Complex Conjugates

Complex Conjugates

How to rationalize the complex denominators by using the complex conjugates: definition, formula, examples, and their solutions.

Definition, Formula

The conjugate of [a + bi] is [a - bi]. The conjugate of [a - bi] is [a + bi]. (a + bi)(a - bi) = a^2 + b^2 is used to rationalize a complex denominator.

The conjugate of [a + bi] is [a - bi].
The conjugate of [a - bi] is [a + bi].

So [a + bi] and [a - bi]
are the conjugates of each other.

The product of these two conjugates is a2 + b2.

This formula is used
to rationalize a complex denominator.

Example 1: Simplify 2/(1 + 3i)

Simplify the given expression. 2/(1 + 3i)

The conjugate of [1 + 3i] is [1 - 3i].

So, to rationalize the denominator,
multiply (1 - 3i)/(1 - 3i).

(1 + 3i)(1 - 3i) = 12 + 32

The denominator's middle sign is (+), not (-).

2(1 - 3i) = 2 - 6i

12 + 32 = 1 + 9

1 + 9 = 10

Divide the numerator by 2
and reduce the denominator 10 to 5.

Then (given) = (1 - 3i)/5.

Example 2: Simplify (1 - 7i)/(2 - i)

Simplify the given expression. (1 - 7i)/(2 - i)

The conjugate of [2 - i] is [2 + i].

So, to rationalize the denominator,
multiply (2 + i)/(2 + i).

(2 - i)(2 + i) = 22 + 12

The denominator's middle sign is (+), not (-).

(1 - 7i)(2 + i) = 2 + i - 14i - 7⋅(-1)

Multiplying complex numbers

22 + 12 = 4 + 1

+i - 14i = -13i
-7⋅(-1) = +7

4 + 1 = 5

2 + 7 = 9

So (given) = (9 - 13i)/5.

Example 3: Multiplicative Inverse of 7 + 2i

Find the multiplicative inverse of the given expression. 7 + 2i

The multiplicative inverse (= reciprocal) of [7 + 2i] is
1/(7 + 2i).

So it says to simplify 1/(7 + 2i).

The conjugate of [7 + 2i] is [7 - 2i].

So, to rationalize the denominator,
multiply (7 - 2i)/(7 - 2i).

(7 + 2i)(7 - 2i) = 72 + 22

The denominator's middle sign is (+), not (-).

72 + 22 = 49 + 4

49 + 4 = 53

So (given) = (7 - 2i)/53.