Remember, a complex number is very similar to a binomial. We’re dealing with imaginary and real numbers at the same time. We already took a look at addition and subtraction, so let’s move on to multiplication and division. These are a little trickier, but only division involves a skill you may not have used yet. Take a look!

## Multiplication

Do you remember having to learn how to FOIL? You multiply the terms of a binomial or complex number in this order: First, Outer, Inner, Last. Let’s take a look at how to do it with a complex number.

That leaves us with this:

Now remember, , as we already covered. So we get this:

## Division

All right, here’s where things get a little tricky, but stick with me. I promise, we’ll come out on the other side (mostly) unscathed.

Let’s say you had to divide 5 + 2*i* by 6 + 3*i*.

Now, remember, *i* is just another way of writing √-1. And, according to the ancient laws of math, we can’t have a radical in the denominator (or bottom part) of a fraction. So, it looks like we have to simplify in order to solve this problem.

*Step One: Conjugate*

In order to divide complex numbers, what you have to do is **multiply by the complex conjugate of the denominator.** I heard about half of you get sudden migraines there, but I promise, that’s not as complicated as it sounds. The **complex conjugate** is just *the same exact denominator* with one tiny change. Instead of 6 + 3*i*, we take 6 – 3*i.*

So our problem now looks like this:

Really, all we’re doing is multiplying by a fancy form of 1, so we’re not actually *changing* the problem; we’re just simplifying it.

*Step Two: Multiply*

It looks like we’re out of plastic wrap, which is okay, because all we need is FOIL. Yes, the good old First-Outer-Inner-Last method of multiplying binomials and complex numbers is back again. *And this time, it’s personal.*

Okay, not really. But let’s FOIL anyway. We’ll do the numerator first.

That leaves us with this:

And now, do the denominator the same way:

*Step Three: Simplify*

Here’s our problem so far:

We already know that , so let’s change that in both the numerator and the denominator.

And now, combine like terms! *Watch the magic!*

Notice how the denominator suddenly doesn’t have any more *i* in it. We’ve fully simplified this problem! Woo-hoo! Take a nice deep breath, Magooshers! You’ve earned it.