# Simplifying Complex Numbers: Addition of Like Terms

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

We will talk about complex numbers - their uses and their parts. Then, we will look at simplifying these types of numbers by adding like terms. Lastly, we will practice this process through various examples.

## Why Study Complex Numbers?

Ahhh, complex numbers. These numbers, that have both a real part and an imaginary part, tend to leave many students confused. Not because the numbers themselves are hard to understand, but because they leave the student wondering how they could possibly be applicable in the real world. I mean, if a number has an imaginary part, meaning it contains the number i = √-1, how can we possibly use it to represent anything in the physical world?

Well, I'm here to tell you that these numbers are very applicable in the real world! For instance, in engineering, they are used to represent the slowing of a pendulum's swinging motion. Electric engineers use them to represent alternating currents. In physics, complex numbers are used anytime a force gets divided into two or more components. In fact, these numbers even show up in weather forecasting! These are just to name a few applications involving these types of numbers.

Of course, the mathematics involved with the mentioned applications is well beyond the scope of this lesson. However, by studying complex numbers and how they work together, we get closer and closer to being able to study their uses and actually perform the mathematics involved in these applications as well as many more! Therefore, let's get ourselves one step closer by talking about complex numbers, and seeing how to add complex numbers by adding like terms.

## Complex Numbers

As we said, complex numbers have both a real part and an imaginary part. These types of numbers take on the form a + bi, where i = √-1. We call a the real part of the number, and we call bi the imaginary part of the number.

For example, the number 2 + 3i is a complex number with real part 2 and imaginary part 3i.

Both real numbers and imaginary numbers are complex numbers. This is because we can always think of a real number a as a + 0i, and we can always think of an imaginary number bi as 0 + bi.

See? I told you the numbers themselves aren't all that hard to understand. They're actually quite straightforward! Now, let's talk about simplifying complex numbers through adding like terms!

## Adding Like Terms to Simplify

You know how in a mathematical expression, we can combine terms that have the same variables and exponents? For instance, we can add x + 2x to get 3x or 2x 2 + 4x 2 to get 6x 2, because in each of these instances, the two terms have the same variable parts. On the other hand, we can't add 2x + 4x 2, because the two terms don't have like variable parts.

Adding terms that have the same variable part is called adding like terms, and the same concept applies to simplifying the sum of two complex numbers. When we are adding two complex numbers, we can add their real parts together, and we can add their imaginary parts together, because these are like parts. Technically, we have that (a + bi) + (c + di) = (a + c) + (bi + di) = (a + c) = (b + d)i.

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