View source for Number

{{Short description|Used to count, measure, and label}}
{{Other uses}}
{{pp-semi-indef}}
{{pp-move-indef}}
{{Use dmy dates|date=December 2022}}

[[File:NumberSetinR2.svg|thumb|[[Set inclusion]]s between the [[natural number]]s {{bug workaround|(ℕ), the [[integer]]s (ℤ), the [[decimal fraction]]s (𝔻), the [[rational number]]s (ℚ), the [[real number]]s (ℝ), and the [[complex number]]s (ℂ)}}]]

A '''number''' is a [[mathematical object]] used to [[Counting|count]], [[Measurement|measure]], and label. The most basic examples are the [[natural number]]s: 1, 2, 3, 4, 5, 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> Individual numbers can be represented in language with number words or by dedicated symbols called ''numerals''; for example, "five" is a number word and "5" is the corresponding numeral. As only a limited list of symbols can be memorized, a [[numeral system]] is used to represent any number in an organized way. The most common representation is the [[Hindu–Arabic numeral system]], which can display any [[Integer|non-negative integer]] using a combination of ten symbols, called [[numerical 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".}} Numerals can be used for counting (as with [[cardinal number]] of a collection or [[Set (mathematics)|set]]), 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|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=2 June 2005 | 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 | chapter=The Mathematical Accomplishments of Ancient Indian Mathematics | first=T. K. | last=Puttaswamy | title=Mathematics across cultures: the history of non-western mathematics | editor-first=Helaine | editor-last=Selin | date=2012 | publisher=Springer Science & Business Media |location=Dordrecht |isbn=978-94-011-4301-1 | pages=409–422 | chapter-url=https://books.google.com/books?id=DbsqBgAAQBAJ&pg=PA408 }}</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.

Viewing the concept of zero as a number required a fundamental shift in philosophy, identifying nothingness with a value. During the 19th century, mathematicians began to develop the various systems now called [[algebraic structure]]s, which share certain properties of numbers, and may be seen as extending the concept. Some algebraic structures are explicitly referred to as numbers (such as the [[p-adic number|{{mvar|p}}-adic numbers]] and [[hypercomplex number]]s) while others are not, but this is more a matter of convention than a mathematical distinction.<ref>{{cite book | last=Gouvêa | first=Fernando Q. | title=[[The Princeton Companion to Mathematics]] | chapter=II.1, The Origins of Modern Mathematics | page=82 | publisher=Princeton University Press | date=28 September 2008 | isbn=978-0-691-11880-2 | quote=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}}
[[File:Ishango bone (cropped).jpg|thumb|right|upright=1|The Ishango bone on exhibit at the Belgian [[Museum of Natural Sciences]]<ref>{{cite web | title=The Ishango Bone | publisher=Institute of Natural Sciences | url=https://www.naturalsciences.be/en/museum/exhibitions-activities/exhibitions/250-years-of-natural-sciences/the-ishango-bone | access-date=2025-10-23 }}</ref>]]
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 | edition=1st |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">{{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 | pages=2–3 | isbn=978-981-12-3685-3 | url=https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA2 }}</ref><ref>{{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 | page=164 | isbn=978-3-030-22180-5 | url=https://books.google.com/books?id=EWSsDwAAQBAJ&pg=PA164 }}</ref> These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of [[Quantity|quantities]], such as of animals.<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>  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.<ref>{{cite journal | title=The writing of numbers: recounting and recomposing numerical notations | first=Stephen | last=Chrisomalis | language=en | journal=[[Terrain (journal)|Terrain]] | volume=70 | year=2018 | doi=10.4000/terrain.17506 }}</ref>

The earliest unambiguous numbers in the archaeological record are the [[Ancient Mesopotamian units of measurement|Mesopotamian base&nbsp;60]] (sexagesimal) 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 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 | first=Scott W. | last=Williams | url=http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin | title=Egyptian Mathematical Papyri | work=Mathematicians of the African Diaspora | publisher=Mathematics Department, State University of New York at Buffalo | 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> A Babylonian clay tablet dated to {{Val|1900|-|1600|u=BC}} provides an estimate of the circumference of a circle to its diameter of <math display="inline">3\frac{1}{8}</math> = 3.125, possibly the oldest approximation of π.<ref name=Arndt_Haenel_2001>{{cite book | title=Pi - Unleashed | first1=Jörg | last1=Arndt | first2=Christoph | last2=Haenel | publisher=Springer Science & Business Media | year=2001 | isbn=978-3-540-66572-4 | page=167 | url=https://books.google.com/books?id=QwwcmweJCDQC&pg=PA167 }}</ref>

===Numerals===
{{main|Numeral system}}
[[File:Numeral Systems of the World.svg|right|thumb|From the top, showing [[braille]], hindu-arabic, [[Devanagari numerals|Devanagari]], [[Eastern Arabic numerals|Eastern Arabic]], [[Chinese numerals|Chinese]], Chinese financial, and [[Roman numerals]]]]
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=September 2003 |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> (However, in 300 BC, [[Archimedes]] first demonstrated the use of a [[positional numeral system]] to display extremely large numbers in ''[[The Sand Reckoner]]''.<ref>{{cite journal | title=The Archimedean Origin of Modern Positional Number Systems | first=Vincenzo | last=Manca | journal=Algorithms | year=2024 | volume=17 | issue=1 | page=11 | doi=10.3390/a17010011 | doi-access=free }}</ref>) Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the [[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" />

===<span class="anchor" id="History of zero"></span> Zero===
[[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.<ref name=Aczel_2014>{{cite journal | title=The Origin of the Number Zero | first=Amir | last=Aczel | journal=Smithsonian Magazine | date=December 2014 | url=https://www.smithsonianmag.com/history/origin-number-zero-180953392 | access-date=2025-10-20 }}</ref>]]
The first known recorded use of [[zero]] as an [[integer]] dates to AD 628, and appeared in the ''[[Brāhmasphuṭasiddhānta]]'', the main work of the [[Indian mathematician]] [[Brahmagupta]]. He is usually considered the first to formulate the mathematical concept of zero. Brahmagupta treated 0 as a number and discussed operations involving it, including [[division by 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". By this time (the 7th&nbsp;century), the concept had clearly reached Cambodia in the form of [[Khmer numerals]],<ref name=Aczel_2014/> and documentation shows the idea later spreading to China and the [[Islamic world]]. The concept began reaching Europe through Islamic sources around the year 1000.<ref>{{cite journal | title=Gerbert of Aurillac and the Transmission of Arabic Numerals to Europe | first=Thomas | last=Freudenhammer | journal=Sudhoffs Archiv | volume=105 | issue=1 | year=2021 | pages=3–19 | doi=10.25162/sar-2021-0001 | jstor=48636817 }}</ref>

There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the ''Brāhmasphuṭasiddhānta''.<ref name=Pranoto_Nair_2020/> The earliest uses of zero was as simply a placeholder numeral in [[place-value system]]s, representing another number as was done by the Babylonians.<ref>{{cite journal | last=Nath | first=R. | date=April 2012 | title=The Mighty Zero | journal=Science Reporter | pages=19–22 | url=https://www.academia.edu/download/54827361/The_Mighty_Zero.pdf | access-date=2025-10-20 }}</ref> Many ancient texts used&nbsp;0, including Babylonian and Egyptian texts. 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 | first=Kim | last=Plofker | author-link=Kim Plofker | title=Historia Matematica Mailing List Archive: Re: [HM] The Zero Story: a question | publisher=Dept. of History of Mathematics, Brown University | date=26 April 1999 | url=http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html | 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]]'',<ref name=Pranoto_Nair_2020>{{cite book | chapter=Zero | first1=Iwan | last1=Pranoto | first2=Ranjit | last2=Nair | title=Keywords for India: A Conceptual Lexicon for the 21st Century | editor1-first=Rukmini Bhaya | editor1-last=Nair | editor2-first=Peter Ronald | editor2-last=deSouza | publisher=Bloomsbury Publishing | year=2020 | pages=73–74 | isbn=978-1-3500-3925-4 | chapter-url=https://books.google.com/books?id=u6XFDwAAQBAJ&pg=PA74 }}</ref> an early example of an [[formal grammar|algebraic grammar]] for the Sanskrit language (also see [[Pingala]]).

