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:
\(\ds i^3\) | \(=\) | \(\ds i^2 \times i\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1} \times i\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -i\) | Definition of Negative of Complex Number |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems: $(1.3)$