New SAT Math: How to Handle Complex Numbers

Don’t be fooled by the name! Complex numbers are not so complicated once you figure out how to deal with them. They are sometimes called imaginary numbers and are represented by the variable letter i, which is equivalent to the square root of -1.

There are only a limited amount of ways you can be tested on this topic, so cementing this knowledge in your head is a great way to win some easy points on the test. After reading this guide, you’ll be doing a fist pump each time you see a complex numbers problem on the new SAT.

Complex Numbers -magoosh

New SAT Math: Simplifying Complex Numbers

On the test, you might see large negative numbers within a radical. If you take a glance at the answer choices and see that they are littered with i’s, chances are good that you need to simplify in order to get to the right answer.

For example, let’s take a look at the square root of negative 36.

In order to simplify this, use the imaginary letter i to take the negative out of the radical. Then reduce the radical down even further if possible.

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This leaves us with 6i.

New SAT Math: Simplifying Fractions With Complex Numbers

When dealing with fractions, it is customary to make sure that you don’t leave any imaginary numbers in the denominator. If you have any stray i’s hanging there, multiply the numerator and denominator by i in order to get rid of it.

If you have something more complicated in the denominator, such as 3 + 4i, then you will need to multiply the top and bottom of the fraction by something called the conjugate. Don’t worry about the name; all you have to do is take the exact same numbers and flip the sign (from addition to subtraction or vice-versa).

In this case, the conjugate of 3 + 4i is 3 – 4i.

New SAT Math: The Powers of Complex Numbers

You will need to know how to reduce a power of i down to its simplest terms. You don’t need to memorize this chart, but it’ll help speed up the process so that you don’t have to think about it if you need it during the test.

i = i
i^2 = -1
i^3 = -i
i^4 = 1
i^5 = i

Since the pattern repeats in sets of 4, all you have to do is divide i’s power by 4 to figure out what it reduces down to. For example, if the quotient ends in .5, you know that it reduces to -1.

New SAT Math: Complex Numbers on a Coordinate Plane

Plotting imaginary numbers on a complex coordinate plane is a piece of cake. The x-axis represents the real numbers while the y-axis represents the imaginary numbers.

For example, (3, -4i) lies on the same position as (3, -4) on a coordinate plane. Another way of representing (3, -4i) is 3 – 4i.

In order to find the magnitude (the distance from the origin to the coordinate) of a complex coordinate, simply use Pythagorean theorem to calculate the length of the diagonal side (starting from the origin). Thus, the magnitude of (3, -4i) is 5.


  • Minh Nguyen

    Minh's passion for helping students succeed grew during his time as a career counselor at the University of California, Irvine. Now, he's helping students all over the world by spilling SAT/ACT secrets through blog posts on Magoosh. When he's not busy tutoring or writing, he enjoys playing guitar, traveling, and talking about himself in third-person.

By the way, Magoosh can help you study for both the SAT and ACT exams. Click here to learn more!

4 Responses to New SAT Math: How to Handle Complex Numbers

  1. josh June 25, 2016 at 4:07 am #

    Hi Minh, thanks for tips. This website also helped
    me with the math part of the SAT.

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert June 25, 2016 at 4:52 am #

      Thanks for the tip, Josh 🙂

  2. Sam August 25, 2016 at 9:10 pm #

    Hi Minh,

    So, I’m helping my son prepare for the SAT and was interested to see that they now may apparently have imaginary numbers.

    One of the questions on a practice test did have an i in it, but it wasn’t clear whether it was intended to represent an imaginary number or just a regular variable (I answered it assuming it was a regular variable and got the answer correct — but it was an invented practice test by a test-prep company, and those tend to be a little sketchy anyway).

    So, any advice? Do you have any actual SAT questions with imaginary numbers so we can see some samples?


    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert September 6, 2016 at 11:12 am #

      Hi Sam,

      The real SAT exam, as designed by the College Board, does have some questions with imaginary numbers, expressed as i. And often, those imaginary numbers can behave just like regular variables, in the same way that you saw in the third-party test question. This is one of the reason that Mihn said complex numbers aren’t as complex as you might assume.

      As Mihn also mentioned, this kind of imaginary number is rare in SAT Math. But you still see problems like this– there will often be one imaginary number problem on a real College Board SAT math section. To find examples of these problems, I recommend combing through the math sections of the free full-length practice tests on the College Board’s official SAT website. Be sure to check the math sections in the Official Guide to the SAT too.

      I found at least one example for you. You can see a typical SAT imaginary number problem in SAT Practice Test # 1. It’s the second question in section 3 of the test (page 35). You can read an answer explanation to the question on page 26 of Practice Test #1’s answer explanation PDF.

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