicate whether the set should start with 0 or 1, respectively.

In the [[base 10]] numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten [[numerical digit|digits]]: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The [[Radix|radix or base]] 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&nbsp;10 system, the rightmost digit of a natural number has a [[place value]] of&nbsp;1, and every other digit has a place value ten times that of the place value of the digit to its right.

In [[set theory]], which is capable of acting as an axiomatic foundation for modern mathematics,<ref>{{Cite book |last=Suppes |first=Patrick |author-link=Patrick Suppes |title=Axiomatic Set Theory |publisher=Courier Dover Publications |year=1972 |page=[https://archive.org/details/axiomaticsettheo00supp_0/page/1 1] |isbn=0-486-61630-4 |url=https://archive.org/details/axiomaticsettheo00supp_0/page/1 }}</ref> natural numbers can be represented by classes of equivalent sets. For instance, the number&nbsp;3 can be represented as the class of all sets that have exactly three elements. Alternatively, in [[Peano Arithmetic]], the number&nbsp;3 is represented as sss0, where s is the "successor" function (i.e.,&nbsp;3 is the third successor of&nbsp;0). Many different representations are possible; all that is needed to formally represent&nbsp;3 is to inscribe a certain symbol or pattern of symbols three times.

===Integers===
{{Main|Integer}}
The negative of a positive integer is defined as a number that produces&nbsp;0 when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a [[minus sign]]). As an example, the negative of&nbsp;7 is written&nbsp;−7, and {{nowrap|7 + (−7) {{=}} 0}}. When the [[set (mathematics)|set]] of negative numbers is combined with the set of natural numbers (including&nbsp;0), the result is defined as the set of [[integer]]s, '''Z''' also written [[Blackboard bold|<math>\mathbb{Z}</math>]]. Here the letter Z comes {{ety|de|Zahl|number}}. The set of integers forms a [[ring (mathematics)|ring]] with the operations addition and multiplication.<ref>{{Mathworld|Integer|Integer}}</ref>

The natural numbers form a subset 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 '''positive integers''', and the natural numbers with zero are referred to as '''non-negative integers'''.

===Rational numbers===
{{Main|Rational number}}
A rational number is a number that can be expressed as a fraction 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 them. The fraction {{sfrac|''m''|''n''}} represents ''m'' parts of a whole divided into ''n'' equal parts. Two different fractions may correspond to the same rational number; for example {{sfrac|1|2}} and {{sfrac|2|4}} are equal, that is:
:<math>{1 \over 2} = {2 \over 4}.</math>

In general,
:<math>{a \over b} = {c \over d}</math> if and only if <math>{a \times d} = {c \times b}.</math>

If the [[absolute value]] of ''m'' is greater than ''n'' (supposed to be positive), then the absolute value of the fraction is greater than&nbsp;1. Fractions can be greater than, less than, or equal to&nbsp;1 and can also be positive, negative, or&nbsp;0. The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator&nbsp;1. For example&nbsp;−7 can be written&nbsp;{{sfrac|−7|1}}. The symbol for the rational numbers is '''Q''' (for ''[[quotient]]''), also written [[Blackboard bold|<math>\mathbb{Q}</math>.]]

===Real numbers===
{{Main|Real number}}

The symbol for the real numbers is '''R''', also written as <math>\mathbb{R}.</math> They include all the measuring numbers. Every real number corresponds to a point on the [[number line]]. 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 [[minus sign]], e.g. −123.456.

Most real numbers can only be ''approximated'' by [[decimal]] numerals, in which a [[decimal point]] is placed to the right of the digit with place value&nbsp;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 {{sfrac|123456|1000}}, 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 [[fractional part]] 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 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 [[repeating decimal]]. Thus {{sfrac|3}} can be written as 0.333..., with an ellipsis to indicate that the pattern continues. Forever repeating 3s are also written as 0.{{overline|3}}.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Repeating Decimal|url=https://mathworld.wolfram.com/RepeatingDecimal.html|access-date=2020-07-23|website=Wolfram MathWorld |language=en|archive-date=2020-08-05|archive-url=https://web.archive.org/web/20200805170548/https://mathworld.wolfram.com/RepeatingDecimal.html|url-status=live}}</ref>

