Complex Numbers

SAT Math Overview. Topic: Complex Numbers

Overview

Expect to see some questions on the topic of Complex Numbers on the SAT. In this section, we will review the following aspects of this topic:

  • Definition of a Complex Number
  • Complex Addition and Subtraction
  • Complex Multiplication and Division
  • Graphing Complex Numbers
  • Comparing Complex Numbers
  • Closure Under the Complex Number System
  • Complex Roots of Equations

Let’s look at each of these in more detail:

Definition of a Complex Number

The real number system consists of quantities that can be graphed on a one-dimensional number line. These are examples of graphs of real numbers.

A number line showing positive and negative real numbers.

There is another one-dimensional number system that can be graphed on a number line. This is the imaginary number system, and here is such a number line:

A number line showing positive and negative imaginary numbers.

Each imaginary number is a multiple of i. What is i? It is defined by the following identity:

i equals square root of negative 1

The square root of negative one is not a real number. All numbers that are a multiple of i are imaginary numbers.

Complex number. A non-real number that can be written in these forms.

We can now combine both number systems to define the complex number system:

Complex Coordinate Systems. A two dimensional coordinate system that tracks a point along the Real axis and the imaginary axis.

Complex numbers are two dimensional and require a two-dimensional coordinate system to graph them. Think of the complex number system as taking the number line of real numbers and using that as the horizontal axis, then taking the number line of imaginary numbers as and using it as the vertical axis. 

Complex Coordinate System

A complex number can be written in three different forms, as shown in the definition of a complex number, but for the purposes of the SAT, this is the form to focus on:

An analysis of a complex number, showing the real and imaginary parts.

Note that a complex number has a real number part and an imaginary part. To learn more about complex numbers, click on this link. 

Properties of Complex Numbers

In this section we go over the properties of complex numbers, including the following:

  • Imaginary Numbers to Odd or Even Powers
  • Graphs of Complex Numbers
  • Complex Conjugates and Their Graphs
  • Magnitude of a Complex Number

Imaginary Numbers to Odd or Even Powers

The imaginary number i raised to integer powers alternates from real to imaginary, with alternating signs. Here are the results of raising i to the first five integer powers.

The effect of even and odd powers on the imaginary number I.

Expect SAT Math questions that rely on an understanding of this property of i.

Graphs of Complex Numbers 

Since complex numbers are two-dimensional, representing them graphically involves a two-dimensional coordinate system. A complex number of the form a + bi is graphed as if it were a pair of coordinates, (a, b) on the Cartesian system.

The graph of an imaginary number of the form A plus B I.

The real part, a, is graphed along the horizontal axis. The imaginary part, bi, is graphed along the vertical axis.

To see examples of plotting complex numbers, click on the following link. 

Complex Conjugates and Their Graphs 

Every complex number has what is known as a complex conjugate. This is what such a pair of numbers look like.

Complex Number

Complex Conjugate

a plus b i

a minus b i

Here are some examples of complex numbers and their conjugates:

 

Complex Number

Complex Conjugate

2 plus 3 i

2 minus 3 i

negative 4 plus 5 i

negative 4 minus 5 i

This is what the graph of a complex number and its conjugate look like.

The graph of a complex number of the form a plus b i, along with its complex conjugate.

Notice that a complex number and its conjugate are mirror images of each other across the horizontal axis.

Magnitude of a Complex Number

Because complex numbers require a two-dimensional coordinate system to represent them graphically, we can measure the distance the number is from the origin.

 

Finding the magnitude of a complex number.

This distance is known as the magnitude of the complex number. To learn more about magnitude and comparing complex numbers, click on this link.

SAT Skill: Working with Imaginary Numbers

Example 1

What is the value of this expression if m = 4 and n = 3.

i to the power of m times i to the power of n

Combine the exponents and plug in the values:

Using the laws of exponents with imaginary numbers.

Recall the repeating pattern starting at an exponent of 1: i, -1, -i, 1. For an exponent of 7, this is the result:

i to the power of 7 equals negative i


 

Complex Number Operations

In this section we go over the properties of complex numbers, including the following:

  • Adding and Subtracting Complex Numbers
  • Multiplying and Dividing Complex Numbers
  • Closure Property

Adding and Subtracting Complex Numbers

Adding and subtracting complex numbers involves adding two-dimensional numbers. Remember, a complex number has a real part and an imaginary part. This means that adding or subtracting complex numbers involves real operating on real and imaginary operating on imaginary.

The sum of two complex numbers.Subtracting complex numbers.

 

To see examples of adding and subtracting complex numbers, click on this link.

Multiplying and Dividing Complex Numbers

When multiplying two complex numbers, use the FOIL method to find the separate products. Then combine real terms and imaginary terms. Note that there will be an i-squared term, that will become a real term.

The product of complex numbers.

Multiplying a complex number and its conjugate results in a real number, with no imaginary part. 

The product of a complex number and its conjugate.

Dividing two complex numbers results in another complex number in standard form. However, to get that form, multiply the numerator and denominator by the conjugate of the denominator.

Dividing two complex numbers.

To see examples of multiplying and dividing complex numbers, click on this link.

 

SAT Skill: Working with Complex Numbers

Example 1

What is the sum of these numbers?

open parentheses 3 plus 4 i close parentheses plus open parentheses negative 7 plus 9 i close parentheses

Combine the real parts and the imaginary parts:

The sum of two complex numbers.

Example 2

Subtract these numbers.

open parentheses 2 minus 5 i close parentheses minus open parentheses negative 7 minus open parentheses negative 8 i close parentheses close parentheses

Combine the real parts and the imaginary parts:

The difference of two complex numbers.

Example 3

Combine these numbers.

open parentheses 3 minus 4 i close parentheses plus open parentheses negative 5 plus 5 i squared close parentheses

Combine the real parts and the imaginary parts. Note that the i2 term becomes a real number:

The sum of two complex numbers.

Complex Roots of Equations

With certain quadratic equations, the solutions to the equations are complex numbers.

a x squared plus b x plus c equals 0

How do we know when a quadratic equation has complex roots? Calculate the discriminant of the quadratic:

Discriminant equals b squared minus 4 a c

If the value of the discriminant is less than zero, then the quadratic will have complex roots.

To see examples of finding the complex roots of quadratic equations, click on this link. This slide show includes a video tutorial and worked-out examples.


 

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