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