Argument of Complex Number/Examples/-i

Example of Argument of Complex Number

$\map \arg {-i} = -\dfrac \pi 2$

Proof

We have that:

\(\ds \cmod {-i} = 1\)

\(=\)

\(\ds \)

Example of Complex Modulus: $-i$

Hence:

\(\ds \map \cos {\map \arg {-i} }\)

\(=\)

\(\ds \dfrac 0 1\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-i}\)

\(=\)

\(\ds \pm \dfrac \pi 2\)

Cosine of Half-Integer Multiple of Pi

\(\ds \map \sin {\map \arg {-i} }\)

\(=\)

\(\ds \dfrac {-1} 1\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds -1\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-i}\)

\(=\)

\(\ds -\dfrac \pi 2\)

Sine of Half-Integer Multiple of Pi

Hence:

$\map \arg {-i} = -\dfrac \pi 2$

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Example $\text{(vi)}$