How to Find the Conjugate of a Complex Number

Here you will learn how to find the conjugate of a complex number and properties of conjugate with examples.

Let’s begin –

How to Find the Conjugate of a Complex Number

Let z = a + ib be a complex number. Then the conjugate of z is denoted by $$\bar{z}$$ and is equal to a – ib.

Thus, z = a + ib $$\implies$$ $$\bar{z}$$ = a – ib

It follows from this definition that the conjugate of a complex number is obtained by replacing i by -i.

For Example : If z = 3 + 4i, then $$\bar{z}$$ = 3 – 4i.

Properties of Conjugate

If $$z$$, $$z_1$$, $$z_2$$ are complex numbers, then

(i) $$\bar{\bar{z}}$$ = z

(ii) z + $$\bar{z}$$ = 2 Re (z)

(iii) z – $$\bar{z}$$ = 2i Im (z)

(iv) z = $$\bar{z}$$ $$\iff$$ z is purely real

(v) z + $$\bar{z}$$ = 0 $$\implies$$ z is purely imaginary

(vi) z$$\bar{z}$$ = $$[Re (z)]^2$$ + $$[Im (z)]^2$$

(vii) $$\bar{z_1 + z_2}$$ =  $$\bar{z_1}$$ + $$\bar{z_2}$$

(viii) $$\bar{z_1 – z_2}$$ =  $$\bar{z_1}$$ – $$\bar{z_2}$$

(ix) $$\bar{z_1z_2}$$ =  $$\bar{z_1}$$ $$\bar{z_2}$$

(x) $$\bar{z_1\over z_2}$$ =  $$\bar{z_1}\over \bar{z_2}$$

Example : Multiply 3 – 2i by its conjugate.

Solution : The conjugate of 3 – 2i is 3 + 2i.

Hence, required product is = (3 – 2i)(3 + 2i) = $$9 – 4i^2$$ = 9 + 4 = 13

Example : Find the conjugate of $$1\over 3 + 4i$$.

Solution : Let z = $$1\over 3 + 4i$$. Then,

z = $$1\over 3 + 4i$$ $$\times$$ $$3 – 4i\over 3 – 4i$$ = $$3 – 4i\over 9 + 16$$ = $${3\over 25} – {4\over 25}i$$

$$\therefore$$ Conjugate of z is $$\bar{z}$$ = $${3\over 25} – {4\over 25}i$$.