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Complex numbers look like the scariest thing in the SAT Math section and behave like the tamest. Every question that uses the imaginary unit runs on one fact: i^2 = -1. Treat i like an ordinary variable while you distribute and combine, then use that single rule to clean up at the end, and these turn into some of the most reliable points on the test.
A complex number has the form a + bi, where a is the real part and b is the imaginary part. The SAT asks you to add them, subtract them, multiply them, and now and then simplify a high power of i. None of it needs new algebra. It needs the algebra you already have, plus the discipline to replace every i^2 with -1.
Key takeaways
- The only rule to memorize: i = \sqrt{-1}, so i^2 = -1. Everything else follows from it.
- Add and subtract by part: combine real with real and imaginary with imaginary, exactly like combining like terms.
- Multiply with FOIL, then swap every i^2 for -1 and recombine. That last swap is where the point is won or lost.
- Powers of i cycle every four: i, -1, -i, 1. Divide the exponent by 4 and read the remainder.
What is the imaginary unit i?
The imaginary unit i is defined as \sqrt{-1}, which means i^2 = -1. It exists because no real number squares to a negative, so the square root of a negative needs a symbol of its own. That is the whole definition, and every complex-number question on the SAT is a consequence of it. These are real-number exponent rules with one extra fact bolted on, so if squaring and multiplying powers already feel automatic, this topic is a short add-on rather than a new unit.
i = \sqrt{-1} and i^2 = -1. Write that at the top of your scratch work before you touch a complex-number question, because the entire problem is usually just finding the i^2 and turning it into -1.
Adding and subtracting complex numbers
Adding and subtracting is the same move as combining like terms: real parts group with real parts, imaginary parts group with imaginary parts, and the two groups never mix. So (5 + 3i) + (2 - 7i) = 7 - 4i. For subtraction, the only trap is the one you already know from algebra, the dropped negative: (5 + 3i) - (2 - 7i) = 5 - 2 + 3i + 7i = 3 + 10i, because the minus sign hits both the 2 and the -7i inside the parentheses.
Multiplying complex numbers
Multiplication is FOIL, exactly as with any two binomials, followed by the one swap. Expand the product, find the term with i^2, replace that i^2 with -1, and then collect the real and imaginary parts. Skipping the swap is the single most common error, because it leaves an i^2 sitting in the answer as though it were just another variable.
If i = \sqrt{-1}, which expression is equal to (3 + 2i)(1 - 4i)?
The powers of i cycle
Raise i to higher powers and the values repeat in a loop of four. Each step is just the previous value times i, simplified with i^2 = -1.
| Power | Value |
|---|---|
| i^1 | i |
| i^2 | -1 |
| i^3 | -i |
| i^4 | 1 |
For a large exponent, divide it by 4 and keep only the remainder. A remainder of 1 gives i, a remainder of 2 gives -1, a remainder of 3 gives -i, and a remainder of 0 lands on i^4 = 1. You never have to multiply i by itself thirty times; you only have to divide the exponent by four.
What is the value of i^{38}?
Where complex numbers fit in SAT Math
Complex numbers live in the Advanced Math part of the section, and they are a small, well-defined slice of it. The broader SAT Advanced Math guide covers the whole nonlinear landscape, choosing forms that reveal roots, vertices, and factors; this page is only the i rules within it. You also meet i indirectly on quadratics and parabolas: when a quadratic has no real solutions, its graph never crosses the x-axis, and the roots are a complex pair. To drill the mechanics under real timing, work the nonlinear question bank.
The trap
Treating i as a mystery symbol, or forgetting that i^2 = -1, so an i^2 term survives into the answer and the real and imaginary parts never get combined.
The move
Do the ordinary algebra first, then replace every i^2 with -1 and collect the real part and the imaginary part separately into the form a + bi.
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Practice the Advanced Math that trips people up →Frequently asked questions
- What is i on the SAT?
- i is the imaginary unit, defined as the square root of negative one, so i squared equals negative one. It lets you work with the square roots of negative numbers, which have no real-number value. Every SAT complex-number question is built on that single definition.
- Does the digital SAT test complex numbers?
- Yes. Complex numbers appear in the Advanced Math part of the Math section, usually as a question that asks you to add, subtract, multiply, or simplify an expression containing i. They are a small but learnable slice of the test, and the rules never change.
- How do you simplify powers of i?
- Powers of i repeat in a cycle of four: i, negative one, negative i, and one. Divide the exponent by four and use the remainder to pick which of the four values it equals. A remainder of zero lands on one.
- How do you multiply complex numbers?
- Use FOIL exactly as you would with binomials, then replace every i squared with negative one and combine the real parts and the imaginary parts separately. Write the final answer in the form a plus bi.
- Are complex numbers hard on the SAT?
- Not once you know the one rule. Most students find these questions easier than the hardest algebra, because the steps are mechanical: do ordinary algebra, swap i squared for negative one, and collect like terms. The main risk is a dropped sign.