Multiplying Complex Numbers & Conjugates

Q: Why does $i^2$ become $-1$?

Because $i = \sqrt{-1}$, squaring it gives $i^2 = -1$ by definition. That substitution is what turns a complex product back into $a + bi$ form.

Multiplying complex numbers uses the same FOIL you use on binomials, with one extra move: wherever $i^2$ appears, replace it with $-1$ .

Conjugates — pairs like $a + bi$ and $a - bi$ — are special: their product is always a real number. That fact is the key to dividing complex numbers later.

FOIL, then simplify $i^2$

Multiply $(a + bi)(c + di)$ by FOIL to get four terms. One term will contain $i^2$ ; replace it with $-1$ and combine.

The result rearranges into $a + bi$ form. The $i^2 = -1$ substitution is what makes complex multiplication different from ordinary binomial multiplication.

Conjugates give real products

The conjugate of $a + bi$ is $a - bi$ — same real part, opposite imaginary sign. Their product is $(a)^2 + (b)^2$ , a real number with no $i$ .

This happens because the middle imaginary terms cancel and $-b^2 i^2$ becomes $+b^2$ . Conjugates are the tool for clearing $i$ out of a denominator.

Worked examples

Example 1: a product

Multiply $(1 + 4i)(2 + i)$ .

FOIL

2 + i + 8i + 4i^2

Replace i-squared with -1

2 + 9i + 4(-1)

Combine

-2 + 9i

Answer: $-2 + 9i$

Example 2: conjugates

Multiply $(3 + 2i)(3 - 2i)$ .

FOIL, middle terms cancel

9 - 4i^2

Replace i-squared with -1

9 + 4

Simplify

13

Answer: $13$

Try one yourself

Common questions

Why does

i^2

become

-1

Because $i = \sqrt{-1}$ , squaring it gives $i^2 = -1$ by definition. That substitution is what turns a complex product back into $a + bi$ form.

What is a complex conjugate?

The number with the same real part and opposite imaginary sign: the conjugate of $a + bi$ is $a - bi$ . Their product is always real.

Where are conjugates used?

To divide complex numbers — multiply the top and bottom by the denominator's conjugate to clear $i$ from the denominator.

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Multiplying Complex Numbers & Conjugates

FOIL, then simplify $i^2$

Conjugates give real products

Worked examples

Example 1: a product

Example 2: conjugates

Try one yourself

Common questions

Keep learning

Adding & Subtracting Complex Numbers

The Imaginary Unit i & Powers of i

The Quadratic Formula over the Complex Numbers

Multiplying Complex Numbers & Conjugates

FOIL, then simplify i2i^2i2

Conjugates give real products

Worked examples

Example 1: a product

Example 2: conjugates

Try one yourself

Common questions

Keep learning

Adding & Subtracting Complex Numbers

The Imaginary Unit i & Powers of i

The Quadratic Formula over the Complex Numbers

FOIL, then simplify $i^2$