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:

\(\displaystyle \left\lvert{3}\right\rvert\) \(=\) \(\displaystyle \sqrt{3^2 + 0^2}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle 3\)


Hence:

\(\displaystyle \cos \left({\arg \left({3}\right)}\right)\) \(=\) \(\displaystyle \dfrac 3 3\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle 1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \arg \left({3}\right)\) \(=\) \(\displaystyle 0\) Cosine of Zero is One


\(\displaystyle \sin \left({\arg \left({3}\right)}\right)\) \(=\) \(\displaystyle \dfrac 0 1\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \arg \left({3}\right)\) \(=\) \(\displaystyle 0 \text { or } \pi\) Sine of Multiple of Pi


Hence:

$\arg \left({3}\right) = 0$

$\blacksquare$


Sources