Complex Numbers: Making Sense of the Square Root of −1

For a long time, "the square root of a negative number" was treated as impossible. Mathematicians fixed that by defining a new number, i, whose square is −1. That single definition opens up the complex numbers — and the good news for test-takers is that you already know how to work with them. A complex number a + bi behaves exactly like a binomial; the only new move is replacing i² with −1 whenever it appears.

This guide builds from the imaginary unit through the four operations and the cycle of powers, with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why Complex Numbers Matter

Complex numbers appear on the SAT and ACT in a handful of predictable ways: simplifying powers of i, multiplying two complex numbers, and recognizing the roots of a quadratic with a negative discriminant. They're Algebra II content beyond the SSAT, but the mechanics lean entirely on skills you built in factoring and binomial multiplication.

The Imaginary Unit i

The imaginary unit is defined by one equation:

This lets you simplify the square root of any negative number: √−16 = √16 · √−1 = 4i.

The Form a + bi

Every complex number can be written as a + bi, where a is the real part and b is the imaginary part. A pure real number (like 5) is just 5 + 0i; a pure imaginary number (like 3i) is 0 + 3i.

Adding & Subtracting

Treat i like a variable and combine like terms — reals with reals, imaginaries with imaginaries:

Multiplying

Multiply complex numbers with FOIL, exactly as you would two binomials — then replace any i² with −1 and simplify:

Powers of i

The powers of i repeat in a cycle of four:

Power	Value
i¹	i
i²	−1
i³	−i
i⁴	1

After i⁴ the pattern restarts. To simplify a high power, divide the exponent by 4 and use the remainder: i²³ has remainder 3 (since 23 = 20 + 3), so i²³ = i³ = −i.

Where You'll See This — Test by Test

No reference sheet covers this. The SAT and ACT test simplifying powers of i and multiplying complex numbers. It's Algebra II material beyond the SSAT.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

1. Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
2. Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
3. Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
4. Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.

And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.