Polar Form of Complex Number/Examples/-i

Example of Polar Form of Complex Number

The imaginary number $-i$ can be expressed in polar form as $\polar {1, \dfrac {3 \pi} 2}$.

Proof

 $\displaystyle \cmod {-i}$ $=$ $\displaystyle \sqrt {0^2 + \paren {-1}^2}$ Definition of Complex Modulus $\displaystyle$ $=$ $\displaystyle 1$

Then:

 $\displaystyle \map \cos {\map \arg {-i} }$ $=$ $\displaystyle \dfrac 0 1$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \map \arg {-i}$ $=$ $\displaystyle \dfrac \pi 2 \text { or } \dfrac {3 \pi} 2$ Cosine of Half-Integer Multiple of Pi

 $\displaystyle \map \sin {\map \arg {-i} }$ $=$ $\displaystyle \dfrac {-1} 1$ Definition of Argument of Complex Number $\displaystyle$ $=$ $\displaystyle -1$ $\displaystyle \leadsto \ \$ $\displaystyle \map \arg {-i}$ $=$ $\displaystyle \dfrac {3 \pi} 2$ Sine of Half-Integer Multiple of Pi

Hence:

$\map \arg {-i} = \dfrac {3 \pi} 2$

and hence the result.

$\blacksquare$