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.<ref name=Riviere_2025>{{cite book | title=Zero – Much to Do About Nothing? | first=Jim E. | last=Riviere | publisher=Springer Nature | year=2025 | isbn=978-3-031-82998-7 | pages=12, 22–23 | url=https://books.google.com/books?id=qd5TEQAAQBAJ&pg=PA12 }}</ref> (The ancient Greeks even questioned whether&nbsp;{{num|1}} was a number.<ref>{{cite journal | title=Is One A Number? According to 'Mathematicks Made Easie,' Yes | first=Kat | last=Eschner | date=8 August 2017 | journal=Smithsonian Magazine | url=https://www.smithsonianmag.com/smart-news/one-number-according-mathematicks-made-easie-yes-180964318/ | access-date=2025-10-20 }}</ref>)

[[File:Maya.svg|thumb|The [[Maya numerals]] are an example of a base-20 numeral system.<ref name=Kiely_2022/>]]
The late [[Olmec]] people of south-central Mexico began to use a placeholder symbol for zero, a shell [[glyph]], in the New World, by 38&nbsp;BC.<ref>{{cite book | title=Zero: A Landmark Discovery, the Dreadful Void, and the Ultimate Mind | first1=Syamal K. | last1=Sen | first2=Ravi P. | last2=Agarwal | publisher=Academic Press | year=2015 | page=95 | isbn=978-0-12-804624-1 | url=https://books.google.com/books?id=fwBaCgAAQBAJ&pg=PA95 }}</ref> It would be the [[Maya peoples|Maya]] who developed zero as a cardinal number, employing it in their [[Maya numerals|numeral system]] and in the [[Maya calendar]].<ref>{{cite journal | title=Non-power positional number representation systems, bijective numeration, and the Mesoamerican discovery of zero | display-authors=1 | first1=Berenice | last1=Rojo-Garibaldia | first2=Costanza | last2=Rangonib | first3=Diego L. | last3=González | first4=Julyan H.E. | last4=Cartwright | journal=Heliyon | volume=7 | issue=3 | article-number=e06580 | date=March 2021 | doi=10.1016/j.heliyon.2021.e06580 | pmid=33851058 | pmc=8022160 | doi-access=free | arxiv=2005.10207 | bibcode=2021Heliy...706580R }}</ref> Maya used a [[Vigesimal|base 20 numerical system]] by combining a number of dots (base&nbsp;5) with a number of bars (base&nbsp;4).<ref name=Kiely_2022>{{cite book | chapter=Numbers and the Classical Maya | title=Numbers: A Cultural History | first=Robert | last=Kiely | publisher=Bloomsbury Publishing USA | year=2022 | isbn=979-8-216-12409-2 | chapter-url=https://books.google.com/books?id=JQTHEAAAQBAJ&pg=PT151 }}</ref> [[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><ref>{{cite journal | title=''Arithmetic in Maya''. George I. Sánchez. Privately printed | first=Linton | last=Satterthwaite | journal=American Antiquity | volume=28 | issue=2 | page=256 | doi=10.2307/278400 | jstor=278400 }}</ref>

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]].<ref>{{cite book | title=The Calculus: A Genetic Approach | first=Otto | last=Toeplitz | publisher=University of Chicago Press | year=2024 | isbn=978-0-226-80669-3 | pages=16–17 | url=https://books.google.com/books?id=189kAkcrpYQC&pg=PA17 }}</ref> 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]]<ref>{{cite book | title=A Survey of the Almagest: With Annotation and New Commentary by Alexander Jones | series=Sources and Studies in the History of Mathematics and Physical Sciences: Mathematics and Statistics | first=Olaf | last=Pedersen | editor-first=Alexander | editor-last=Jones | publisher=Springer Science & Business Media | year=2010 | isbn=978-0-387-84826-6 | url=https://books.google.com/books?id=8eaHxE9jUrwC&pg=PA52 }}</ref> (otherwise meaning&nbsp;70 in [[isopsephy]]<ref>{{cite book | title=Greek and Latin Roots of Medical and Scientific Terminologies | first=Todd A. | last=Curtis | publisher=John Wiley & Sons | year=2024 | isbn=978-1-118-35863-4 | url=https://books.google.com/books?id=gHgZEQAAQBAJ&pg=PA98 }}</ref>).

A 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.<ref>{{cite book | title=The Easter Computus and the Origins of the Christian Era | series=Oxford Early Christian Studies | first=Alden A. | last=Mosshammer | publisher=OUP Oxford | year=2008 | isbn=978-0-19-954312-0 | pages=8, 33 | url=https://books.google.com/books?id=9gkUDAAAQBAJ&pg=PA33 }}</ref> 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).{{cn|date=October 2025}} 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.

===<span class="anchor" id="History of negative numbers"></span> 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 | last1=Staszkow | first1=Ronald | first2=Robert | last2=Bradshaw | title=The Mathematical Palette | edition=3rd | 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.<ref>{{cite journal | title=The Symbolic and Mathematical Influence of Diophantus's Arithmetica | first=Cyrus | last=Hettle | journal=Journal of Humanistic Mathematics | volume=5 | issue=1 | date=January 2015 | pages=139–166 | doi=10.5642/jhummath.201501.08 | doi-access=free }}</ref>

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".<ref name=Agarwal_2024/>

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th&nbsp;century,<ref name=Agarwal_2024/> 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.<ref>{{cite journal | title=The Development of Number Systems | first=Roger | last=Knott | journal=Mathematics in School | volume=8 | issue=4 | date=September 1979 | pages=23–25 | jstor=30213485 }}</ref> 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> An early European experimenter with negative numbers was [[Nicolas Chuquet]] during the 15th&nbsp;century. He used them as [[exponent]]s,<ref>{{cite book | title=Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick | first=Helena M. | last=Pycior | publisher=Cambridge University Press | year=1997 | isbn=978-0-521-48124-3 | url=https://books.google.com/books?id=TJUJol1Qak4C&pg=PA18 }}</ref> 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.

===<span class="anchor" id="History of rational numbers"></span> Rational numbers===
[[File:Archimedes pi.svg|right|thumb|upright=1.1|[[Archimedes' doubling method|Archimedes' method]] of confining the value of pi using the perimeters of circumscribed and inscribed polygons results in rational number estimates.<ref>{{cite journal | title=Modernizing Archimedes' Construction of π | first=David | last=Weisbart | journal=Mathematics | year=2020 | volume=8 | issue=12 | article-number=2204 | doi=10.3390/math8122204 | doi-access=free }}</ref>]]
It is likely that the concept of fractional numbers dates to [[prehistoric times]].<ref name=Agarwal_2024/> The [[Ancient Egyptians]] used their [[Egyptian fraction]] notation for rational numbers in mathematical texts such as the [[Rhind Mathematical Papyrus]] and the [[Kahun Papyrus]].<ref>{{cite book | chapter=Egyptian mathematics | first=C. S. | last=Roero | title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences | series=A Johns Hopkins paperback | editor-first=I. | editor-last=Grattan-Guinness | publisher=JHU Press | year=2003 | isbn=978-0-8018-7396-6 | pages=30–36 | volume=1 | chapter-url=https://books.google.com/books?id=2hDvzITtfdAC&pg=PA30 }}</ref> The Rhind Papyrus includes an example of deriving the area of a circle from its diameter, which yields an estimate of π as <math display="inline">\bigl(\frac{16}{9}\bigr)^2</math> ≈ 3.16049....<ref name=Arndt_Haenel_2001/> 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><ref name=Agarwal_2024/> A particularly influential example of these is [[Euclid's Elements|Euclid's ''Elements'']], dating to roughly 300&nbsp;BC.<ref>{{cite book | title=Math Makers: The Lives and Works of 50 Famous Mathematicians | first1=Alfred S. | last1=Posamentier | first2=Christian | last2=Spreitzer | publisher=Jaico Publishing House | year=2024 | isbn=978-93-48098-11-5 | url=https://books.google.com/books?id=qts6EQAAQBAJ&pg=PT29 }}</ref> Of the Indian texts, the most relevant is the [[Sthananga Sutra]], which also covers number theory as part of a general study of mathematics.<ref name=Agarwal_2024>{{cite book | title=Mathematics Before and After Pythagoras: Exploring the Foundations and Evolution of Mathematical Thought | first=Ravi P. | last=Agarwal | publisher=Springer Nature | year=2024 | isbn=978-3-031-74224-8 | pages=46–47 | url=https://books.google.com/books?id=CZU0EQAAQBAJ&pg=PA46 }}</ref>

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.<ref>{{cite web | title=History of fractions | first=Liz | last=Pumfrey | date=2 January 2011 | website=NRich | publisher=University of Cambridge | url=https://nrich.maths.org/articles/history-fractions | access-date=2025-10-21 }}</ref>