It turns out that these repeating decimals (including the [[Trailing zero|repetition of zeroes]]) 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 [[irrational number|irrational]]. A famous irrational real number is the [[pi|{{pi}}]], the ratio of the [[circumference]] of any circle to its [[diameter]]. When pi is written as
:<math>\pi = 3.14159265358979\dots,</math>
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 [[proof that pi is irrational|{{pi}} is irrational]]. Another well-known number, proven to be an irrational real number, is
:<math>\sqrt{2} = 1.41421356237\dots,</math>
the [[square root of 2]], that is, the unique positive real number whose square is 2. Both these numbers have been approximated (by computer) to trillions {{nowrap|( 1 trillion {{=}} 10<sup>12</sup> {{=}} 1,000,000,000,000 )}} of digits.

Not only these prominent examples but [[almost all]] 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 [[rounding|rounded]] or [[truncation|truncated]] real numbers. Any rounded or truncated number is necessarily a rational number, of which there are only [[countably many]]. All measurements are, by their nature, approximations, and always have a [[margin of error]]. Thus 123.456 is considered an approximation of any real number greater or equal to {{sfrac|1234555|10000}} and strictly less than {{sfrac|1234565|10000}} (rounding to 3 decimals), or of any real number greater or equal to {{sfrac|123456|1000}} and strictly less than {{sfrac|123457|1000}} (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 [[significant digits]]. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 [[Metre|m]]. If the sides of a rectangle are measured as 1.23&nbsp;m and 4.56&nbsp;m, then multiplication gives an area for the rectangle between {{nowrap|5.614591 m<sup>2</sup>}} and {{nowrap|5.603011 m<sup>2</sup>}}. Since not even the second digit after the decimal place is preserved, the following digits are not ''significant''. Therefore, the result is usually rounded to 5.61.

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, [[0.999...]], 1.0, 1.00, 1.000, ..., all represent the natural number&nbsp;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.

The real numbers also have an important but highly technical property called the [[least upper bound]] property.

It can be shown that any [[ordered field]], which is also [[completeness of the real numbers|complete]], is isomorphic to the real numbers. The real numbers are not, however, an [[algebraically closed field]], because they do not include a solution (often called a [[square root of minus one]]) to the algebraic equation <math> x^2+1=0</math>.

===Complex numbers===
{{Main|Complex number}}
Moving to a greater level of abstraction, the real numbers can be extended to the [[complex number]]s. This set of numbers arose historically from trying to find closed formulas for the roots of [[cubic function|cubic]] and [[quadratic function|quadratic]] polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a [[square root]] of&nbsp;−1, denoted by ''[[imaginary unit|i]]'', a symbol assigned by [[Leonhard Euler]], and called the [[imaginary unit]]. The complex numbers consist of all numbers of the form
:<math>\,a + b i</math>
where ''a'' and ''b'' are real numbers. Because of this, complex numbers correspond to points on the [[complex plane]], a [[vector space]] of two real [[dimension]]s. In the expression {{nowrap|''a'' + ''bi''}}, the real number ''a'' is called the [[real part]] and ''b'' is called the [[imaginary part]]. If the real part of a complex number is&nbsp;0, then the number is called an [[imaginary number]] or is referred to as ''purely imaginary''; if the imaginary part is&nbsp;0, then the number is a real number. Thus the real numbers are a [[subset]] of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a [[Gaussian integer]]. The symbol for the complex numbers is '''C''' or <math>\mathbb{C}</math>.

The [[fundamental theorem of algebra]] asserts that the complex numbers form an [[algebraically closed field]], meaning that every [[polynomial]] with complex coefficients has a [[zero of a function|root]] in the complex numbers. Like the reals, the complex numbers form a [[field (mathematics)|field]], which is [[complete space|complete]], but unlike the real numbers, it is not [[total order|ordered]]. That is, there is no consistent meaning assignable to saying that ''i'' is greater than&nbsp;1, nor is there any meaning in saying that ''i'' is less than&nbsp;1. In technical terms, the complex numbers lack a [[total order]] that is [[ordered field|compatible with field operations]].

