Editing Number (section)

==History==

===First use of numbers===
{{main|History of ancient numeral systems}}
Bones and other artifacts have been discovered with marks cut into them that many believe are [[tally marks]].<ref>{{Cite book |last=Marshack |first=Alexander |url=https://books.google.com/books?id=vbQ9AAAAIAAJ |title=The roots of civilization; the cognitive beginnings of man's first art, symbol, and notation (1st ed.) |date=1971 |publisher=McGraw-Hill |isbn=0-07-040535-2 |location=New York |oclc=257105}}</ref> These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

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

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

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

===Zero{{anchor|History of zero}}===
{{refimprove section|date=November 2022}}
The first known recorded use of [[zero]] dates to AD 628, and appeared in the ''[[Brāhmasphuṭasiddhānta]]'', the main work of the [[Indian mathematician]] [[Brahmagupta]]. He treated&nbsp;0 as a number and discussed operations involving it, including [[division by zero]]. By this time (the 7th&nbsp;century), the concept had clearly reached Cambodia in the form of [[Khmer numerals]],<ref>{{Cite magazine |last=Aczel |first=Amir D. |date=2015-05-07 |title=My Quest to Find the First Zero |url=https://time.com/3845786/my-quest-to-find-the-first-zero/ |access-date=2025-02-15 |magazine=TIME |language=en}}</ref> and documentation shows the idea later spreading to China and the [[Islamic world]].

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

Brahmagupta's ''Brāhmasphuṭasiddhānta'' is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". The ''Brāhmasphuṭasiddhānta'' is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.

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

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

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

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

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

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

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

During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician [[Brahmagupta]], in ''[[Brāhmasphuṭasiddhānta]]'' in 628, who used negative numbers to produce the general form [[quadratic formula]] that remains in use today. However, in the 12th&nbsp;century in India, [[Bhāskara II|Bhaskara]] gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots".

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th&nbsp;century, although [[Fibonacci]] allowed negative solutions in financial problems where they could be interpreted as debts (chapter&nbsp;13 of {{Lang|la|[[Liber Abaci]]}}, 1202) and later as losses (in {{lang|la|Flos}}). [[René Descartes]] called them false roots as they cropped up in algebraic polynomials yet he found a way to swap true roots and false roots as well. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.<ref>{{Cite book |last=Smith |first=David Eugene |author-link=David Eugene Smith |title=History of Modern Mathematics |publisher=Dover Publications |year=1958 |page=259 |isbn=0-486-20429-4}}</ref> The first use of negative numbers in a European work was by [[Nicolas Chuquet]] during the 15th&nbsp;century. He used them as [[exponent]]s, but referred to them as "absurd numbers".

As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.

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

The concept of [[decimal fraction]]s is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math [[sutra]] to include calculations of decimal-fraction approximations to [[pi]] or the [[square root of 2]].{{Citation needed|date=September 2020}} Similarly, Babylonian math texts used sexagesimal (base&nbsp;60) fractions with great frequency.

===Irrational numbers {{anchor|History of irrational numbers}}===
{{further|History of irrational numbers}}
The earliest known use of irrational numbers was in the [[Indian mathematics|Indian]] [[Sulba 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=451 |isbn=0-7923-6481-3}}</ref>{{Better source needed|reason=Source may be unreliable it garbles both the history and the mathematics. Source only says the mathematics in the Shulba Sutras "leads to the concept of irrational numbers". Since good approximations of irrational numbers appeared in earlier times, it's not clear what special role is being claimed for the Shulba Sutras in the history of irrational numbers. Also, should page reference be to p. 412 rather than p. 451?|date=September 2020}} The first existence proofs of irrational numbers is usually attributed to [[Pythagoras]], more specifically to the [[Pythagoreanism|Pythagorean]] [[Hippasus|Hippasus of Metapontum]], who produced a (most likely geometrical) proof of the irrationality of the [[square root of 2]]. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news.<ref>{{cite book |title=Harvard Studies in Classical Philology |chapter=Horace and the Monuments: A New Interpretation of the Archytas ''Ode'' |author=Bernard Frischer |editor=D.R. Shackleton Bailey |editor-link=D. R. Shackleton Bailey |page=83 |publisher=Harvard University Press |year=1984 |isbn=0-674-37935-7}}</ref>{{Better source needed|reason=Hippasus is mentioned only briefly in passing in this work. Entire books have been written on Pythagoras and Pythagoreanism; surely a reference could be provide to one of those? But any serious work will say that everything in this paragraph is unreliable myth, and some is outright modern fabrication, e.g. Pythagoras sentencing Hippasus to death.|date=September 2020}}

The 16th century brought final European acceptance of negative integral and fractional numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. It had remained almost dormant since [[Euclid]]. In 1872, the publication of the theories of [[Karl Weierstrass]] (by his pupil E. Kossak), [[Eduard Heine]],<ref>Eduard Heine, [[doi:10.1515/crll.1872.74.172|"Die Elemente der Functionenlehre"]], ''[Crelle's] Journal für die reine und angewandte Mathematik'', No. 74 (1872): 172–188.</ref> [[Georg Cantor]],<ref>Georg Cantor, [[doi:10.1007/BF01446819|"Ueber unendliche, lineare Punktmannichfaltigkeiten", pt. 5]], ''Mathematische Annalen'', 21, 4 (1883‑12): 545–591.</ref> and [[Richard Dedekind]]<ref>Richard Dedekind, ''[https://books.google.com/books?id=n-43AAAAMAAJ Stetigkeit & irrationale Zahlen] {{Webarchive|url=https://web.archive.org/web/20210709184745/https://books.google.ca/books?id=n-43AAAAMAAJ |date=2021-07-09 }}'' (Braunschweig: Friedrich Vieweg & Sohn, 1872). Subsequently published in: ''———, Gesammelte mathematische Werke'', ed. Robert Fricke, Emmy Noether & Öystein Ore (Braunschweig: Friedrich Vieweg & Sohn, 1932), vol. 3, pp. 315–334.</ref> was brought about. In 1869, [[Charles Méray]] had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by [[Salvatore Pincherle]] (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by [[Paul Tannery]] (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a [[Dedekind cut|cut (Schnitt)]] in the system of [[real number]]s, separating all [[rational number]]s into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, [[Leopold Kronecker|Kronecker]],<ref>L. Kronecker, [[doi:10.1515/crll.1887.101.337|"Ueber den Zahlbegriff"]], ''[Crelle's] Journal für die reine und angewandte Mathematik'', No. 101 (1887): 337–355.</ref> and Méray.

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

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

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

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

===Infinity and infinitesimals {{anchor|History of infinity and infinitesimals}}===
{{further|History of infinity}}
The earliest known conception of mathematical [[infinity]] appears in the [[Yajur Veda]], an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the [[Jain]] mathematicians c. 400&nbsp;BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. The symbol <math>\text{∞}</math> is often used to represent an infinite quantity.

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

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

A modern geometrical version of infinity is given by [[projective geometry]], which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in [[perspective (graphical)|perspective]] drawing.

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

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

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

The existence of complex numbers was not completely accepted until [[Caspar Wessel]] described the geometrical interpretation in 1799. [[Carl Friedrich Gauss]] rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in [[John Wallis|Wallis]]'s ''De algebra tractatus''.

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

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

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

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

In 1796, [[Adrien-Marie Legendre]] conjectured the [[prime number theorem]], describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the [[Goldbach conjecture]], which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the [[Riemann hypothesis]], formulated by [[Bernhard Riemann]] in 1859. The [[prime number theorem]] was finally proved by [[Jacques Hadamard]] and [[Charles de la Vallée-Poussin]] in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.