===<span class="anchor" id="History of irrational numbers"></span> Real numbers and irrational numbers===
[[File:YBC-7289-OBV-labeled.jpg|right|thumb|upright=1.1|Babylonian clay tablet YBC 7289 showing the first four [[sexagesimal]] [[Positional notation|place values]] for an approximation of the square root of 2:<ref name=Fowler_Eleanor_1998/> {{nowrap|1 24 51 10}}]]
{{further|Real number#History|History of irrational numbers}}
The Babylonians, as early as 1800 BCE, demonstrated numerical approximations of irrational quantities such as √2 on clay tablets, with an accuracy analogous to six decimal places, as in the tablet [[YBC 7289]].<ref name=Fowler_Eleanor_1998>{{cite journal | title=Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context | first1=David | last1=Fowler | first2=Eleanor | last2=Robson | journal=Historia Mathematica | volume=25 | issue=4 | date=November 1998 | pages=366–378 | publisher=Elsevier | doi=10.1006/hmat.1998.2209 }}</ref> These values were primarily used 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=((978-0-486-23356-7)) | pages=36–38 | url=https://books.google.com/books?id=JVhTtVA2zr8C&pg=PA36 }}</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://archive.org/details/mathematicsacrossculturesthehistoryofnonwesternmathematicshelaineselin1946 }}</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]].<ref>{{cite journal | title=The Discovery of Incommensurability by Hippasus of Metapontum | first=Kurt | last=Von Fritz | journal=Annals of Mathematics | volume=46 | issue=2 | date=April 1945 | pages=242–264 | doi=10.2307/1969021 | jstor=1969021 }}</ref> 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'' | first=Bernard | last=Frischer | editor-first=D. R. Shackleton | editor-last=Bailey | editor-link=D. R. Shackleton Bailey | page=83 | publisher=Harvard University Press | year=1984 | volume=88 | isbn=0-674-37935-7 | doi=10.2307/311446 | jstor=311446 }}</ref>

The 16th century brought final European acceptance of negative integers and fractional numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. The concept of [[real number]]s was introduced in the 17th century by [[René Descartes]].<ref>{{cite book
 | last=Borthwick | first=D. | year=2025 | chapter=Real Numbers | title=A Primer for Mathematical Analysis | series=Synthesis Lectures on Mathematics & Statistics | pages=1–15 | publisher=Springer, Cham. | doi=10.1007/978-3-031-91713-4_1 | isbn=978-3-031-91712-7 }}</ref> While studying [[compound interest]], in 1683 [[Jacob Bernoulli]] found that as the compounding intervals grew ever shorter, the rate of [[exponential growth]] converged to a [[Base (exponentiation)|base]] of 2.71828...; this key mathematical constant would later be named [[Euler's number]] ({{mvar|e}}).<ref>{{cite journal | title=A Number between 2 and 3 | first=Graham | last=Winter | journal=Mathematics in School | volume=36 | issue=5 | date=November 2007 | pages=30–32 | publisher=The Mathematical Association | jstor=30216078 }}</ref> Irrational numbers began to be studied systematically in the 18th century, with [[Leonhard Euler]] who proved that the irrational numbers are those numbers whose [[simple continued fraction]]s is not finite and that Euler's number ({{mvar|e}}) is irrational.<ref>{{cite journal | title=The origins of Euler's early work on continued fractions | first=Rosanna | last=Cretney | journal=Historia Mathematica | volume=41 | issue=2 | date=May 2014 | pages=139–156 | publisher=Elsevier | doi=10.1016/j.hm.2013.12.004 }}</ref> The [[Proof that π is irrational|irrationality of {{pi}} was proved]] in 1761 by [[Johann Lambert]].<ref name=Laczkovich_1997>{{cite journal | title=On Lambert's Proof of the Irrationality of π | first=M. | last=Laczkovich | author-link=Miklós Laczkovich | journal=The American Mathematical Monthly | volume=104 | issue=5 | date=May 1997 | pages=439–443 | publisher=Taylor & Francis, Ltd. | doi=10.2307/2974737 | jstor=2974737 }}</ref>

It is in the second half of the 19th century that real numbers, and thus irrational numbers, were rigorously defined, with the work of [[Augustin-Louis Cauchy]], [[Charles Méray]] (1869), [[Karl Weierstrass]] (1872), [[Eduard Heine]] (1872),<ref>{{cite journal | first=Eduard | last=Heine | doi=10.1515/crll.1872.74.172 | title=Die Elemente der Functionenlehre | journal=[Crelle's] Journal für die reine und angewandte Mathematik | issue=74 | volume=1872 | pages=172–188 | date=December 14, 2009 }}</ref> [[Georg Cantor]] (1883),<ref>{{cite journal | first=Georg | last=Cantor | doi=10.1007/BF01446819 | title=Ueber unendliche, lineare Punktmannichfaltigkeiten, pt. 5 | journal=Mathematische Annalen | volume=21 | issue=4 | date=December 1883 | pages=545–591 }}</ref> and [[Richard Dedekind]] (1872).<ref>{{cite book | first=Richard | last=Dedekind | url=https://books.google.com/books?id=n-43AAAAMAAJ | title=Stetigkeit und irrationale Zahlen | location=Braunschweig | publisher=Friedrich Vieweg & Sohn | year=1872 }} Subsequently published in: {{cite book | title=Gesammelte mathematische Werke | editor1-first=Robert | editor1-last=Fricke | editor2-first=Emmy | editor2-last=Noether | editor3-first=Öystein | editor3-last=Ore | location=Braunschweig | publisher=Friedrich Vieweg & Sohn | year=1932 | volume=3 | pages=315–334 }}</ref>

===<span class="anchor" id="History of transcendental numbers and reals"></span> Transcendental numbers and reals===
{{further|History of π|Liouville number}}

A [[transcendental number]] is a numerical value that is not the root of a [[polynomial]] with integer coefficients. This means it is not [[Algebraic number|algebraic]] and thus excludes all rational numbers.<ref name=Church>{{cite web | title=Transcendental Numbers | first=Benjamin | last=Church | publisher=Stanford University | url=https://web.stanford.edu/~bvchurch/assets/files/talks/Liouville.pdf | access-date=2025-10-22 }}</ref> 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.<ref name=NIE_14>{{cite encyclopedia | title=Number | display-editors=1 | editor1-first=Daniel Coit | editor1-last=Gilman | editor2-first=Harry Thurston | editor2-last=Peck | editor3-first=Frank Moore | editor3-last=Colby | page=676 | encyclopedia=The New International Encyclopaedia | volume=14 | publisher=Dodd, Mead | year=1906 | url=https://books.google.com/books?id=RTorAAAAMAAJ&pg=PA676 }}</ref> 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.<ref name=Johnson_1972>{{cite journal | title=The Genesis and Development of Set Theory | first=Phillip E. | last=Johnson | journal=The Two-Year College Mathematics Journal | volume=3 | year=1972 | issue=1 | pages=55–62 | publisher=Taylor & Francis | doi=10.2307/3026799 | jstor=3026799 }}</ref>

===<span class="anchor" id="History of infinity and infinitesimals"></span> Infinity and infinitesimals===
{{further|History of infinity}}
In mathematics, [[infinity]] is considered an abstract [[concept]] rather than a number; instead of being "greater than any number", infinite is the property of having no end.<ref>{{cite book | title=The Language of Mathematics: Utilizing Math in Practice | first=Robert L. | last=Baber | publisher=John Wiley & Sons | year=2011 | isbn=978-1-118-06176-3 | pages=102–103 | url=https://books.google.com/books?id=xHLG8cNQ14wC&pg=PA102 }}</ref> The earliest known conception of mathematical infinity appears in the [[Yajurveda]], an ancient Indian script, which at one point states, "If &lsqb;the whole&rsqb; was subtract from &lsqb;the whole&rsqb;, the leftover will still be &lsqb;the whole&rsqb;".<ref>{{cite book | title=History of Ancient India, From the Last Ice Age to The Mahabharata War (≈9000–1400 BCE) | first=Omesh K. | last=Chopra | publisher=Blue Rose Publishers | year=2023 | page=201 | url=https://books.google.com/books?id=xOmmEAAAQBAJ&pg=PA201 }} The word 'purna' is used, which can mean whole.</ref> 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.<ref>{{cite book | title=Infinity: A Very Short Introduction | series=Very Short Introductions | first=Ian | last=Stewart | publisher=Oxford University Press | year=2017 | isbn=978-0-19-107151-5 | url=https://books.google.com/books?id=HDNdDgAAQBAJ&pg=PT141 }}</ref>

