# Argument of Complex Number/Examples/3

## 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:

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

$\blacksquare$