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

\(\ds \cmod {-i}\)

\(=\)

\(\ds \sqrt {0^2 + \paren {-1}^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds 1\)

Then:

\(\ds \map \cos {\map \arg {-i} }\)

\(=\)

\(\ds \dfrac 0 1\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-i}\)

\(=\)

\(\ds \dfrac \pi 2 \text { or } \dfrac {3 \pi} 2\)

Cosine of Half-Integer Multiple of Pi

\(\ds \map \sin {\map \arg {-i} }\)

\(=\)

\(\ds \dfrac {-1} 1\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds -1\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-i}\)

\(=\)

\(\ds \dfrac {3 \pi} 2\)

Sine of Half-Integer Multiple of Pi

Hence:

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

and hence the result.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $81 \ \text {(d)}$