Before you read

Answer first — see what you already know.

What is the conjugate of $4 - 3 i$ ?

A
$- 4 - 3 i$
B
$- 4 + 3 i$
C
$4 + 3 i$
D
$3 - 4 i$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

What is $(2 + 5 i) (2 - 5 i)$ ?

A
$4 + 25$
B
$29$
C
$4 - 25 i^{2}$
D
$- 21$

Check your answer

Answer:

Theory

Complex Conjugates

The conjugate of $a + bi$ is $a - bi$ . You flip the sign of the imaginary part.

Key property: $(a + bi)(a - bi) = a^2 + b^2$ (always a real number!)

This is because: $a^2 - abi + abi - b^2i^2 = a^2 + b^2$ .

The conjugate is used to eliminate $i$ from denominators.

$(3+2i)/(1-i)$
$(3+2i)(1+i) / (1-i)(1+i)$ multiply by conjugate
$(3+3i+2i+2i²) / (1-i²)$
$(3+5i-2) / (1+1)$ i $^{2}$ = -1
$(1+5i) / 2$
$1/2 + 5i/2$

Dividing complex numbers using the conjugate

Example 1

Find the conjugate of $3 + 7i$ and multiply them.

Conjugate: $3 - 7i$

$(3+7i)(3-7i) = 9 + 49 = 58$

Conjugate =

3 - 7i

, Product =

58

Theory

Dividing Complex Numbers

To divide $\frac{a + bi}{c + di}$ , multiply top and bottom by the conjugate of the denominator:
$\frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{(a+bi)(c-di)}{c^2 + d^2}$

The denominator becomes a real number, and you simplify.

Example 1

Simplify $\frac{5 + i}{2 - 3i}$

Multiply by conjugate: $\frac{(5+i)(2+3i)}{(2-3i)(2+3i)}$

Numerator: $10 + 15i + 2i + 3i^2 = 10 + 17i - 3 = 7 + 17i$

Denominator: $4 + 9 = 13$

Result: $\frac{7}{13} + \frac{17}{13}i$

\frac{7}{13} + \frac{17}{13}i

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