Complex numbers on the ACT boil down to i = √(-1), so i² = -1, and every complex number has the form a + bi where a is real and b is imaginary. Add, subtract, multiply, and divide using a handful of rules—nothing beyond early algebra.
The imaginary unit
i is defined so that i² = -1. That single fact drives every complex-number problem. Powers of i cycle: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, then the pattern repeats. For i²⁰, divide 20 by 4—the remainder 0 means i²⁰ = 1.
Adding and subtracting
Combine real parts and imaginary parts separately. (3 + 2i) + (5 - 4i) = 8 - 2i. Treat i like a variable you cannot merge with plain numbers—same idea as combining like terms.
Multiplying with FOIL
Use FOIL, then replace every i² with -1. Example: (2 + i)(3 - 2i) = 6 - 4i + 3i - 2i² = 6 - i + 2 = 8 - i. If exponents show up inside the expression, refresh ACT exponent rules first.
Dividing with conjugates
To divide by a complex number, multiply top and bottom by the conjugate of the denominator—the same real part, opposite imaginary part. For (1 + i) / (2 - i), multiply by (2 + i) / (2 + i). The denominator becomes a real number (2² + 1² = 5), and you simplify from there.
How much to study
Complex numbers appear rarely—often zero or one question per test. They sit with the hardest ACT Math topics, so learn the rules once and do not over-invest. For where they fit in the bigger picture, see ACT Math topics.
Start practicing
Start with a free diagnostic, then drill your weak spots with 15-question quizzes and track how you're doing across Reading, English, and Math. Compare plans whenever you're ready to go further.
This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.