Editing Pi (section)

=== Complex numbers and Euler's identity ===
[[File:Euler's formula.svg|thumb|alt=A diagram of a unit circle centred at the origin in the complex plane, including a ray from the centre of the circle to its edge, with the triangle legs labelled with sine and cosine functions.|The association between imaginary powers of the number {{math|''e''}} and [[Point (geometry)|points]] on the [[unit circle]] centred at the [[Origin (mathematics)|origin]] in the [[complex plane]] given by [[Euler's formula]]]]

Any [[complex number]], say {{Mvar|z}}, can be expressed using a pair of [[real number]]s. In the [[Polar coordinate system#Complex numbers|polar coordinate system]], one number ([[radius]] or {{Mvar|r}}) is used to represent {{Mvar|z}}'s distance from the [[Origin (mathematics)|origin]] of the [[complex plane]], and the other (angle or {{Mvar|φ}}) the counter-clockwise [[rotation]] from the positive real line:{{sfn|Abramson|2014|loc=[https://openstax.org/books/precalculus/pages/8-5-polar-form-of-complex-numbers Section 8.5: Polar form of complex numbers]}}

<math display=block>z = r\cdot(\cos\varphi + i\sin\varphi),</math>

where {{Mvar|i}} is the [[imaginary unit]] satisfying <math>i^2=-1</math>. The frequent appearance of {{pi}} in [[complex analysis]] can be related to the behaviour of the [[exponential function]] of a complex variable, described by [[Euler's formula]]:{{sfn|Bronshteĭn|Semendiaev|1971|p=592}}

<math display=block>e^{i\varphi} = \cos \varphi + i\sin \varphi,</math>

where [[E (mathematical constant)|the constant {{math|''e''}}]] is the base of the [[natural logarithm]]. This formula establishes a correspondence between imaginary powers of {{math|''e''}} and points on the [[unit circle]] centred at the origin of the complex plane. Setting <math>\varphi=\pi</math> in Euler's formula results in [[Euler's identity]], celebrated in mathematics due to it containing five important mathematical constants:{{sfn|Bronshteĭn|Semendiaev|1971|p=592}}<ref>{{cite book |last=Maor |first=Eli |author-link=Eli Maor |title=E: The Story of a Number |publisher=Princeton University Press |year=2009 |page=160 |isbn=978-0-691-14134-3}}</ref>

<math display=block>e^{i \pi} + 1 = 0.</math>

There are {{math|''n''}} different complex numbers {{Mvar|z}} satisfying <math>z^n=1</math>, and these are called the "{{math|''n''}}-th [[root of unity|roots of unity]]"{{sfn|Andrews|Askey|Roy|1999|p=14}} and are given by the formula:

<math display=block>e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).</math>