[[Aristotle]] defined the traditional Western notion of mathematical infinity. He distinguished between [[Actual and potential infinity|actual infinity and potential infinity]]—the general consensus being that only the latter had true value.<ref>{{cite journal | title=Aristotelian Infinity | first=Jaakko | last=Hintikka | author-link=Jaakko Hintikka | journal=The Philosophical Review | volume=75 | issue=2 | date=April 1966 | pages=197–218 | publisher=Duke University Press | doi=10.2307/2183083 | jstor=2183083 }}</ref> [[Galileo Galilei]]'s ''[[Two New Sciences]]'' discussed the idea of [[bijection|one-to-one correspondences]] between infinite sets, known as [[Galileo's paradox]].<ref>{{Cite book | last=Galilei | first=Galileo | author-link=Galileo Galilei | translator-last=Crew and de Salvio | title=[[Dialogues concerning two new sciences]] | year=1954 | orig-year = 1638 | publisher=[[Dover Publications|Dover]] | location=New York  | pages=31–33}}</ref> 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]].<ref name=Johnson_1972/> The symbol <math>\text{∞}</math>, often used to represent an infinite quantity, was first introduced in a mathematical context by [[John Wallis]] in 1655.<ref>{{cite book | title=Zero and infinity: Mathematics without frontiers | first=Ilija | last=Barukcic | edition=2nd | publisher=BoD – Books on Demand | year=2020 | isbn=9-783-7519-1873-2 | url=https://books.google.com/books?id=BazdDwAAQBAJ&pg=PA134 }}</ref>

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.<ref>{{cite web | title=Abraham Robinson | first1=J. J. | last1=O'Connor | first2=E. F. | last2=Robertson | date=July 2000 | website=MacTutor | url=https://mathshistory.st-andrews.ac.uk/Biographies/Robinson/ | access-date=2025-10-22 }}</ref><ref>{{cite web | title=An Introduction to Nonstandard Analysis | first=Isaac | last=Davis | publisher=Department of Mathematics, The University of Chicago | url=https://math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Davis.pdf | access-date=2025-10-22 }}</ref> 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]].<ref>{{cite book | chapter=The Hyperreals | first=Michael | last=Henle | title=Which Numbers are Real? | series=Classroom Resource Materials | publisher=Mathematical Association of America | year=2012 | pages=125–170 | isbn=978-1-61444-107-6 | doi=10.5948/UPO9781614441076.010 | url=https://www.cambridge.org/core/books/abs/which-numbers-are-real/hyperreals/D7D72569C82CC6F454549E59C440E105 | access-date=2025-10-22 }}</ref>

A modern geometrical version of infinity is given by [[projective geometry]], which introduces "ideal [[Point at infinity|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.<ref>{{cite book | title=Geometry from Euclid to Knots | series=Dover Books on Mathematics | first=Saul | last=Stahl | publisher=Courier Corporation | year=2012 | page=191 | isbn=978-0-486-13498-7 | url=https://books.google.com/books?id=jLk7lu3bA1wC&pg=PA191 }}</ref>

===<span class="anchor" id="History of complex numbers"></span> 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]].<ref>{{cite web | title=A complex mistake? | website=Nrich | publisher=University of Cambridge | url=https://nrich.maths.org/complex-mistake | access-date=2025-10-23 }}</ref> 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.<ref>{{cite journal | title=A Short History of Imaginary Numbers | journal=International Journal of Fundamental Physical Sciences | volume=9 | issue=1 | pages=01–05 | date=March 2019 | first=Misha | last=Nikouravan | doi=10.14331/ijfps.2019.330121 | url=https://www.fundamentaljournals.com/index.php/ijfps/article/view/126 | access-date=2025-10-22 <!-- Note: the doi=10.14331/ijfps.2019.330121 is broken --> | doi-access=free }}</ref>

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. [[René Descartes]] is sometimes credited with coining the term "imaginary" for these quantities in 1637, intending it as derogatory.<ref>{{cite book | title=Descartes's Imagination: Proportion, Images, and the Activity of Thinking | first=Dennis L. | last=Sepper | publisher=University of California Press | year=1996 | page=71 | isbn=978-0-520-20050-0 | url=https://books.google.com/books?id=bDS1cCdw7oEC&pg=PA71 }}</ref> (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.

[[File:Euler's formula caimi.svg|right|thumb|[[Argand diagram]] of Euler's formula in the [[complex plane]], showing re&lsqb;al&rsqb; and im&lsqb;aginary&rsqb; coordinates]]
The 18th century saw the work of [[Abraham de Moivre]] and [[Leonhard Euler]]. [[De Moivre's formula]] (1730) states:<ref>{{cite book | title=Abraham De Moivre: Setting the Stage for Classical Probability and Its Applications | series=An A K Peters book | first=David R. | last=Bellhouse | publisher=CRC Press | year=2011 | page=142 | isbn=978-1-4398-6578-1 | url=https://books.google.com/books?id=PPTRBQAAQBAJ&pg=PA142 }}</ref>
:<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>
A special case of this formula yields [[Euler's identity]]:
:<math>e ^{i\pi} + 1 = 0</math>
showing a profound connection between the most fundamental numbers in mathematics.<ref>{{cite journal | title=An Appreciation of Euler's Formula | first=Caleb | last=Larson | year=2017 | journal=Rose-Hulman Undergraduate Mathematics Journal | volume=18 | issue=1 | article-number=17 | url=https://scholar.rose-hulman.edu/rhumj/vol18/iss1/17 | access-date=2025-10-23 }}</ref>

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.<ref>{{cite book | title=A History of Vector Analysis: The Evolution of the Idea of a Vectorial System | series=Dover Books on Mathematics Series | first=Michael J. | last=Crowe | publisher=Courier Corporation | year=1994 | pages=5–12 | isbn=978-0-486-67910-5 | url=https://books.google.com/books?id=iVFAVqA91h4C&pg=PA5 }}</ref> However, the idea of the graphic representation of complex numbers had appeared as early as 1685, in [[John Wallis|Wallis]]'s ''De algebra tractatus''.<ref>{{cite conference | title=Argand and the Early Work on Graphical Representation: New Sources and Interpretations | first=Gert | last=Schubring | conference=Around Caspar Wessel and the Geometric Representation of Complex Numbers. Proceedings of the Wessel Symposium at The Royal Danish Academy of Sciences and Letters, Copenhagen, August 11-15 1998. Invited Papers | series=Mathematisk-fysiske meddelelser | editor-first=Jesper | editor-last=Lützen | publisher=Kgl. Danske Videnskabernes Selskab | year=2001 | isbn=978-87-7876-236-8 | pages=140–142 | url=https://books.google.com/books?id=BUxedd5rxFoC&pg=PA140 }}</ref>

In the same year, Gauss provided the first generally accepted proof of the [[fundamental theorem of algebra]],{{cn|date=October 2025|reason=This claim does not match the information on the linked article}} 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.<ref>{{cite book | title=An Episodic History of Mathematics: Mathematical Culture Through Problem Solving | volume=19 | series=Mathematical Association of America Textbooks | first=Steven G. | last=Krantz | publisher=Mathematical Association of America | year=2010 | isbn=978-0-88385-766-3 | page=189 | url=https://books.google.com/books?id=ulmAH-6IzNoC&pg=PA189 }}</ref> 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]].

