Editing Nth root (section)

===''n''th roots===
{{visualisation_complex_number_roots.svg}}
Every complex number has ''n'' different ''n''th roots in the complex plane.  These are

<math display="block">\eta,\;\eta\omega,\;\eta\omega^2,\;\ldots,\;\eta\omega^{n-1},</math>

where ''η'' is a single ''n''th root, and 1,&nbsp;''ω'',&nbsp;''ω''{{sup|2}},&nbsp;...&nbsp;''ω''{{sup|''n''−1}} are the ''n''th roots of unity.  For example, the four different fourth roots of 2 are

<math display="block">\sqrt[4]{2},\quad i\sqrt[4]{2},\quad -\sqrt[4]{2},\quad\text{and}\quad -i\sqrt[4]{2}.</math>

In [[polar form]], a single ''n''th root may be found by the formula

<math display="block">\sqrt[n]{re^{i\theta}} = \sqrt[n]{r} \cdot e^{i\theta/n}.</math>

Here ''r'' is the magnitude (the modulus, also called the [[absolute value]]) of the number whose root is to be taken; if the number can be written as ''a+bi'' then <math>r=\sqrt{a^2+b^2}</math>. Also, <math>\theta</math> is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that <math>\cos \theta = a/r,</math>  <math> \sin \theta = b/r,</math> and <math> \tan \theta = b/a.</math>

Thus finding ''n''th roots in the complex plane can be segmented into two steps. First, the magnitude of all the ''n''th roots is the ''n''th root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of the ''n''th roots is <math>\theta / n</math>, where <math>\theta</math> is the angle defined in the same way for the number whose root is being taken. Furthermore, all ''n'' of the ''n''th roots are at equally spaced angles from each other.

If ''n'' is even, a complex number's ''n''th roots, of which there are an even number, come in [[additive inverse]] pairs, so that if a number ''r''<sub>1</sub> is one of the ''n''th roots then ''r''<sub>2</sub> =  −''r''<sub>1</sub> is another. This is because raising the latter's coefficient −1 to the ''n''th power for even ''n'' yields 1: that is, (−''r''<sub>1</sub>){{sup|''n''}} = (−1){{sup|''n''}} × ''r''<sub>1</sub>{{sup|''n''}} = ''r''<sub>1</sub>{{sup|''n''}}.

As with square roots, the formula above does not define a [[continuous function]] over the entire complex plane, but instead has a [[branch cut]] at points where ''θ''&nbsp;/&nbsp;''n'' is discontinuous.