Polar Form of Complex Number/Examples/-4
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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: $81 \ \text {(e)}$