===<span class="anchor" id="History of prime numbers"></span> Prime numbers===
[[Prime number]]s may have been studied throughout recorded history. They are natural numbers that  are not a product of two smaller natural numbers. It has been suggested that the Ishango bone includes a list of the prime numbers between 10 and 20.<ref>{{cite book | chapter=The Ishango Bone | last=Overmann | first=K. A. | year=2025 | pages=53–58 | title=Cultural Number Systems | series=Interdisciplinary Contributions to Archaeology | publisher=Springer, Cham. | doi=10.1007/978-3-031-83383-0_8 | isbn=978-3-031-83382-3 }}</ref> The Rhind papyrus display different forms for prime numbers. But the formal study of prime numbers is first documented by the ancient Greek. 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.<ref name=Deza_2021>{{cite book | title=Mersenne Numbers and Fermat Numbers | volume=1 | series=Selected Chapters Of Number Theory: Special Numbers | first=Elena | last=Deza | publisher=World Scientific | year=2021 | pages=39–40 | isbn=978-981-123-033-2 | url=https://books.google.com/books?id=-Wo-EAAAQBAJ&pg=PA39 }}</ref>

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. At around 1000 AD, [[Ibn al-Haytham]] discovered [[Wilson's theorem]]. [[Ibn al-Banna' al-Marrakushi]] found a way to speed up the Sieve of Eratosthenes by only testing up to the square root of the number. Fibonacci communicated Islamic mathematical contributions to Europe, and in 1202 was the first to describe the method of [[trial division]].<ref name=Deza_2021/>

In 1796, [[Adrien-Marie Legendre]] conjectured the [[prime number theorem]], describing the [[Asymptote|asymptotic]] distribution of primes.<ref name=Agarwal_Sen_2014>{{cite book | title=Creators of Mathematical and Computational Sciences | first1=Ravi P. | last1=Agarwal | first2=Syamal K. | last2=Sen | publisher=Springer | year=2014 | isbn=978-3-319-10870-4 | pages=218–219 | url=https://books.google.com/books?id=bENTBQAAQBAJ&pg=PA219 }}</ref> Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges,<ref>{{cite journal | title=New Proof That the Sum of the Reciprocals of Primes Diverges | first1=Vicente | last1=Jara-Vera | first2=Carmen | last2=Sánchez-Ávila | journal=Mathematics | year=2020 | volume=8 | issue=9 | article-number=1414 | doi=10.3390/math8091414 | doi-access=free }}</ref> and the [[Goldbach conjecture]], which claims that any sufficiently large even number is the sum of two primes.<ref name=Weisstein_Goldbach>{{cite web | title=Goldbach Conjecture | last=Weisstein | first=Eric W. | work=MathWorld–A Wolfram Resource | url=https://mathworld.wolfram.com/GoldbachConjecture.html | access-date=2025-10-24 }}</ref> Yet another conjecture related to the distribution of prime numbers is the [[Riemann hypothesis]], formulated by [[Bernhard Riemann]] in 1859.<ref>{{cite journal | last=Conrey | first=J. B. | date=March 2003 | title=The Riemann Hypothesis | journal=Notices of the American Mathematical Society | volume=50 | issue=3 | pages=341–353 | url=https://www.ams.org/journals/notices/200303/fea-conrey-web.pdf?adat=March%202003&trk=200303fea-conrey-web&cat=feature&galt=feature | access-date=2025-10-23 }}</ref> The [[prime number theorem]] was finally proved by [[Jacques Hadamard]] and [[Charles de la Vallée-Poussin]] in 1896.<ref name=Agarwal_Sen_2014/> Goldbach and Riemann's conjectures remain unproven and unrefuted.

=== Cultural and symbolic significance ===
[[File:ShanghaiMissingFloors.jpg|right|thumb|upright=0.5|A Shanghai apartment is missing floors 0, 4, 13, and 14]]
Numbers have held cultural, symbolic and religious significance throughout history and in many cultures.<ref name="Ore" /><ref>{{Cite book |last=Kalvesmaki |first=Joel |url=https://chs.harvard.edu/book/kalvesmaki-joel-the-theology-of-arithmetic-number-symbolism-in-platonism-and-early-christianity/ |title=The Theology of Arithmetic: Number Symbolism in Platonism and Early Christianity |publisher=Hellenic Studies Series 59 |year=2013 |location=Washington, DC}}</ref><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> In Ancient Greece, [[Numerology|number symbolism]] 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" /> According to [[Plato]], [[Pythagoreanism|Pythagoreans]] attributed specific characteristics and meaning to particular numbers, and believed that "things themselves are numbers".<ref>{{cite book | title=Non-diophantine Arithmetics In Mathematics, Physics And Psychology | first1=Mark | last1=Burgin | first2=Marek | last2=Czachor | publisher=World Scientific | year=2020 | isbn=978-981-12-1432-5 | page=38 | url=https://books.google.com/books?id=nVcNEAAAQBAJ&pg=PA38 }}</ref>

Folktales in different cultures exhibit preferences for particular numbers, with three and seven holding special significance in European culture, while four and five are more prominent in Chinese folktales.<ref>{{Cite journal |last=Zhmud |first=Leonid |date=29 August 2019 |title=From Number Symbolism to Arithmology |url=https://www.academia.edu/40206385 |journal=Zahlen- und Buchstabensysteme im Dienste religiöser Bildung | editor-first=L. | editor-last=Schimmelpfennig  | location=Tübingen | publisher=Seraphim | volume=25 | page=45 | isbn=978-3-16-156930-2 }}</ref> Numbers are sometimes associated with luck: in Western society, the [[13 (number)|number 13]] is considered [[unlucky]] while in Chinese culture the [[Chinese numerology#Eight|number eight]] is considered auspicious.<ref>{{cite journal | title='Lucky' numbers, unlucky consumers | first=Zili | last=Yang | journal=The Journal of Socio-Economics | volume=40 | issue=5 | date=October 2011 | pages=692–699 | publisher=Elsevier | doi=10.1016/j.socec.2011.05.008 }}</ref>

==<span class="anchor" id="Classification"></span><span class="anchor" id="Classification of numbers"></span> Main classification==
{{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:<ref name=Bass_2023>{{cite book | title=The Mathematical Neighborhoods of School Mathematics | series=Miscellaneous Book Series | first=Hyman | last=Bass | publisher=American Mathematical Society | year=2023 | isbn=978-1-4704-7247-4 | page=6 | url=https://books.google.com/books?id=UxnREAAAQBAJ&pg=PA6 }}</ref>
{|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:<ref name=Bass_2023/>
:<math>\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}</math>.

[[File:Venn Diagram of Numbers Expanded.svg|right|thumb|[[Venn diagram]] of the number systems]]

===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, various 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>,<ref name=Bass_2023/> and sometimes <math>\mathbb{N}_0</math><ref>{{cite book | title=Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction | last1=Fokas | first1=Athanassios | last2=Kaxiras | first2=Efthimios | date=12 December 2022 | publisher=World Scientific | isbn=978-1-80061-182-5 | page=4 | language=en | url=https://books.google.com/books?id=QwuhEAAAQBAJ&pg=PA4 }}</ref> or <math>\mathbb{N}_1</math><ref>{{cite book | title=Formal Software Development: From VDM to Java | first1=Quentin | last1=Charatan | first2=Aaron | last2=Kans | publisher=Bloomsbury Publishing | year=2003 | page=26 | isbn=978-0-230-00586-0 | url=https://books.google.com/books?id=OyJIEAAAQBAJ&pg=PA26 }}</ref> 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.<ref>{{cite book | title=Basic Electronics Math | first=Clyde | last=Herrick | publisher=Newnes | year=1997 | isbn=978-0-7506-9727-9 | page=26 | url=https://books.google.com/books?id=KVaKP3y1t8MC&pg=PA26 }}</ref>

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 ''S''(''S''(''S''(0))), where ''S'' is the "successor" function (i.e.,&nbsp;3 is the third successor of&nbsp;0).<ref>{{cite book | title=Logic and How it Gets That Way | first=Dale | last=Jacquette | publisher=Routledge | year=2014 | isbn=978-1-317-54653-5 | url=https://books.google.com/books?id=wkuPBAAAQBAJ&pg=PT190 }}</ref> 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}}
[[File:The Ancient Quipu Plate XXI.jpg|right|thumb|The [[Inca Empire]] used knotted strings, or [[quipu]]s, for numerical records and other uses<ref>{{cite journal | title=The Ancient Quipu, a Peruvian Knot Record | first=L. Leland | last=Locke | journal=American Anthropologist, New Series | publisher=Wiley | volume=14 | issue=2 | date=April–June 1912 | pages=325–332 | doi=10.1525/aa.1912.14.2.02a00070 | jstor=659935 }}</ref>]]
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>]].<ref name=Bass_2023/> 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.<ref name=Renshaw_Ireland_2021>{{cite book | title=Maths for Economics | first1=Geoffrey | last1=Renshaw | first2=Norman J. | last2=Ireland | publisher=Oxford University Press | year=2021 | pages=25–27 | isbn=978-0-19-257591-3 | url=https://books.google.com/books?id=k5whEQAAQBAJ&pg=PA25 }}</ref> 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:<ref>{{cite book | title=Mathematical Logic: On Numbers, Sets, Structures, and Symmetry | volume=4 | series=Springer Graduate Texts in Philosophy | first=Roman | last=Kossak | edition=2nd | publisher=Springer Nature | year=2024 | isbn=978-3-031-56215-0 | pages=48–49 | url=https://books.google.com/books?id=ohQDEQAAQBAJ&pg=PA48 }}</ref>
:<math>{1 \over 2} = {2 \over 4}.</math>

In general,{{efn|This follows from the [[substitution property of equality]], by multiplying both fractions with the product of their denominators: <math>{{b \times d}}</math>. Likewise, the converse is true by dividing with the product.}}
:<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 and it is termed an [[improper fraction|improper or top-heavy fraction]].<ref>{{cite book | last=Greer | first=A. | title=New comprehensive mathematics for 'O' level | date=1986 | publisher=Thornes | location=Cheltenham | isbn=978-0-85950-159-0 | page=5 | edition=2nd, reprinted | url=https://books.google.com/books?id=wX2dxeDahAwC&pg=PA5 }}</ref> Fractions can be greater than, less than, or equal to&nbsp;1<ref name=Renshaw_Ireland_2021/> 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>.]]<ref name=Bass_2023/>

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

The symbol for the real numbers is '''R''', also written as <math>\mathbb{R}.</math><ref name=Bass_2023/> They include all the measuring numbers. Every real number corresponds to a point on the [[number line]]. 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.

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.

====Repeating decimal====
If the fractional part of a real number has an infinite sequence of digits that follows a cyclical 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|11}} can be written as 0.272727..., with an ellipsis to indicate that the pattern continues. Forever repeating 27s are also written as 0.{{overline|27}}.<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> These recurring decimals, including the [[Trailing zero|repetition of zeroes]], denote exactly the rational numbers, i.e., all rational numbers are real numbers, but it is not the case that every real number is rational.<ref>{{cite book | title=Basic Concepts in Modern Mathematics | series=Dover Books on Mathematics | first=John Edward | last=Hafstrom | publisher=Courier Corporation | year=2013 | isbn=978-0-486-49729-7 | pages=142–144 | url=https://books.google.com/books?id=DcR51Bv1g3sC&pg=PA142 }}</ref>

For a fractional part with a repeating decimal of consecutive nines, they may be replaced by incrementing the last digit before the nines. Thus, 3.7399999999... or 3.73{{overline|9}} is equivalent to 3.74. A fractional part with an unlimited number of 0s can be rewritten by dropping the 0s to the right of the rightmost nonzero digit.<ref name=Heaton_2017>{{cite book | title=A Brief History of Mathematical Thought | first=Luke | last=Heaton | publisher=Oxford University Press | year=2017 | page=80 | isbn=978-0-19-062179-7 | url=https://books.google.com/books?id=cF7ODQAAQBAJ&pg=PA80 }}</ref> 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,<ref name=Heaton_2017/> 1.00, 1.000, ..., all represent the natural number&nbsp;1.

====Irrational numbers====
For real numbers that are not rational numbers, representing them as decimals would require an infinite sequence of varying digits to the right of the decimal point. These real numbers are called [[irrational number|irrational]]. A famous irrational real number is the [[pi|{{pi}}]],<ref name=Laczkovich_1997/> 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.<ref>{{cite book | title=The Mathematics of Infinity: A Guide to Great Ideas | volume=80 | series=Pure and Applied Mathematics | first=Theodore G. | last=Faticoni | publisher=John Wiley & Sons | year=2006 | isbn=978-0-470-04913-6 | pages=130–131 | url=https://books.google.com/books?id=TJuvjR4YM2kC&pg=PA130 }}</ref> Both these numbers have been approximated (by computer) to trillions {{nowrap|( 1 trillion {{=}} 10<sup>12</sup> {{=}} 1,000,000,000,000 )}} of digits.<ref>{{cite journal | title=Computation of the 100 quadrillionth hexadecimal digit of π on a cluster of Intel Xeon Phi processors | first=Daisuke | last=Takahashi | journal=Parallel Computing | volume=75 | date=July 2018 | pages=1–10 | publisher=Elsevier | doi=10.1016/j.parco.2018.02.002 | hdl=2241/00153370 }}</ref><ref>{{cite journal | title=Origin of Irrational Numbers and Their Approximations | first1=Ravi P. | last1=Agarwal | first2=Hans | last2=Agarwal | journal=Computation | year=2021 | volume=9 | issue=3 | page=29 | doi=10.3390/computation9030029 | doi-access=free }}</ref>

[[File:SimilarGoldenRectangles.svg|right|thumb|Euclid's [[golden ratio]], defined here by <math>{\color{OliveGreen}a + b}</math> is to <math>{\color{Blue}a}</math> as <math>{\color{Blue}a}</math> is to <math>{\color{Red}b}</math>, is an irrational number 𝜙=1.61803… that tends to appear in many aspects of both art and science.<ref>{{cite journal | title=The Golden Ratio in Nature: A Tour across Length Scales | first1=Callum Robert | last1=Marples | first2=Philip Michael | last2=Williams | journal=Symmetry | year=2022 | volume=14 | issue=10 | article-number=2059 | doi=10.3390/sym14102059 | bibcode=2022Symm...14.2059M | doi-access=free }}</ref>]]
[[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, in which a [[decimal point]] is placed to the right of the digit with place value&nbsp;1. 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 in the [[Interval (mathematics)|interval]]:
: <math>\left[\tfrac{12345\mathit{55}}{10000}, \tfrac{12345\mathit{65}}{10000} \right)</math>
when rounding to three decimals, or of any real number in the interval:
: <math>\left[\tfrac{123456}{1000}, \tfrac{123457}{1000} \right)</math>
when truncating after the third 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 subsequent digits are not ''significant''. Therefore, the result is usually rounded to {{nowrap|5.61 m<sup>2</sup>}}.<ref>{{cite book | title=Engineering Mathematics | first=John | last=Bird | edition=6th, revised | publisher=Routledge | year=2010 | page=28 | isbn=978-1-136-40640-9 | url=https://books.google.com/books?id=dOCAixjvUVkC&pg=PA28-IA1 }}</ref>

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

It can be shown that any [[completeness of the real numbers|complete]], [[ordered field]] is isomorphic to the real numbers.<ref>{{cite book | title=The Real Number System | series=Dover Books on Mathematics | first=John M. H. | last=Olmsted | publisher=Courier Dover Publications | year=2018 | isbn=978-0-486-83474-0 | pages=128–129 | url=https://books.google.com/books?id=UitnDwAAQBAJ&pg=PA128 }}</ref> 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>.<ref name=Bădescu_Carletti_2024>{{cite book | title=Lectures on Geometry | series=Mathematics and Statistics | first1=Lucian | last1=Bădescu | first2=Ettore | last2=Carletti | publisher=Springer Nature | year=2024 | isbn=978-3-031-51414-2 | page=9 | url=https://books.google.com/books?id=rhQDEQAAQBAJ&pg=PA9 }}</ref>

===Complex numbers===
{{Main|Complex number}}
[[File:Mandel zoom 00 mandelbrot set.jpg|right|thumb|The [[Mandelbrot set]] is a [[fractal]] on the [[complex plane]] with values that are [[uncountable set|not recursively enumerable]]<ref>{{cite journal | title=The Irrationals are not Recursively Enumerable | first=Richard | last=Mansfield | journal=Proceedings of the American Mathematical Society | volume=110 | issue=2 | date=October 1990 | pages=495–497 | doi=10.2307/2048094 | jstor=2048094 }}</ref>]]
Moving to a greater level of abstraction, the real numbers can be extended to the [[complex number]]s. The complete solution set of a polynomial of [[Quadratic function|degree two]] or higher can include the square roots of negative numbers. (An example is <math>x^2+1=0</math>.<ref name=Bădescu_Carletti_2024/>) To conveniently represent this, the [[square root]] of&nbsp;−1 is denoted by ''i'', a symbol assigned by [[Leonhard Euler]] called the [[imaginary unit]].<ref name=Magalhães_2025/> Hence, complex numbers consist of all values 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]].<ref name=Magalhães_2025>{{cite book | title=Complex Analysis and Dynamics in One Variable with Applications | series=Mathematics and Statistics | first=Luis T. | last=Magalhães | publisher=Springer Nature | year=2025 | pages=1–2 | isbn=978-3-03164999-8 | url=https://books.google.com/books?id=H-RYEQAAQBAJ&pg=PA1 }}</ref>

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'';<ref name=Magalhães_2025/> 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]].<ref>{{cite journal | title=Exploring the Gaussian Integers | first=Robert G. | last=Stein | journal=The Two-Year College Mathematics Journal | volume=7 | issue=4 | pages=4–10 | doi=10.1080/00494925.1976.11974454 | doi-broken-date=26 October 2025 }}</ref> The symbol for the complex numbers is {{math|'''C'''}} or <math>\mathbb{C}</math>.<ref name=Bass_2023/>

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.<ref>{{cite book | title=Introduction to Modern Algebra and Its Applications | first=Nadiya | last=Gubareni | publisher=CRC Press | year=2021 | isbn=978-1-000-20947-1 | pages=172–173 | url=https://books.google.com/books?id=dpoIEQAAQBAJ&pg=PA173 }}</ref> 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]].<ref>{{cite book | title=From Numbers to Analysis | first=Inder K. | last=Rana | publisher=World Scientific | year=1998 | page=327 | isbn=978-981-02-3304-4 | url=https://books.google.com/books?id=CIsOVPm6zE8C&pg=PA327 }}</ref> 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]].