==Subclasses of the integers==

===Even and odd numbers===
{{main|Even and odd numbers}}
An '''even number''' is an integer that is "evenly divisible" by two, that is [[Euclidean division|divisible by two without remainder]]; an '''odd number''' is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "[[divisibility|divisible]]".) Any odd number ''n'' may be constructed by the formula {{nowrap|''n'' {{=}} 2''k'' + 1,}} for a suitable integer ''k''. Starting with {{nowrap|''k'' {{=}} 0,}} the first non-negative odd numbers are {1, 3, 5, 7, ...}. Any even number ''m'' has the form {{nowrap|''m'' {{=}} 2''k''}} where ''k'' is again an [[integer]]. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}.

===Prime numbers===
{{main|Prime number}}
A '''prime number''', often shortened to just '''prime''', 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 [[number theory]]. [[Goldbach's conjecture]] is an example of a still unanswered question: "Is every even number the sum of two primes?"

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 [[fundamental theorem of arithmetic]]. A proof appears in [[Euclid's Elements]].

===Other classes of integers===
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 [[Fibonacci number]]s and [[perfect number]]s. For more examples, see [[Integer sequence]].

==Subclasses of the complex numbers==

===Algebraic, irrational and transcendental numbers===
[[Algebraic number]]s are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called [[irrational number]]s. Complex numbers which are not algebraic are called [[transcendental number]]s. The algebraic numbers that are solutions of a [[monic polynomial]] equation with integer coefficients are called [[algebraic integer]]s.

===Periods and exponential periods===
{{Main|Period (algebraic geometry)}}
A period is a complex number that can be expressed as an [[integral]] of an [[algebraic function]] over an algebraic [[Domain of a function|domain]]. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known [[Mathematical constant|mathematical constants]] such as the [[Pi|number ''π'']]. The set of periods form a countable [[Ring (mathematics)|ring]] and bridge the gap between algebraic and transcendental numbers.<ref name=":1">{{Citation |last1=Kontsevich |first1=Maxim |title=Periods |date=2001 |work=Mathematics Unlimited — 2001 and Beyond |pages=771–808 |editor-last=Engquist |editor-first=Björn |url=https://link.springer.com/chapter/10.1007/978-3-642-56478-9_39 |access-date=2024-09-22 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-56478-9_39 |isbn=978-3-642-56478-9 |last2=Zagier |first2=Don |editor2-last=Schmid |editor2-first=Wilfried|url-access=subscription }}</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Algebraic Period |url=https://mathworld.wolfram.com/AlgebraicPeriod.html |access-date=2024-09-22 |website=mathworld.wolfram.com |language=en}}</ref> 

The periods can be extended by permitting the integrand to be the product of an algebraic function and the [[Exponential function|exponential]] of an algebraic function. This gives another countable ring: the exponential periods. The [[E (mathematical constant)|number ''e'']] as well as [[Euler's constant]] are exponential periods.<ref name=":1" /><ref>{{Cite journal |last=Lagarias |first=Jeffrey C. |date=2013-07-19 |title=Euler's constant: Euler's work and modern developments |journal=Bulletin of the American Mathematical Society |volume=50 |issue=4 |pages=527–628 |doi=10.1090/S0273-0979-2013-01423-X |arxiv=1303.1856 |issn=0273-0979}}</ref> 

===Constructible numbers===
Motivated by the classical problems of [[Straightedge and compass construction|constructions with straightedge and compass]], the [[constructible number]]s 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.

===Computable numbers===
{{Main|Computable number}}
A '''computable number''', also known as ''recursive number'', is a [[real number]] such that there exists an [[algorithm]] which, given a positive number ''n'' as input, produces the first ''n'' digits of the computable number's decimal representation. Equivalent definitions can be given using [[μ-recursive function]]s, [[Turing machine]]s or [[λ-calculus]]. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a [[polynomial]], and thus form a [[real closed field]] that contains the real [[algebraic number]]s.

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 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.

The set of computable numbers has the same cardinality as the natural numbers. Therefore, [[almost all]] real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.

==Extensions of the concept==

===''p''-adic numbers===
{{main|p-adic number|l1=''p''-adic number}}
The ''p''-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 [[radix|base]] is used for the digits: any base is possible, but a [[prime number]] base provides the best mathematical properties. The set of the ''p''-adic numbers contains the rational numbers, but is not contained in the complex numbers.

The elements of an [[algebraic function field]] over a [[finite field]] and algebraic numbers have many similar properties (see [[Function field analogy]]). Therefore, they are often regarded as numbers by number theorists. The ''p''-adic numbers play an important role in this analogy.

