Reciprocal of a Complex Number

How to find the reciprocal of a complex number?

Let z = x + iy be a non-zero complex number. Then

$\frac{1}{z}$

= $\frac{1}{x + iy}$

= $\frac{1}{x + iy}$ × $\frac{x - iy}{x - iy}$, [Multiplying numerator and denominator by conjugate of denominator i.e., Multiply both numerator and denominator by conjugate of x + iy]

= $\frac{x - iy}{x^{2} - i^{2}y^{2}}$

= $\frac{x - iy}{x^{2} + y^{2}}$

=  $\frac{x}{x^{2} + y^{2}}$ +  $\frac{i(-y)}{x^{2} + y^{2}}$

Clearly, $\frac{1}{z}$ is equal to the multiplicative inverse of z. Also,

$\frac{1}{z}$ = $\frac{x - iy}{x^{2} + y^{2}}$ = $\frac{\overline{z}}{|z|^{2}}$

Therefore, the multiplicative inverse of a non-zero complex z is equal to its reciprocal and is represent as

$\frac{Re(z)}{|z|^{2}}$ + i$\frac{(-Im(z))}{|z|^{2}}$= $\frac{\overline{z}}{|z|^{2}}$

Solved examples on reciprocal of a complex number:

1. If a complex number z = 2 + 3i, then find the reciprocal of z? Give your answer in a + ib form.

Solution:

Given z = 2 + 3i

Then, $\overline{z}$ = 2 - 3i

And |z| = $\sqrt{x^{2} + y^{2}}$

= $\sqrt{2^{2} + (-3)^{2}}$

= $\sqrt{4 + 9}$

= $\sqrt{13}$

Now, |z|$^{2}$ = 13

Therefore, $\frac{1}{z}$ = $\frac{\overline{z}}{|z|^{2}}$ = $\frac{2 - 3i}{13}$ = $\frac{2}{13}$ + (-$\frac{3}{13}$)i, which is the required a + ib form.

2. Find the reciprocal of the complex number z = -1 + 2i. Give your answer in a + ib form.

Solution:

Given z = -1 + 2i

Then, $\overline{z}$ = -1 - 2i

And |z| = $\sqrt{x^{2} + y^{2}}$

= $\sqrt{(-1)^{2} + 2^{2}}$

= $\sqrt{1 + 4}$

= $\sqrt{5}$

Now, |z|$^{2}$= 5

Therefore, $\frac{1}{z}$ = $\frac{\overline{z}}{|z|^{2}}$ = $\frac{-1 - 2i}{5}$ = (-$\frac{1}{5}$) + (-$\frac{2}{5}$)i, which is the required a + ib form.

3. Find the reciprocal of the complex number z = i. Give your answer in a + ib form.

Solution:

Given z = i

Then, $\overline{z}$ = -i

And |z| = $\sqrt{x^{2} + y^{2}}$

= $\sqrt{0^{2} + 1^{2}}$

= $\sqrt{0 + 1}$

= $\sqrt{1}$

= 1

Now, |z|$^{2}$= 1

Therefore, $\frac{1}{z}$ = $\frac{\overline{z}}{|z|^{2}}$ = $\frac{-i}{1}$ = -i = 0 + (-i), which is the required a + ib form.

Note: The reciprocal of i is its own conjugate - i.

11 and 12 Grade Math 

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