Simplifying Complex Numbers With Multiple Steps

Instructor: Lauren Link

Lauren has taught high school Math and has a master's degree in education.

Performing the basic operations with complex numbers isn't much different than doing so with polynomials. In this lesson, we'll go over what exactly you must know about complex numbers, including some key facts that'll help you perform these operations.

What Is a Complex Number?

A complex number is ''complex'' because it's made up of more than one part. A complex number is written in the form a + bi, where a is the real part and bi is the imaginary part. For example, 2 + 3i is a complex number, where 2 is the real part and 3i is the imaginary part. But, is 5i a complex number? Most people will assume that it's not because it only has one part. However, isn't 5i the same thing as 0 + 5i? Yes, so 5i is a complex number! We know that in algebra i represents an imaginary number, which comes from the fact that:


By squaring each side we get:

rule 2

This fact is necessary to remember as we move forward.

Adding Complex Numbers

In order to add complex numbers, we'll treat them as polynomials and use the same rules to add. For example, we would treat (3 + 2i) + (-4 + 6i) as (3 + 2x) + (-4 + 6x). Let's use this as our first example.

Example 1

(3 + 2i) + (-4 + 6i)

Since we're asked to perform addition and the coefficients before the second complex number, (-4 + 6i), is an implied positive 1, we can drop all of those parenthesis. So, we have 3 + 2i + -4 + 6i or 3 + 2i - 4 + 6i.

Next, we'll combine all like terms by adding the coefficients of the like terms (like we would if these were polynomials); that is, combine constants 3 - 4 and i terms 2i + 6i. We get our answer of -1 + 8i.

Lastly, we check to see if our answer is in proper a + bi form. Yes, it is. We're done!

Subtracting Complex Numbers

When subtracting complex numbers, we can use the same strategy as with adding. Let's look at an example.

Example 2

(-3 - i) - (4 - 5i)

Subtraction is almost exactly like addition except for an additional first step:

(-3 - i) - (4 - 5i) really means (-3 - i) -1(4 - 5i)

Since there's that implied -1 before the second set of parenthesis, we first have to distribute the -1 prior to dropping the parenthesis. After distributing -1 to the second set of parenthesis, we'll get -3 - i - 4 + 5i (notice the signs only switched for each term).

Now, we can go ahead and combine like terms like with did in Example 1. Combine constants -3 - 4 and i terms -i + 5i. Remember that -i is actually -1i. So, -3 - i - 4 + 5i = -7 + 4i.

Once again, we'll check to see if our answer is in a + bi form. As long as it is, we're done!

Multiplying Complex Numbers

Multiplying complex numbers is a little different than adding and subtracting them, but we can follow the same procedure that we would if we were multiplying polynomials. We'll use ''FOIL'' (First, Outer, Inner, Last) to make sure we multiply every term in the first complex number by every term in the second complex number.

Example 3

(2 + i)(4 + 3i)

First: (2)(4) = 8

Outer: (2)(3i) = 6i

Inner: (i)(4) = 4i

Last: (i)(3i) = 3i2

Then, combine like terms and get 8 + 10i + 3i2

Now, what did we say about a key fact that we have to remember? This is where -1 = i2 comes in. Since we know that these are equivalent values, we'll replace i2 with -1.

8 + 10i + 3(-1)

Then, simply this to 8 + 10i - 3. Then, again, to get our final answer of 5 +10i.

Dividing Complex Numbers

In order to divide complex numbers, we need to know something about conjugates. The conjugate of a complex number requires you to change the sign of the bi term. For example, the conjugate of 2 + 3i is 2 - 3i, and the conjugate of -2 - i is -2 + i. Make sure to only change the sign of the bi term!

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