Argument of Complex Number/Examples/3
Jump to navigation
Jump to search
Example of Argument of Complex Number
- $\arg 3 = 0$
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg 3\) | \(=\) | \(\ds 0\) | Cosine of Zero is One |
\(\ds \map \sin {\map \arg 3}\) | \(=\) | \(\ds \dfrac 0 1\) | 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:
- $\map \arg 3 = 0$
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Example $\text{(i)}$