Polar Form of Complex Number/Examples/-1

Example of Polar Form of Complex Number

The real number $-1$ can be expressed as a complex number in polar form as $\polar {1, \pi}$.

Proof

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

\(=\)

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

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds 1\)

Then:

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

\(=\)

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

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds -1\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \pi\)

Cosine of Multiple of Pi

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

\(=\)

\(\ds \dfrac 0 1\)

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:

$\map \arg {-1} = \pi$

and hence the result.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations