Polar Form of Complex Number/Examples/i
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Example of Polar Form of Complex Number
The imaginary unit $i$ can be expressed in polar form as $\polar {1, \dfrac \pi 2}$.
Proof
\(\ds \cmod i\) | \(=\) | \(\ds \sqrt {0^2 + 1^2}\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
Then:
\(\ds \map \cos {\map \arg i}\) | \(=\) | \(\ds \dfrac 0 1\) | Definition of Argument of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg i\) | \(=\) | \(\ds \pm \dfrac \pi 2\) | Cosine of Half-Integer Multiple of Pi |
\(\ds \map \sin {\map \arg i}\) | \(=\) | \(\ds \dfrac 1 1\) | Definition of Argument of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg i\) | \(=\) | \(\ds \dfrac \pi 2\) | Sine of Half-Integer Multiple of Pi |
Hence:
- $\map \arg i = \dfrac \pi 2$
and hence the result.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations