Argument of Complex Number/Examples/-3

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 $\text{(ii)}$