# 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.