Question

Find the multiplicative inverse of the complex number 2-3i.
   (a) $2+3i$
   (b) $\dfrac{2+3i}{13}$
   (c) $-2-3i$
   (d) $3i$

Answer 1

Hint: Think of a complex number which, when multiplied by 2-3i, gives the product 1. Make sure that you convert your answer to suitable form matching with the option. Before moving to the question, let us talk about the multiplicative inverse. The multiplicative inverse of a number is defined as the reciprocal of the number itself. In other words, an expression is said to be the multiplicative inverse of the other if on multiplying, their product is 1.

Complete step-by-step answer:
For example: if $xy=1$ , then, x and y are the multiplicative inverses of each other.
Now, moving to the solution to the question mentioned above.
We are asked to find the multiplicative inverse of 2 – 3i. Let the given complex number be equal to ‘m’.
$\therefore m=2-3i$
Let the multiplicative inverse of m be ‘n.’
Now, according to the definition of the multiplicative inverse:
$mn=1$
$\Rightarrow n=\dfrac{1}{m}$
$\Rightarrow n=\dfrac{1}{2-3i}$
Now, we will multiply and divide the right-hand side of the by (2 + 3i). So, we get:
$n=\dfrac{1}{2-3i}\times \dfrac{2+3i}{2+3i}$
$n=\dfrac{2+3i}{\left( 2+3i \right)\left( 2-3i \right)}$
We know that ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ . Using this formula in the denominator of n, we get:
$n=\dfrac{2+3i}{{{2}^{2}}-{{\left( 3i \right)}^{2}}}$
$\Rightarrow n=\dfrac{2+3i}{4-9{{i}^{2}}}$
We know; $i=\sqrt{-1}$
$\therefore {{i}^{2}}=-1$
So, we will substitute $\therefore {{i}^{2}}=-1$ in $n=\dfrac{2+3i}{4-9{{i}^{2}}}$ . So, we get:
$n=\dfrac{2+3i}{4-9(-1)}$
$\Rightarrow n=\dfrac{2+3i}{4+9}$
$\Rightarrow n=\dfrac{2+3i}{13}$
So, the multiplicative inverse of the complex number $2-3i$ is $\dfrac{2+3i}{13}$ .
Therefore, the answer is option (b).

Note: The inverse, in general, refers to multiplicative inverse, but there is a possibility of additive inverse being asked. An expression is said to be the additive inverse of other if both expressions add up, giving 0 as the sum. For example: if $x+y=0$ , then, x and y are the additive inverses of each other. Also, multiplicative inverse and reciprocal are the same in case of numbers. Generally, students make a mistake in the expression of ${{2}^{2}}-{{\left( 3i \right)}^{2}}$ . They calculate the value as $4-9=-5$ , which is wrong.