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A complex Number is a number that is of the form
Where and . -
A quaternion
can be represented as the rotation corresponding to -
Euler’s Formula states that
Intuitively, this says that an imaginary number captures the rotation on the unit circle. This proof follows from expanding the Taylor series of
and extracting the terms corresponding to and . From this we derive Euler’s Identity
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A root of unity is a complex number that when raised to a positive integer, results in
. More formally, the
-th roots of unity are the complex solutions to the equation -
The closed form of the
-th roots of unity can be obtained using Euler’s Formula. We have that if is the set of -th roots of unity, then This follows immediately from Euler’s identity .
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