Question

Find the modulus and amplitude of the complex number $-5i.$

Answer 1

Hint: For the above question we will have to know about the complex number. A complex number is a number that can be expressed in the form of \[a+bi\], where a and b are real numbers, and “ i ” is a solution of the equation . Because no real number satisfies this equation,” i ” is called an imaginary number.

Complete step-by-step solution -
If we have a complex number \[a+bi\] where a and b are real numbers, then the non- negative square root of (\[{{a}^{2}}+{{b}^{2}}\]) is known as modulus or absolute value of the complex number and the tangent value of the ratio of \[\left| \dfrac{b}{a} \right|\] is known as the amplitude of the complex number where the values must lie between zero and pi for the principal value of the amplitude.
Mathematically, it is shown as belows:
\[\begin{align}
  & \text{modulus=}\sqrt{{{a}^{2}}+{{b}^{2}}} \\
 & \text{amplitude = }\theta \text{ =}{{\tan }^{-1}}\left| \dfrac{b}{a} \right|\text{ where 0}\le \theta \le \pi \text{ for principal value}\text{.} \\
\end{align}\]

In the above question we have been given a =0 and b = -5.
So, the value of modulus and amplitude are as follows:
\[\begin{align}
  & \bmod ulus=\sqrt{{{0}^{2}}+{{(-5)}^{2}}} \\
 & \text{ = }\sqrt{25} \\
 & \text{ = 5} \\
 & \text{amplitude=ta}{{\text{n}}^{-1}}\left| \dfrac{-5}{0} \right| \\
 & \text{ = ta}{{\text{n}}^{-1}}(\infty ) \\
 & \text{ = }\dfrac{\pi }{2} \\
\end{align}\]
Therefore, the value of the modulus and amplitude for the given complex number are 5 and \[\dfrac{\pi }{2}\] respectively.

Note: Just remember the formulae of the modulus and amplitude of a complex number as it will help you a lot in these types of questions. Sometimes amplitude is called as an argument so we need to keep this in mind . for the argument and amplitude concept to be the same.