===Hypercomplex numbers===
{{main|hypercomplex number}}
Some number systems that are not included in the complex numbers may be constructed from the real numbers <math>\mathbb{R}</math> in a way that generalize the construction of the complex numbers. They are sometimes called [[hypercomplex number]]s. They include the [[quaternion]]s <math>\mathbb{H}</math>, introduced by Sir [[William Rowan Hamilton]], in which multiplication is not [[commutative]], the [[octonion]]s <math>\mathbb{O}</math>, in which multiplication is not [[associative]] in addition to not being commutative, and the [[sedenion]]s <math>\mathbb{S}</math>, in which multiplication is not [[Alternative algebra|alternative]], neither associative nor commutative. The hypercomplex numbers include one real unit together with <math>2^n-1</math> imaginary units, for which ''n'' is a non-negative integer. For example, quaternions can generally represented using the form

<math display=block>a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k,</math>

where the coefficients {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are real numbers, and {{math|'''i''', '''j'''}}, {{math|'''k'''}} are 3 different imaginary units.

Each hypercomplex number system is a [[subset]] of the next hypercomplex number system of double dimensions obtained via the [[Cayley–Dickson construction]]. For example, the 4-dimensional quaternions <math>\mathbb{H}</math> are a subset of the 8-dimensional quaternions <math>\mathbb{O}</math>, which are in turn a subset of the 16-dimensional sedenions <math>\mathbb{S}</math>, in turn a subset of the 32-dimensional [[trigintaduonion]]s <math>\mathbb{T}</math>, and ''[[ad infinitum]]'' with <math>2^n</math> dimensions, with ''n'' being any non-negative integer. Including the complex and real numbers and their subsets, this can be expressed symbolically as:
:<math>\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \mathbb{T} \subset \cdots</math>

Alternatively, starting from the real numbers <math>\mathbb{R}</math>, which have zero complex units, this can be expressed as

:<math>\mathcal C_0 \subset \mathcal C_1 \subset \mathcal C_2 \subset \mathcal C_3 \subset \mathcal C_4 \subset \mathcal C_5 \subset \cdots \subset C_n</math>

with <math>C_n</math> containing <math>2^n</math> dimensions.<ref name="Saniga">{{cite journal | last1=Saniga | first1=Metod | last2=Holweck | first2=Frédéric | last3=Pracna | first3=Petr | title=From Cayley-Dickson Algebras to Combinatorial Grassmannians | journal=Mathematics | publisher=MDPI AG | volume=3 | issue=4 | date=2015 | issn=2227-7390 | arxiv=1405.6888 | doi=10.3390/math3041192 | doi-access=free | pages=1192–1221}}</ref>

===Transfinite numbers===
{{main|transfinite number}}
For dealing with infinite [[set (mathematics)|sets]], the natural numbers have been generalized to the [[ordinal number]]s and to the [[cardinal number]]s. 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.

===Nonstandard numbers===
[[Hyperreal number]]s are used in [[non-standard analysis]]. The hyperreals, or nonstandard reals (usually denoted as *'''R'''), denote an [[ordered field]] that is a proper [[Field extension|extension]] of the ordered field of [[real number]]s '''R''' and satisfies the [[transfer principle]]. This principle allows true [[first-order logic|first-order]] statements about '''R''' to be reinterpreted as true first-order statements about *'''R'''.

[[Superreal number|Superreal]] and [[surreal number]]s extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form [[field (mathematics)|fields]].

==See also==
{{Portal|Mathematics}}
{{cols|colwidth=21em}}
* [[Concrete number]]
* [[List of numbers]]
* [[List of types of numbers]]
* {{annotated link|List of books on history of number systems}}
* {{Annotated link|Mathematical constant}}
* [[Complex number]]s
* [[Numerical cognition]]
* [[Orders of magnitude]]
* {{Annotated link|Physical constant}}
* {{Annotated link|Physical quantity}}
* {{Annotated link|Pi}}
* {{Annotated link|Positional notation}}
* {{Annotated link|Prime number}}
* {{Annotated link|Scalar (mathematics)}}
* [[Subitizing and counting]]
{{colend}}

