Copyright

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:

z1+z1

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

associative

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

commutative

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:

z2+z3+z4

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

associative

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):

commutative

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

Substituting:

z2-z3

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support