Euler's Identity - Generalizations

Generalizations

Euler's identity is also a special case of the more general identity that the nth roots of unity, for n > 1, add up to 0:

Euler's identity is the case where n = 2.

In another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions. Let {i, j, k} be the basis elements, then,

In general, given real a_n such that, then,

For octonions, with real a_n such that and the octonion basis elements {i₁, i₂,..., i₇}, then,