If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## SAT (Fall 2023)

### Course: SAT (Fall 2023)>Unit 6

Lesson 6: Additional Topics in Math: lessons by skill

# Complex numbers | Lesson

## What are complex numbers, and how frequently do they appear on the test?

Note: On your official SAT, you'll likely see at most 1 question that tests your knowledge of complex numbers. Make sure you understand the more frequently-tested skills on the SAT before you spend time practicing this skill.

The square roots of negative numbers are not real numbers, but instead imaginary numbers. We use the notation i to indicate square root of, minus, 1, end square root, and all imaginary numbers are real number multiples of i.
A complex number, then, has the form a, plus, b, i, where a is the real component and b, i is the imaginary component.
In this lesson, we'll learn to add, subtract, multiply, and divide complex numbers.
This lesson builds upon an understanding of Polynomial operations, which has its own lesson.
For more comprehensive coverage on the topic of complex numbers that goes beyond what you need to know for the SAT, check out our Algebra 2 unit on complex numbers.
You can learn anything. Let's do this!

## How do I add and subtract complex numbers?

### Adding and subtracting complex numbers

We can treat complex numbers like
when adding and subtracting them. This means we need to combine like terms (real with real, imaginary with imaginary) and write the result in the form a, plus, b, i.

Example: What is the sum of 3, plus, 2, i and 5, plus, 7, i ?

When subtracting two complex numbers, we need to be careful with negative signs just like we would when subtracting two polynomials.

Example: If a, equals, 6, minus, 3, i and b, equals, minus, 2, minus, 9, i, what is a, minus, b ?

### Try it!

try: add and subtract two complex numbers
\begin{aligned} &a=7-3i \\\\ &b=2+i \end{aligned}
Two complex numbers, a and b, are shown above.
What is the sum of a and b ? Enter your answer as a complex number.
What is b, minus, a ? Enter your answer as a complex number.

## How do I multiply and divide complex numbers?

Note: Neither multiplying nor dividing complex numbers has appeared on more recent SAT practice tests.

### Multiplying and dividing complex numbers

Both multiplying and dividing complex numbers make use of the fact that since i, equals, square root of, minus, 1, end square root, i, squared, equals, minus, 1.
To multiply two complex numbers:
1. Multiply the two numbers using FOIL as you would when multiplying two binomials. The result looks like a quadratic expression in terms of i.
2. Replace i, squared with minus, 1. This changes the i, squared-term into a real number term.
3. Combine like terms and write the result in the form a, plus, b, i.

Example: What is the product of 3, plus, 2, i and 5, plus, 7, i ?

To divide two complex numbers:
1. If necessary, write the division as a rational expression with the dividend in the numerator and the divisor in the denominator.
2. Multiply both the numerator and the denominator of the rational expression by the complex conjugate of the denominator. The complex conjugate of a complex number has the same real component and the opposite imaginary component. For example, the complex conjugate of 3, plus, 2, i is 3, minus, 2, i.
3. Find the product of the numerator and the conjugate from step 2. The result should be a complex number.
4. Find the product of the denominator and the conjugate from step 2. The result should be a real number because multiplying a complex number by its conjugate eliminates the imaginary terms.
5. Divide the result of step 3 by the result of step 4 and write the quotient in the form a, plus, b, i.

Example: If a, equals, 4, minus, 3, i and b, equals, 2, plus, i, what is start fraction, a, divided by, b, end fraction ?

### Try it!

try: identify and use the complex conjugate
start fraction, 4, minus, i, divided by, 4, minus, 2, i, end fraction
For the rational expression shown above, what is the complex conjugate of the denominator? Enter your answer as a complex number.
The product of the numerator and the complex conjugate of the denominator is 18, plus, 4, i. What is the product of the denominator and its complex conjugate?

What is the sum of 13, minus, 5, i and 2, minus, 6, i ?

Practice: multiply two complex numbers
left parenthesis, 2, minus, i, right parenthesis, left parenthesis, 5, plus, 5, i, right parenthesis, equals, a, plus, b, i
In the equation above, a and b are constants. What is the value of a ?

## Want to join the conversation?

• if i^2 =-1,then (2-3i)^2=
• You just do (2-3i) times (2-3i). You should get -5-12i.
• How will i know if the question is using a complex number?
Will it be always i? or
will it be mentioned in the question?
(1 vote)
• Complex numbers will always be in the form a + bi on the SAT, where a and b are two numbers and i is the square root of -1. If you see an i, then the problem relates to complex numbers. Some example complex number questions would ask you to raise i to a certain power, or to add or subtract or multiply complex numbers together.