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 onedimensional number line. These are examples of graphs of real numbers.
There is another onedimensional number system that can be graphed on a number line. This is the imaginary number system, and here is such a number line:
Each imaginary number is a multiple of i. What is i? It is defined by the following identity:
The square root of negative one is not a real number. All numbers that are a multiple of i are imaginary numbers.
We can now combine both number systems to define the complex number system:
Complex numbers are two dimensional and require a twodimensional 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.
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:
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.
Expect SAT Math questions that rely on an understanding of this property of i.
Graphs of Complex Numbers
Since complex numbers are twodimensional, representing them graphically involves a twodimensional 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 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.




Here are some examples of complex numbers and their conjugates:






This is what the graph of a complex number and its conjugate look like.
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 twodimensional coordinate system to represent them graphically, we can measure the distance the number is from the origin.
This distance is known as the magnitude of the complex number. To learn more about magnitude and comparing complex numbers, click on this link.

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 twodimensional 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.
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 isquared term, that will become a real term.
Multiplying a complex number and its conjugate results in a real number, with no imaginary part.
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.
To see examples of multiplying and dividing complex numbers, click on this link.

Complex Roots of Equations
With certain quadratic equations, the solutions to the equations are complex numbers.
How do we know when a quadratic equation has complex roots? Calculate the discriminant of the quadratic:
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 workedout examples.
About Media4Math
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