What is a Complex Number? - Definition & Properties

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Applications of complex numbers include signal processing filters in engineering and vector length calculations in physics. After defining complex numbers, this lesson explores how to combine them using addition, subtraction and multiplication.

Complex Numbers

Imagine keeping track of two things at once, like eating lunch and knowing the time. This is a complex situation somewhat related to complex numbers. Another example is that a signal can have a magnitude and a phase. Complex numbers are great for describing signals. In this lesson we define complex numbers and then use math properties to add, subtract and multiply complex numbers.

Defining Complex Numbers

Indeed, a complex number really does keep track of two things at the same time. One of those things is the real part while the other is the imaginary part. For example, z = 3 + 2i is a complex number. The real part of z is 3 and the imaginary part of z is 2. The everyday meaning of ''imaginary'' is something which doesn't exist. The meaning in math is quite different. Identifying the imaginary part of a complex number is easy because it has a label. The imaginary part is the number multiplying the label i'. That's right, the imaginary part of 3 + 2i is the 2. Be careful because the imaginary part is not 2i. The imaginary does not include the label.

Another example? Let's take the complex number z = -15 - 32i. Which part is real and which imaginary? The real part is -15 while the imaginary part is -32.

The great Swiss mathematician Euler invented i in 1777. The value of i is the square root of negative one. For the most part, we will use i as the label identifying the imaginary part of a complex number. Still, we may need to evaluate i2 from time to time. If i is the square root of negative one, then i2 is the square root of negative one times the square root of negative one. Thus, i2 is -1.

Number Properties

Remember the commutative property? The commutative property is about ordering. When adding or multiplying, changing the order does not change the result. For example, 3 + 6i is the same as 6i + 3. How about the associative property? The associative property is about grouping. When adding or multiplying, we can group the terms in any way without changing the result. For example, (2 + 3i) + (3 - 4i) is the same as (2 + 3) + (3i - 4i) which gives us 5 - i. Lastly, do you remember the distributive property? The distributive property is about distributing a multiplication over an addition. When multiplying a number times a parenthesis containing the sum of two or more numbers, the multiplication applies to every number in the parenthesis. For example, 2(3 - 5i) is the same as 2(3) + 2(-5i) which gives us 6 - 10i.

In the following examples we will use these four complex numbers:

  • z1 = 2 + 3i
  • z2 = -3 + 2i
  • z3 = 4 - 2i
  • z4 = -2 - 4i

Adding Complex Numbers

Example: Add z1 to itself

Start by substituting:


The associative property allows any grouping, so we can eliminate the parentheses:


The commutative property allows any ordering for adding. The goal is to add the real parts and imaginary parts separately:


Adding the real parts gives us 2 + 2 = 4. Adding the imaginary parts gives 3 + 3 = 6. The answer is 4 + 6i.

Example: Calculate z2 + z3 + z4.

First, substitute:


Then, we can remove the parentheses (associative property):


Did you see where the +(-2 - 4i) became -2 - 4i ? We can think of +(-2 - 4i) as +1 times (-2 - 4i). The +1 times the (-2 - 4i) gives -2 - 4i.

Now, we can reorder the addition (commutative property):


Adding the real parts gives -3 + 4 - 2 = -1. Adding the imaginary parts, 2 - 2 - 4 = -4. The answer is -1 - 4i.

Subtracting Complex Numbers

Subtracting complex numbers is very much like adding complex numbers. Here's an example:

Example: Calculate z1 - z3



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