Polar Form of Complex Number/Examples/2 - 2i

Example of Polar Form of Complex Number

The complex number $2 - 2 i$ can be expressed as a complex number in polar form as $\polar {2 \sqrt 2, \dfrac {7 \pi} 4}$.

Proof

\(\ds \cmod {2 - 2 i}\)

\(=\)

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

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {2 \times 4}\)

\(\ds \)

\(=\)

\(\ds 2 \sqrt 2\)

Then:

\(\ds \map \cos {\map \arg {2 - 2 i} }\)

\(=\)

\(\ds \dfrac 2 {2 \sqrt 2}\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds \frac {\sqrt 2} 2\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {2 - 2 i}\)

\(=\)

\(\ds \dfrac {\pi} 4 \text { or } \dfrac {7 \pi} 4\)

Cosine of $45 \degrees$, Cosine of $315 \degrees$

\(\ds \map \sin {\map \arg {2 - 2 i} }\)

\(=\)

\(\ds \dfrac {-2} {2 \sqrt 2}\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds -\frac {\sqrt 2} 2\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {2 - 2 i}\)

\(=\)

\(\ds \dfrac {5 \pi} 4 \text { or } \dfrac {7 \pi} 4\)

Sine of $225 \degrees$, Sine of $315 \degrees$

Hence:

$\map \arg {2 - 2 i} = \dfrac {7 \pi} 4$

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