Powers of Imaginary Unit

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Theorem

The (integer) powers of the imaginary unit $i$ are:

\(\ds i^0\) \(=\) \(\ds 1\)
\(\ds i^1\) \(=\) \(\ds i\)
\(\ds i^2\) \(=\) \(\ds -1\)
\(\ds i^3\) \(=\) \(\ds -i\)
\(\ds i^4\) \(=\) \(\ds 1\)

$\blacksquare$


Proof

By definition:

$i^2 = -1$


Then we have:

\(\ds i^0\) \(=\) \(\ds e^{0 \ln i}\) Definition of Power to Complex Number
\(\ds \) \(=\) \(\ds e^0\)
\(\ds \) \(=\) \(\ds 1\)


Then:

\(\ds i^1\) \(=\) \(\ds e^{1 \ln i}\) Definition of Power to Complex Number
\(\ds \) \(=\) \(\ds e^{\ln i}\)
\(\ds \) \(=\) \(\ds i\) Definition of Exponential Function


Finally:

\(\ds i^3\) \(=\) \(\ds i^2 \times i\)
\(\ds \) \(=\) \(\ds \left({-1}\right) \times i\)
\(\ds \) \(=\) \(\ds -i\) Definition of Negative of Complex Number

$\blacksquare$


Sources