Complex Numbers on the SAT: i, Operations, and Conjugates

Complex numbers show up a few times on every SAT. The questions aren't hard once you know the rules — and there are only a handful of rules to learn.

What Is ?

is defined as:

This means:

That single fact is the key to almost every complex number question.

Powers of

The powers of follow a cycle of 4:

The pattern repeats:

Example 1: What is ?

Divide 23 by 4:

So

Adding and Subtracting Complex Numbers

Combine real parts with real parts, imaginary parts with imaginary parts.

Example 2:

Example 3:

Multiplying Complex Numbers

Use FOIL, then replace with .

Example 4:

Dividing: Using the Conjugate

To divide complex numbers, multiply top and bottom by the conjugate of the denominator.

The conjugate of is .

Example 5: Simplify

Multiply by :

The Product of Conjugates

(always a real number)

Example 6:

Practice Problems

Problem 1: Simplify

Problem 2: Multiply

Problem 3: Simplify

Key Takeaways

• is the most important rule
• Powers of cycle every 4:
• Add/subtract: combine like terms (real with real, imaginary with imaginary)
• Multiply: use FOIL, then replace with
• Divide: multiply by the conjugate of the denominator

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