Question

Find the multiplicative inverse of the following complex number $1-i$
a) $-\dfrac{1}{2}-\dfrac{1}{2}i$
b) $\dfrac{1}{2}-\dfrac{1}{2}i$
c) $\dfrac{1}{2}+\dfrac{1}{2}i$
d) None of these

Answer 1

Here we have been given a complex number and we have to find the multiplicative inverse of it. Firstly we will write the formula for finding the multiplicative inverse then we will find the values used in the formula for the complex number given. Finally we will put the values in the formula and simplify it to get the desired answer.

Complete step-by-step solution:
We have to find the multiplicative inverse of,
$1-i$
Now as we know the multiplicative inverse formula is as follows:
${{z}^{-1}}=\dfrac{\overline{z}}{{{\left| z \right|}^{2}}}$…..$\left( 1 \right)$
Where, $z=ax+by$
$\overline{z}=$ Conjugate of $z$ and we find it by changing the sign of the complex number which means
$\overline{z}=ax-by$
$\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$ (Modulus)
So we have,
$z=1-i$….$\left( 2 \right)$
So the conjugate of the above value will be,
$\overline{z}=\overline{\left( 1-i \right)}$
$\Rightarrow \overline{z}=1+i$…..$\left( 3 \right)$
Next we will find the modulus as follows:
On comparing equation (2) by $z=ax+by$ we get
$a=1$ and $b=-1$ so,
$\left| z \right|=\sqrt{{{1}^{2}}+{{\left( -1 \right)}^{2}}}$
$\Rightarrow \left| z \right|=\sqrt{2}$……$\left( 4 \right)$
Substitute the value from equation (2), (3) and (4) in equation (1) we get,
$\Rightarrow {{\left( 1-i \right)}^{-1}}=\dfrac{1+i}{{{\left( \sqrt{2} \right)}^{2}}}$
$\Rightarrow {{\left( 1-i \right)}^{-1}}=\dfrac{1+i}{2}$
So we get,
${{\left( 1-i \right)}^{-1}}=\dfrac{1}{2}+\dfrac{1}{2}i$
Hence correct option is (C).

Note: Complex numbers are those numbers that can’t be located in the real line. They are usually found when the sign of a number inside the square root is negative. Multiplicative inverse is simply the reciprocal of the number given. When we multiply the number by its multiplicative inverse we get the answer as $1$. The non-negative square root of the coefficient of real and imaginary value of a complex number is known as its modulus. Conjugate is calculated by simply changing the sign of the imaginary value and keeping the rest thing the same.