Argument of Complex Number/Examples/3

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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