==Notes==
{{notelist}}
{{reflist}}

==References==
* {{cite book
 | last = Burgin
 | first = Mark
 | title = Trilogy of Numbers and Arithmetic - Book 1: History of Numbers and Arithmetic: An Information Perspective
 | publisher = World Scientific Publishing Company
 | location = Singapore
 | year = 2022
}}
* Cory, Leo. ''A Brief History of Numbers'', Oxford University Press, 2015, {{isbn|978-0-19-870259-7}}.
* [[Tobias Dantzig|Dantzig, Tobias]], ''Number, the language of science; a critical survey written for the cultured non-mathematician'', New York, The Macmillan Company, 1930.{{ISBN?}}
* Friedman, Erich. ''[http://www.stetson.edu/~efriedma/numbers.html What's special about this number?] {{Webarchive|url=https://web.archive.org/web/20180223062027/http://www2.stetson.edu/~efriedma/numbers.html |date=2018-02-23 }}''
* Galovich, Steven. ''Introduction to Mathematical Structures'', Harcourt Brace Javanovich, 1989, {{isbn|0-15-543468-3}}.
* [[Paul Halmos|Halmos, Paul]]. ''Naive Set Theory'', Springer, 1974, {{isbn|0-387-90092-6}}.
* [[Morris Kline|Kline, Morris]]. ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, 1990. {{isbn|978-0195061352}}
* {{cite book
 | last1 = Thiam
 | first1 = Thierno
 | last2 = Rochon
 | first2 = Gilbert
 | title = Sustainability, Emerging Technologies, and Pan-Africanism
 | publisher = Springer International Publishing
 | location = Germany
 | year = 2019
}}
* [[Alfred North Whitehead|Whitehead, Alfred North]] and [[Bertrand Russell]], ''[[Principia Mathematica]]'' to *56, Cambridge University Press, 1910.{{ISBN?}}

==External links==
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{{Commons|Numbers}}
{{Wikiquote}}
{{Wiktionary|number}}
{{wikiversity|Primary mathematics: Numbers}}
* {{SpringerEOM|title=Number|id=Number|oldid=11869|first=V.I.|last=Nechaev|mode=cs1}}
* {{cite web|last=Tallant|first=Jonathan|title=Do Numbers Exist|url=http://www.numberphile.com/videos/exist.html|work=Numberphile|publisher=[[Brady Haran]]|access-date=2013-04-06|archive-url=https://web.archive.org/web/20160308015528/http://www.numberphile.com/videos/exist.html|archive-date=2016-03-08|url-status=dead}}
* {{cite AV media|url=http://www.bbc.co.uk/programmes/p003hyd9|date=9 March 2006|archive-url=https://web.archive.org/web/20220531120903/https://www.bbc.co.uk/programmes/p003hyd9|archive-date=31 May 2022|publisher=BBC Radio 4|title=In Our Time: Negative Numbers}}
* {{cite web|url=http://www.gresham.ac.uk/lectures-and-events/4000-years-of-numbers|archive-url=https://web.archive.org/web/20220408112133/http://www.gresham.ac.uk/lectures-and-events/4000-years-of-numbers|url-status=live|archive-date=8 April 2022|title=4000 Years of Numbers|author=Robin Wilson|date=7 November 2007|publisher=[[Gresham College]]}}
* {{cite news|url=https://www.npr.org/sections/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1|title=What's the World's Favorite Number?|newspaper=NPR|url-status=live|archive-url=https://web.archive.org/web/20210518141211/https://www.npr.org/sections/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1|archive-date=18 May 2021|access-date=17 September 2011|date=22 July 2011|last1=Krulwich|first1=Robert}}; {{cite web|url=https://www.npr.org/templates/transcript/transcript.php?storyId=139797360|url-status=live|archive-url=https://web.archive.org/web/20181106205912/https://www.npr.org/templates/transcript/transcript.php?storyId=139797360?storyId=139797360|archive-date=6 November 2018|title=Cuddling With 9, Smooching With 8, Winking At 7|website=[[NPR]]|date=21 August 2011|access-date=17 September 2011}}
* [http://oeis.org Online Encyclopedia of Integer Sequences]

{{Number systems}}
{{Number theory}}
{{Authority control}}

[[Category:Numbers| ]]
[[Category:Group theory]]
[[Category:Abstraction]]
[[Category:Mathematical objects]]