Argument of Complex Number/Examples/-3
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Example of Argument of Complex Number
- $\map \arg {-3} = \pi$
Proof
We have that:
- $-3 = -3 + 0 i$
and so:
\(\ds \size {-3}\) | \(=\) | \(\ds \sqrt {3^2 + 0^2}\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds 3\) |
Hence:
\(\ds \map \cos {\map \arg {-3} }\) | \(=\) | \(\ds \dfrac {-3} 3\) | Definition of Argument of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds -1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg {-3}\) | \(=\) | \(\ds \pi\) | Cosine of Multiple of Pi |
\(\ds \map \sin {\map \arg {-3} }\) | \(=\) | \(\ds \dfrac 0 3\) | Definition of Argument of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg {-3}\) | \(=\) | \(\ds 0 \text { or } \pi\) | Sine of Multiple of Pi |
Hence:
- $\map \arg {-3} = \pi$
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Example $\text{(ii)}$