Polar Form of Complex Number/Examples/root 2 i

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Example of Polar Form of Complex Number

The imaginary number $\sqrt 2 i$ can be expressed in polar form as $\polar {\sqrt 2, \dfrac \pi 2}$.


Proof

\(\ds \cmod {\sqrt 2 i}\) \(=\) \(\ds \sqrt {0^2 + \sqrt 2^2}\) Definition of Complex Modulus
\(\ds \) \(=\) \(\ds \sqrt 2\)


Then:

\(\ds \map \cos {\map \arg {\sqrt 2 i} }\) \(=\) \(\ds \dfrac 0 {\sqrt 2}\) Definition of Argument of Complex Number
\(\ds \) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \map \arg {\sqrt 2 i}\) \(=\) \(\ds \dfrac \pi 2 \text { or } \dfrac {3 \pi} 2\) Cosine of Half-Integer Multiple of Pi


\(\ds \map \sin {\map \arg {\sqrt 2 i} }\) \(=\) \(\ds \dfrac {\sqrt 2} {\sqrt 2}\) Definition of Argument of Complex Number
\(\ds \) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds \map \arg {\sqrt 2 i}\) \(=\) \(\ds \dfrac \pi 2\) Sine of Half-Integer Multiple of Pi


Hence:

$\map \arg {\sqrt 2 i} = \dfrac \pi 2$

and hence the result.

$\blacksquare$


Sources