Polar Form of Complex Number/Examples/-1
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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