What is the amplitude of a complex number?

Hint: In this type of question we have to use the concept of complex numbers. We know that a complex number is generally represented as \[z=x+iy\] where \[x\] is known as the real part and \[y\] is known as the imaginary part of the complex number \[z\]. Also we know that the value of \[i=\sqrt{-1}\] and hence the value of \[{{i}^{2}}=-1\].


Complete step-by-step solution:
Now, we have to find the amplitude of a complex number. For this let us assume that, a complex number \[z=x+iy\] where \[x > 0,y > 0\] are real numbers and \[i=\sqrt{-1}\].
Let us substitute \[x=r\cos \theta \] and \[y=r\sin \theta \] to convert \[z=x+iy\] in polar form where \[r\] is the modulus of \[z\] and \[\theta \] is the amplitude of \[z\].
Hence, we get the polar form of complex number \[z=x+iy\] as
\[\Rightarrow z=r\cos \theta +ir\sin \theta =r\left( \cos +i\sin \theta \right)\]
The formulas to find the values of modulus and amplitude that means \[r\] and \[\theta \]
\[\begin{align}
  & \Rightarrow r=\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}} \\
 & \Rightarrow \theta ={{\tan }^{-1}}\left( \dfrac{y}{x} \right) \\
\end{align}\]
Now, from the equations \[x=r\cos \theta \] and \[y=r\sin \theta \] we can say that these equations are true for infinite values of \[\theta \] and that value of \[\theta \] is nothing but the value of amplitude of \[z\].
Also as we know that, \[\cos \left( 2n\pi +\theta \right)=\cos \theta \] and \[\sin \left( 2n\pi +\theta \right)=\sin \theta \] where \[n\in \mathbb{Z}\] i.e. set of integers, hence we can say that, \[Amp\left( z \right)=2n\pi +\theta \] where \[-\pi <\theta \le \pi \].
Thus, for any unique value of \[\theta \] that lies in the interval \[-\pi <\theta \le \pi \] and satisfies the equations \[x=r\cos \theta \] and \[y=r\sin \theta \] is known as the amplitude of \[z\] and it is denoted by \[Amp\left( z \right)\].

Note: In this type of question students have to consider the polar form of a complex number to obtain its amplitude. Also students have to take care that they have to find amplitude and not the argument of complex numbers. Students have to note that there is a slight difference in amplitude and argument, the range of amplitude is \[\left( -\pi ,\pi \right]\] and the range of argument is \[\left[ 0,2\pi \right)\]. Also students have to note that they can find the amplitude of every complex number except \[0\], the amplitude of \[0\] does not exist.