Question

Find the multiplicative inverse of the complex number $-i$.

Answer 1

Hint: The multiplicative inverse of any number, real or complex, is defined as the number which gives a product 1 when multiplied by the original number. The square root of -1 is an imaginary number called iota. A complex number has a real and an imaginary part and is of the form $a + ib$. The multiplicative inverse of a complex number z is $\dfrac{1}{{\text{z}}}$

Complete step-by-step answer:
We have to find the multiplicative inverse of the complex number $-i$. From the properties of complex number, we can write the multiplicative inverse as-
Let $z = -i$ so,
$\begin{align}
  &\dfrac{1}{{\text{z}}} = \dfrac{1}{{ - {\text{i}}}} = - \dfrac{1}{{\text{i}}} \\
  &Multiplying\;and\;dividing\;by\;{\text{i}}\;we\;get - \\
  &\dfrac{1}{{\text{z}}} = - \dfrac{{\text{i}}}{{{{\text{i}}^2}}} \\
  &We\;know\;that\;{{\text{i}}^2} = - 1 \\
  &\dfrac{1}{{\text{z}}} = - \dfrac{{\text{i}}}{{ - 1}} = {\text{i}} \\
\end{align} $

Hence the multiplicative inverse of $-i$ is $i$. This is the required answer.

Note: A common mistake is that students leave their answer without further simplification. Whenever we have complex numbers in fractional form, we have to ensure that the denominator is a real number. Hence, we should always factorize the denominator so that it becomes a purely real number. Here, we multiplied and divided by iota to make the denominator -1, which is a real number.