The conjugate of a posh quantity a + bi is the complicated quantity a – bi or a + -bi. The true a part of the complicated quantity a + bi is a and the imaginary half is b. 

a + bi and a – bi are referred to as complicated conjugates.

Examples displaying how one can discover the complicated conjugate of a posh quantity

Instance #1

Discover the complicated conjugate of 5 + 9i

9 is the imaginary half. The alternative of 9 is -9.

The complicated conjugate of 5 + 9i is 5 + -9i or 5 – 9i

Instance #2

Discover the complicated conjugate of -10 – 3i

-10 – 3i = -10 + -3i.

-3 is the imaginary half. The alternative of -3 is 3.

The complicated conjugate of -10 – 3i is -10 + 3i

Instance #3

Discover the complicated conjugate of i

i = 1i.

1 is the imaginary half. The alternative of 1 is -1.

The complicated conjugate of i is -i

Why do we want the conjugate of a posh quantity?

Allow us to see what’s going to occur once we multiply a posh quantity by its complicated conjugate.

(a + bi)(a – bi) = a² – abi + abi – b²i²

(a + bi)(a – bi) = a² – b²i²

(a + bi)(a – bi) = a²  – b²(-1)

(a + bi)(a – bi) = a² + b²

Since i just isn’t right here anymore, a² + b² is only a actual quantity

Instance

(4 + 3i)(4 – 3i) = 4² + 3²

(4 + 3i)(4 – 3i) = 16 + 9

(4 + 3i)(4 – 3i) = 25

This property of the complicated conjugate will be very helpful if you find yourself attempting to simplify rational expressions or if you find yourself attempting to eliminate the “i” within the denominator of a rational expression.

For instance, simplify 5i / (2 – i).

Multiply the numerator and the denominator of 5i / (2 – i) by the conjugate of two – i

The conjugate of two – i is 2 + i.

[5i(2 + i)] / (2 – i)(2 + i) = (10i + 5i²) / (2² + 1²)

[5i(2 + i)] / (2 – i)(2 + i) = [10i + 5(-1)] / ( 4 + 1)

[5i(2 + i)] / (2 – i)(2 + i) = (10i – 5) / 5

[5i(2 + i)] / (2 – i)(2 + i) = 2i – 1

Subsequently, 5i / (2 – i) = 2i – 1