Hi and welcome. This is Anthony Varela, and today in this tutorial, we're going to multiply complex numbers. So we're going to talk about complex numbers a plus bi, we're going to multiply two complex numbers, and we're going to be using a process called FOIL. Something you might be familiar with.
So first let's review complex numbers. A complex number generally is expressed as a plus bi. It has a real part, it has an imaginary part, and then it has this imaginary unit, the i, the square root of negative 1. And this is an imaginary number, because any real number squared is positive. So to take the square root of a negative number would be not real or imaginary.
Well, when we're multiplying two complex numbers, we're going to be using FOIL, and recall this stands for First, Outside, Inside, Last. So let's quickly go through an example of FOIL in a context you may be more familiar with other than multiplying complex numbers. So here we have 2x plus 1 times 3x minus 2. And to multiply these two together, we're going to use FOIL.
So multiplying the first two terms, so 2x times 3x, we get 6x squared. Then we multiply the outside terms, so this would be 2x times negative 2, well, that would be minus 4x. Then we multiply the two inside terms, so this would be 1 and 3x, so we're going to add a 3x. And finally, we multiply the two last terms. So that would be 1 times negative 2, or minus 2.
Well, we can combine some like terms because I see two x terms, the negative 4x and the positive 3x. So combining those two together, I have 6x squared minus x minus 2. So that's how we can use FOIL, and we're going to be using this when multiplying two complex numbers. So let's multiply 3 plus 2i by negative 2 plus 3i.
So we're going to multiply the two first terms, so 3 times negative 2, that will give us a negative 6. Multiplying the outside terms, this would be 3 times 3i, so I'm going to add 9i. Multiplying the inside terms, 2i times negative 2, so I'm going to subtract a 4i. And lastly, 2i times 3i is plus 6i squared. So once again I see that I can combine those two terms in the middle here plus 9i and minus 4i. So I have negative 6 plus 5i plus 6i squared.
Now let's focus on how to simplify 6i squared. Well, i is the square root of negative 1. So if we were to then square that, what we would get then is negative 1. That's a real number. So i squared is negative 1. So how I like to rewrite something that has an i squared term, is I like to just erase the i squared, but then change the sign out in front. So instead of plus 6i squared, it's minus 6. Well, notice since this is a real number, it can be combined with this real number, so my simplified product is negative 12 plus 5i.
Let's go ahead and go through another example, 2 minus 5i times 1 plus 2i. Multiplying the first terms, 2 times 1, multiplying the outside terms, 2 times 2i, multiplying the inside terms, negative 5i times 1, and multiplying the last two terms, negative 5i times 2i, so you get negative or minus 10 i squared. Once again, we will go ahead and combine these two terms. Plus 4i and minus 5i is a minus i.
And how can we rewrite then this negative 10i squared? Well, i squared is negative 1, so I'm going to remove it and change the sign out in front. It becomes plus 10. Well, 10 can be combined with 2, so my simplified product is 12 minus i. So let's review multiplying complex numbers.
Well, we talked about a complex number containing a real part, and an imaginary part with i being the square root of negative 1. We used FOIL to multiply two complex numbers, so First, Outside, Inside, and Last. And remember, i squared is a real number. It's negative 1. So thanks for watching this tutorial on multiplying complex numbers. Hope to see you next time.
