Question

What is the amplitude of a complex number?

Answer 1

Meaning of amplitude:

The polar representation of a complex number $z=x+iy$ is $z=r(\cos \theta +i\sin \theta )$.

Here, $r$ is the modulus and $\theta$ is the amplitude or argument of a complex number.

Formula: $\mathit{\theta }\mathbf{=}\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\left(\left|\frac{y}{x}\right|\right)$ and $\mathbf{-}\mathbf{\pi }\mathbf{<}\mathit{a}\mathit{m}\mathit{p}\mathbf{\left(}\mathbf{z}\mathbf{\right)}\mathbf{\le }\mathbf{\pi }$

If $z=x+iy$ belongs to the first quadrant, then $\mathit{a}\mathit{m}\mathit{p}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathit{\theta }$.

If $z=x+iy$ belongs to the second quadrant, then $\mathit{a}\mathit{m}\mathit{p}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{\pi }\mathbf{-}\mathit{\theta }$

If $z=x+iy$ belongs to the third quadrant, then $\mathit{a}\mathit{m}\mathit{p}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{-}\mathbf{}\mathbf{(}\mathit{\pi }\mathbf{}\mathbf{-}\mathbf{}\mathit{\theta }\mathbf{)}\mathbf{}$

If $z=x+iy$ belongs to the fourth quadrant, then $\mathit{a}\mathit{m}\mathit{p}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{-}\mathit{\theta }\mathbf{}$

Example: The amplitude of the complex number $z=1-\sqrt{3}i$ is,

$\theta ={\tan }^{-1}\left(\left|\frac{-\sqrt{3}}{1}\right|\right)$ and complex number lies in fourth quadrant so $\mathbf{\text{amp(z) =-θ}}$

$\Rightarrow$ $\theta ={\tan }^{-1}\left(\sqrt{3}\right)$

$\Rightarrow$ $\theta =\frac{\mathrm{\pi }}{3}$

$\Rightarrow$ $amp\left(z\right)=-\frac{\mathrm{\pi }}{3}$