Complex Numbers Complete SAT Study Guide

What this topic tests

Add, multiply, and simplify complex number expressions. The College Board groups this topic inside the Additional Topics in Math content domain. Across a full SAT Math section, you can expect roughly 3–6 questions touching this topic, distributed across the easy, medium, and hard difficulty tiers.

Additional Topics in Math is the smallest official SAT Math domain by raw question count, but it carries outsized weight because the questions are concentrated at the harder end of each section. You will see roughly six of these per test, and they tend to separate students aiming for a 750 from students aiming for an 800. The domain covers right triangle trigonometry, circle theorems, volume formulas, complex number arithmetic, and the geometry of lines in the coordinate plane. Most of the formulas you need are listed at the start of the math section — but the test rewards students who have memorized them anyway, because looking them up costs precious seconds. ScoreReady's Additional Topics drills focus on the application patterns that appear most often: SOH-CAH-TOA on real triangles, arc length and sector area from radians, equation of a circle in standard form, parallel and perpendicular slopes, and i-squared simplifications. Every worked solution draws or describes the figure explicitly, because half the difficulty in geometry questions disappears the moment you re-sketch the figure on your scratch paper.

Sub-skills inside Complex Numbers

ScoreReady breaks this topic into four distinct sub-skills, each of which the College Board tests with its own characteristic question patterns. Mastering each sub-skill in isolation is faster than trying to master the whole topic at once.

Powers of i

The imaginary unit i is defined by i² = −1. The powers of i cycle with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then the cycle repeats. To compute i^n for any positive integer n, divide n by 4 and look up the remainder in the cycle: remainder 1 gives i, remainder 2 gives −1, remainder 3 gives −i, remainder 0 gives 1. SAT questions on powers of i are quick once you recognize the four-step cycle.

Adding and Subtracting Complex Numbers

To add or subtract complex numbers, combine the real parts and the imaginary parts separately. (a + bi) + (c + di) = (a + c) + (b + d)i. The arithmetic is straightforward; the only common error is mishandling the sign when subtracting, particularly when one of the imaginary parts is itself negative. Distribute the subtraction sign explicitly to both the real and imaginary parts of the second complex number.

Multiplying Complex Numbers

To multiply complex numbers, FOIL as you would for binomials, then replace i² with −1 and combine real and imaginary parts. (a + bi)(c + di) = ac + adi + bci + bdi², which simplifies to (ac − bd) + (ad + bc)i. The wrong-answer choices on multiplication problems usually correspond to forgetting the i² substitution or losing track of one of the four FOIL terms.

Complex Conjugates and Division

The complex conjugate of a + bi is a − bi. Multiplying a complex number by its conjugate produces a real number: (a + bi)(a − bi) = a² + b². To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator; this clears the imaginary part from the denominator and produces a quotient in standard a + bi form. The College Board tests this pattern on the harder Additional Topics questions.

Score-band drills

Once you have read through the sub-skills, drill the questions filtered to your current score band. The four bands below correspond to the four roughly-equal scoring ranges on the SAT Math section.

Key formulas

i² = −1
i cycle: i, −1, −i, 1, repeat
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Conjugate of a + bi is a − bi

For longer worked examples that walk through every formula on this list, see the formula reference page.

Common pitfalls

Forgetting to replace i² with −1 after multiplying
Mishandling signs when subtracting complex numbers
Computing (a + bi)² as a² + b²i² without the cross term
Multiplying by the wrong conjugate when dividing complex numbers

Each of these pitfalls maps to a wrong-answer choice the College Board reliably includes on questions in this topic. Read the common pitfalls walkthrough for a worked example of each one.

Suggested study order

Work the four sub-skill drills in the order they are listed above. The first sub-skill is the foundational one, and each subsequent sub-skill assumes fluency with the previous one. After you can clear all four sub-skill drills without notes, take the full topic question bank as a single timed sitting. Aim for at least 90% accuracy at a pace of one question per 75 seconds.