Argument of Complex Number/Examples/2i

Example of Argument of Complex Number

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

Proof

We have that:

\(\ds \cmod {2 i}\)

\(=\)

\(\ds \sqrt {0^2 + 2^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds 2\)

simplifying

Hence:

\(\ds \map \cos {\map \arg {2 i} }\)

\(=\)

\(\ds \dfrac 0 2\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {2 i}\)

\(=\)

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

Cosine of Half-Integer Multiple of Pi

\(\ds \map \sin {\map \arg {2 i} }\)

\(=\)

\(\ds \dfrac 2 2\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds 1\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {2 i}\)

\(=\)

\(\ds \dfrac \pi 2\)

Sine of Half-Integer Multiple of Pi

Hence:

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

$\blacksquare$

Sources

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