Concept 7

Work with complex numbers (basic operations, conjugates, magnitude where relevant).

Use i^2 = -1, combine real and imaginary parts, and rationalize denominators with conjugates.

Core Idea

Use $𝑖 2 = - 1$ , combine real and imaginary parts separately, and multiply by the conjugate when a denominator contains $𝑖$ .

Understanding

Rule: Complex number questions on the ACT are usually algebra questions with one extra rule: $𝑖 2 = - 1$ .

Addition and subtraction work by combining like parts. For multiplication or division, expand carefully and simplify powers of $𝑖$ . If a denominator contains a complex number, use the conjugate to make the denominator real.

Question

Worked Example

What is $3 + 2 𝑖 1 - 𝑖$ written in the form $𝑎 + 𝑏 𝑖$ ?

Select an answer to see the explanation