Polar Form of Complex Number/Examples/-4

Example of Polar Form of Complex Number

The real number $-4$ can be expressed as a complex number in polar form as $\polar {4, \pi}$.

Proof

\(\ds \cmod {-4}\)

\(=\)

\(\ds \sqrt {\paren {-4}^2 + 0^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds 4\)

Then:

\(\ds \map \cos {\map \arg {-4} }\)

\(=\)

\(\ds \dfrac {-4} 4\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds -1\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-1}\)

\(=\)

\(\ds \pi\)

Cosine of Multiple of Pi

\(\ds \map \sin {\map \arg {-4} }\)

\(=\)

\(\ds \dfrac 0 4\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-1}\)

\(=\)

\(\ds 0 \text { or } \pi\)

Sine of Multiple of Pi

Hence:

$\map \arg {-4} = \pi$

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: $81 \ \text {(e)}$