[[Complex analysis]] is the branch of [[mathematical analysis]] that investigates functions of complex numbers. It is useful for the solution of physical problems, and is widely used in modern mathematics, engineering, and the sciences. Examples of applications include [[fluid dynamics]], [[control theory]], [[signal processing]], number theory, and solving [[differential equation]]s.<ref>{{cite book | title=Advanced Engineering Mathematics, International Adaptation | first=Erwin | last=Kreyszig | author-link=Erwin Kreyszig | publisher=John Wiley & Sons | year=2025 | isbn=978-1-394-31946-6 | page=647 | url=https://books.google.com/books?id=IE5aEQAAQBAJ&pg=PA647 }}</ref> Complex numbers appear to be a fundamental aspect of [[quantum mechanics]]; it can not be formulated using only real numbers.<ref>{{cite journal | title=Quantum Mechanics Must Be Complex | first=Alessio | last=Avella | date=January 24, 2022 | journal=Physics | volume=15 | article-number=7 | publisher=American Physical Society | doi=10.1103/Physics.15.7 | bibcode=2022PhyOJ..15....7A | url=https://physics.aps.org/articles/v15/7 | access-date=2025-10-22 | hdl=11696/75499 | hdl-access=free }}</ref>

==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.<ref name=Sidebotham_2003/> (The old-fashioned term "evenly divisible" is now almost always shortened to "[[divisibility|divisible]]".) This property of an integer is called the [[Parity (mathematics)|parity]].<ref>{{cite journal | title=Parity as a Property of Integers | last=Ziobro | first=R. | year=2018 | journal=Formalized Mathematics | volume=26 | issue=2 | pages=91–100 | doi=10.2478/forma-2018-0008 | url=https://journalspress.com/LJRS_Volume23/Pattern-and-Parity-in-Mathematics.pdf | access-date=2025-10-26 }}</ref> 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, ...}. The product of an even number with an integer is another even number; only the product of an odd number with an odd number is another odd number.<ref name=Sidebotham_2003>{{cite book | title=The A to Z of Mathematics: A Basic Guide | first=Thomas H. | last=Sidebotham | publisher=John Wiley & Sons | year=2003 | isbn=978-0-471-46163-0 | page=181 | url=https://books.google.com/books?id=VsAZa5PWLz8C&pg=PA181 }}</ref>

===Prime numbers===
{{main|Prime number}}
[[File:Largest known prime number.svg|right|thumb|Largest known prime numbers by year since 1951<ref>{{cite web | title=The Largest Known prime by Year: A Brief History | first=Chris | last=Caldwell | work=The PrimePages: prime number research & records | url=https://t5k.org/notes/by_year.html | access-date=2025-10-23 }}</ref>]]
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. A special class are the [[Mersenne prime]]s, which are prime numbers of the form {{nowrap|2<sup>''n''</sup> − 1}}, where ''n'' is a positive integer. These hold many records for the largest prime numbers discovered.<ref>{{cite journal | title=The Great Prime Number Record Races | first=Günter M. | last=Ziegler | journal=Monthly Notices of the American Mathematical Society | date=April 2004 | volume=51 | issue=4 | url=https://www.ams.org/journals/notices/200404/comm-ziegler.pdf | access-date=2025-10-26 }}</ref>

The study of primes have led to many questions, only some of which have been answered. The study of these questions belongs to [[number theory]].<ref name="Ore"/> [[Goldbach's conjecture]] is an example of a still unanswered question: "Is every even number the sum of two primes?"<ref name=Weisstein_Goldbach/> 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]].<ref name=Deza_2021/>

In the modern world, prime numbers have a number of important applications, including in [[public-key cryptography]], [[digital signature]], [[pseudorandom number generation]], [[signal processing]], and filtering data for [[digital image processing]].<ref>{{cite book | title=From Great Discoveries in Number Theory to Applications | display-authors=1 | first1=Michal | last1=Křížek | first2=Lawrence | last2=Somer | first3=Alena | last3=Šolcová | publisher=Springer Nature | year=2021 | page=4 | isbn=978-3-030-83899-7 | url=https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA4 }}</ref> Prime numbers are useful in [[hash table]]s<ref>{{cite web | title=Hash table size | work=Advanced Data Structures: CSE 100 | publisher=UC San Diego | url=https://cseweb.ucsd.edu/~kube/cls/100/Lectures/lec16/lec16-8.html | access-date=2025-10-23 }}</ref> and [[error detection]] codes (such as those used in [[ISBN]] and [[ISSN]]).<ref>{{cite book | title=From Great Discoveries in Number Theory to Applications | display-authors=1 | first1=Michal | last1=Křížek | first2=Lawrence | last2=Somer | first3=Alena | last3=Šolcová | publisher=Springer Nature | year=2021 | pages=253–256 | isbn=978-3-030-83899-7 | url=https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA253 }}</ref>

===Other classes of integers===
Many subsets of the natural numbers have been the subject of specific studies and have been named, often eponymously after the first mathematician that has studied them. Examples of such sets of integers are [[Bernoulli number]]s,<ref>{{cite journal | title=An Arithmetical Theory of the Bernoulli Numbers | first=H. S. | last=Vandiver | journal=Transactions of the American Mathematical Society | volume=51 | issue=3 | date=May 1942 | pages=502–531 | publisher=American Mathematical Society | doi=10.2307/1990076 | jstor=1990076 }}</ref> [[Fibonacci number]]s, [[Lucas number]]s,<ref>{{cite journal | title=The Fibonacci Numbers—Exposed | last1=Kalman | first1=D. | last2=Mena | first2=R. | year=2003 | journal=Mathematics Magazine | volume=76 | issue=3 | pages=167–181 | doi=10.1080/0025570X.2003.11953176 }}</ref> and [[perfect number]]s.<ref>{{cite journal | title=On perfect and near-perfect numbers | first1=Paul | last1=Pollack | first2=Vladimir | last2=Shevelev | journal=Journal of Number Theory | volume=132 | issue=12 | date=December 2012 | pages=3037–3046 | publisher=Elsevier | doi=10.1016/j.jnt.2012.06.008 }}</ref> 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.<ref name=Church/> The algebraic numbers that are solutions of a [[monic polynomial]] equation with integer coefficients are called [[algebraic integer]]s.<ref>{{cite book | title=A Physicists Introduction to Algebraic Structures: Vector Spaces, Groups, Topological spaces and more | first=Palash B. | last=Pal | publisher=Cambridge University Press | year=2019 | isbn=978-1-108-49220-1 | pages=47–48 | url=https://books.google.com/books?id=VoOWDwAAQBAJ&pg=PA48 }}</ref>

===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=19 July 2013 |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.<ref>{{cite book | title=Introduction to Number Theory | series=Textbooks in Mathematics | display-authors=1 | first1=Anthony | last1=Vazzana | first2=Martin | last2=Erickson | first3=David | last3=Garth | publisher=CRC Press | year=2007 | isbn=978-1-58488-938-0 | page=100 | url=https://books.google.com/books?id=yJ7SBQAAQBAJ&pg=PA100 }}</ref> A related subject is [[origami]] numbers, which are points constructed through paper folding.<ref>{{cite book | title=Origametry: Mathematical Methods in Paper Folding | first=Thomas C. | last=Hull | publisher=Cambridge University Press | year=2020 | isbn=978-1-108-47872-4 | pages=48–57 | url=https://books.google.com/books?id=LdX7DwAAQBAJ&pg=PA48 }}</ref>

