Editing Euler's identity (section)

==Generalizations==
Euler's identity is also a special case of the more general identity that the {{mvar|n}}th [[roots of unity]], for {{math|''n'' > 1}}, add up to 0:

:<math>\sum_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = 0 .</math>

Euler's identity is the case where {{math|''n'' {{=}} 2}}.

A similar identity also applies to [[quaternion#Exponential, logarithm, and power functions|quaternion exponential]]: let {{math|{{mset|''i'', ''j'', ''k''}}}} be the basis [[quaternion]]s; then,
:<math>e^{\frac{1}{\sqrt 3}(i \pm j \pm k)\pi} + 1 = 0. </math>

More generally, let {{mvar|q}} be a quaternion  with a zero real part and a norm equal to 1; that is, <math>q=ai+bj+ck,</math> with <math>a^2+b^2+c^2=1.</math> Then one has
:<math>e^{q\pi} + 1 = 0. </math>

The same formula applies to [[octonion]]s, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since <math>i</math> and <math>-i</math> are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.