Editing Complex number (section)

===Powers and roots===
{{see also|Square root#Square roots of negative and complex numbers|l1=Square roots of negative and complex numbers}}
The ''n''-th power of a complex number can be computed using [[de Moivre's formula]], which is obtained by repeatedly applying the above formula for the product:
<math display=block> z^{n}=\underbrace{z \cdot \dots \cdot z}_{n \text{ factors}} = (r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).</math>
For example, the first few powers of the imaginary unit ''i'' are <math>i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, \dots</math>.

{{Visualisation complex number roots|1=upright=1.35}}
The {{mvar|n}} [[nth root|{{mvar|n}}th roots]] of a complex number {{mvar|z}} are given by
<math display=block>z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)</math>
for {{math|0 ≤ ''k'' ≤ ''n'' − 1}}. (Here <math>\sqrt[n]r</math> is the usual (positive) {{mvar|n}}th root of the positive real number {{mvar|r}}.) Because sine and cosine are periodic, other integer values of {{mvar|k}} do not give other values. For any <math>z \ne 0</math>, there are, in particular ''n'' distinct complex ''n''-th roots. For example, there are 4 fourth roots of 1, namely
:<math>z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i.</math>
In general there is ''no'' natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. (This is in contrast to the roots of a positive real number ''x'', which has a unique positive real ''n''-th root, which is therefore commonly referred to as ''the'' ''n''-th root of ''x''.) One refers to this situation by saying that the {{mvar|n}}th root is a [[multivalued function|{{mvar|n}}-valued function]] of {{mvar|z}}.