===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.<ref>{{cite journal | title=On computable numbers, with an application to the Druckproblem | display-authors=1 | first1=Sophie | last1=Berthelette | first2=Gilles | last2=Brassard | first3=Xavier | last3=Coiteux-Roy | journal=Theoretical Computer Science | volume=1002 | date=29 June 2024 | article-number=114573 | publisher=Elsevier | doi=10.1016/j.tcs.2024.114573 }}</ref> Equivalent definitions can be given using [[μ-recursive function]]s, [[Turing machine]]s or [[λ-calculus]].<ref>{{cite encyclopedia | title=Computability and Complexity | encyclopedia=Stanford Encyclopedia of Philosophy | date=October 18, 2021 | last=Immerman | first=Neil | edition=Winter 2021 | editor-first=Edward N. | editor-last=Zalta | url=https://plato.stanford.edu/archives/win2021/entries/computability/ | access-date=2025-10-27 }}</ref> 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.<ref>{{cite book | title=Computability in Analysis and Physics | volume=1 | series=Perspectives in Logic | first1=Marian B. | last1=Pour-El | first2=J. Ian | last2=Richards | publisher=Cambridge University Press | year=2017 | isbn=978-1-107-16844-2 | page=44 | url=https://books.google.com/books?id=9jMoDgAAQBAJ&pg=PA44 }}</ref>

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,<ref>{{cite book | chapter=Local and Global in Number Theory | first=Fernando Q. | last=Gouvêa | title=The Princeton Companion to Mathematics | display-editors=1 | editor1-first=Timothy | editor1-last=Gowers | editor2-first=June | editor2-last=Barrow-Green | editor3-first=Imre | editor3-last=Leader | publisher=Princeton University Press | year=2010 | isbn=978-1-4008-3039-8 | pages=242–243 | chapter-url=https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA242 }}</ref><ref>{{cite web | title=Introduction to ''p''-adic Numbers | first1=Benjamin | last1=Church | first2=Matthew | last2=Lerner-Brecher | publisher=Stanford University | url=https://web.stanford.edu/~bvchurch/assets/files/talks/p-adics.pdf | access-date=2025-10-27 }}</ref> 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}}
Higher dimensional number systems 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, and are not included in the set of complex numbers. They include the [[quaternion]]s <math>\mathbb{H}</math>, introduced by Sir [[William Rowan Hamilton]], in which multiplication is not [[commutative]];<ref>{{cite journal | title=The Tragic Downfall and Peculiar Revival of Quaternions | first=Danail |last=Brezov | journal=Mathematics | year=2025 | volume=13 | issue=4 | page=637 | doi=10.3390/math13040637 | doi-access=free }}</ref> the [[octonion]]s <math>\mathbb{O}</math>, in which multiplication is not [[associative]] in addition to not being commutative;<ref name=Yefremov_2019>{{cite book | title=The General Theory of Particle Mechanics: A Special Course | first=Alexander P. | last=Yefremov | publisher=Cambridge Scholars Publishing | year=2019 | isbn=978-1-5275-3292-2 | pages=8–11 | url=https://books.google.com/books?id=3ZqSDwAAQBAJ&pg=PA8 }}</ref> and the [[sedenion]]s <math>\mathbb{S}</math>, in which multiplication is not [[Alternative algebra|alternative]], neither associative nor commutative.<ref>{{cite book | title=Hypercomplex: Trends for a Mathematical Foundation | first1=Manoel Ferreira Borges | last1=Neto | first2=José | last2=Marão | publisher=Editora Appris | year=2023 | isbn=978-65-250-4443-9 | pages=55–56 | url=https://books.google.com/books?id=3Be_EAAAQBAJ&pg=PA56 }}</ref> 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.<ref name=Yefremov_2019/>

Each hypercomplex number system is a [[subset]] of the next hypercomplex number system of double dimensions obtained via the [[Cayley–Dickson construction]].<ref name=Valkova-Jarvis_et_al_2025/> For example, the 4-dimensional quaternions <math>\mathbb{H}</math> are a subset of the 8-dimensional octonions <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:<ref name=Valkova-Jarvis_et_al_2025>{{cite journal | title=Hypercomplex Numbers—A Tool for Enhanced Efficiency and Intelligence in Digital Signal Processing | display-authors=1 | last1=Valkova-Jarvis | first1=Zlatka | last2=Nenova | first2=Maria | last3=Mihaylova | first3=Dimitriya | journal=Mathematics | volume=13 | issue=3 | year=2025 | page=504 | doi=10.3390/math13030504 | doi-access=free }}</ref>
:<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 \mathcal C_n</math>

with <math>\mathcal 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>

Quaternions have proven particularly useful for computation of rotations in three dimensions. For example, they are used in control systems for rockets and aircraft, as well as for robotics, computer visualization, navigation, and animation.<ref>{{cite news | title=The many modern uses of quaternions | first=Peter | last=Lynch | date=4 October 2018 | newspaper=The Irish Times | url=https://www.irishtimes.com/news/science/the-many-modern-uses-of-quaternions-1.3642385 | access-date=2025-10-22 }}</ref> Octonions appear to have a deeper theoretical connection with physics, particularly in [[string theory]], [[M-theory]] and [[supergravity]].<ref>{{cite web | title=The Peculiar Math That Could Underlie the Laws of Nature | first=Natalie | last=Wolchover | date=20 July 2018 | work=Quanta Magazine | url=https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ | access-date=2025-10-22 }}</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.<ref>{{cite book | title=Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective | volume=12 | series=World Scientific Series In Information Studies | first=Mark | last=Burgin | publisher=World Scientific | year=2022 | isbn=978-981-123-685-3 | url=https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA99 }}</ref>

===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'''.<ref>{{cite book | title=Nonstandard Analysis | first=Martin | last=Väth | publisher=Springer Science & Business Media | year=2007 | isbn=978-3-7643-7773-1 | pages=59–61 | url=https://books.google.com/books?id=sxxFDijkS1UC&pg=PA59 }}</ref>

[[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]].<ref name=Kuhlemann_2024>{{cite book | title=Nonstandard Analysis: In Higher Education, Logic and Philosophy | series=De Gruyter STEM | first=Karl | last=Kuhlemann | publisher=Walter de Gruyter GmbH & Co KG | year=2024 | pages=105–106 | isbn=978-3-11-143053-9 | url=https://books.google.com/books?id=gUczEQAAQBAJ&pg=PA105 }}</ref><ref>{{cite journal | title=Conway’s field of surreal numbers | first=Norman L. | last=Alling | journal=Transactions of the American Mathematical Society | volume=287 | year=1985 | pages=365–386 | doi=10.1090/S0002-9947-1985-0766225-7 }}</ref>

==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}}
* [[Numerical cognition]]
* [[Orders of magnitude]]
* {{Annotated link|Physical constant}}
* {{Annotated link|Physical quantity}}
* {{Annotated link|Scalar (mathematics)}}
* [[Subitizing and counting]]
{{colend}}

==Notes==
{{notelist}}

==References==
{{reflist}}

==Further reading==
* {{cite book | last=Cory | first=Leo | title=A Brief History of Numbers | publisher=Oxford University Press | year=2015 | isbn=978-0-19-870259-7 }}
* {{cite book | author-link=Tobias Dantzig | last=Dantzig | first=Tobias | title=Number, the language of science; a critical survey written for the cultured non-mathematician | location=New York | publisher=The Macmillan Company | year=1930 }}
* {{cite web | last=Friedman | first=Erich | publisher=Stetson University | url=http://www.stetson.edu/~efriedma/numbers.html | title=What's special about this number? | archive-url=https://web.archive.org/web/20180223062027/http://www2.stetson.edu/~efriedma/numbers.html | access-date=23 February 2018 | archive-date=2018-02-23 | url-status=live }}
* {{cite book | last=Galovich | first=Steven | title=Introduction to Mathematical Structures | publisher=Harcourt Brace Javanovich | year=1989 | isbn=0-15-543468-3 }}
* {{cite book | author-link=Paul Halmos | last=Halmos | first=Paul | title=Naive Set Theory | publisher=Springer | year=1974 | isbn=0-387-90092-6 }}
* {{cite book | author-link=Morris Kline | last=Kline | first=Morris | title=Mathematical Thought from Ancient to Modern Times | publisher=Oxford University Press | year=1990 | isbn=978-0-19-506135-2 }}
* {{cite book | author-link=Alfred North Whitehead | last1=Whitehead | first1=Alfred North | author-link2=Bertrand Russell | first2=Bertrand | last2=Russel | title=[[Principia Mathematica]] to *56 | publisher=Cambridge University Press | year=1910 }}

==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=https://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|url-status=live|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]]