Conjugates of Complex Numbers

Every complex number $z = a + bi$ has a conjugate — it's written as $\overline{z} = a - bi$ .

You just flip the sign of the imaginary part. That's it! Two complex numbers are the conjugates of each other if their real part is the same and their imaginary parts are additive inverses of each other.

For example, consider $z_1 = 3 - 7i$ and $z_2 = 3 + 7i$ . These two complex numbers are conjugates of each other. Usually, for a complex number, we denote its conjugate with an overline.

Why is it useful? Multiplying a complex number by its conjugate gives a real number which we will prove in this lesson. This helps simplify expressions and solve equations involving complex numbers.

Geometric hint: On the complex plane, the conjugate is a reflection of the number across the real axis which has its own utility. We will look at the graphical representation in the future lessons.

Activity: Match The Columns!

Reorder the Conjugates to match them with the correct Complex Numbers.

Complex Numbers

Conjugates

$3 + 4i$

$-2 + 5i$

$7 - 6i$

$-3 - 2i$

$5i$

$4$

Property 1: Sum of A complex number and its conjugate is a real number

Prove that the sum of a complex number and its conjugate is always real

SKIP

Let $z = a + bi$

SKIP

$z + \overline{z} = \text{Real}$

$L.H.S. = z + \overline{z}$

$L.H.S = (a + bi) + (a - bi)$

$= a + bi + a - bi$

$= (a + a) + (bi - bi)$

$= 2a$

$2a$ is always real

SKIP

$\therefore z + \overline{z}$ is always real.

Property 2: Product of A complex number and its conjugate is a real number

Prove that the product of a complex number and its conjugate is always real

SKIP

Let $z = a + bi$

SKIP

$z \cdot \overline{z} = \text{Real}$

$L.H.S. = z \cdot \overline{z}$

$L.H.S = (a + bi)(a - bi)$

$= (a)^2 - (bi)^2$

$= a^2 - b^2i^2$

$= a^2 - b^2(-1)$

$= a^2 + b^2$

$a^2 + b^2$ is always real

SKIP

$\therefore z \cdot \overline{z}$ is always real.

End of Lesson