Editing Number (section)

===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]].