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Multiply Complex Numbers

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Today we're going to talk about multiplying complex numbers. So we're going to start by reviewing the process of FOIL, because complex numbers have two terms, which makes them binomial, so we can use FOIL to multiply. And then we'll do some examples.

So let's start by looking at complex numbers. Numbers that are complex are written in this form, a plus bi, where a and b are real numbers. And a is the real part of the complex number. b is the imaginary part of the complex number. And i is the imaginary unit, which, if you remember, is equal to the square root of negative 1.

And we call the square root of negative 1 imaginary, because no two real numbers multiplied by themself we'll give you a negative 1. Or no two real numbers squared will give you a value of negative 1. So the square root of negative 1 is not a real number, so we call imaginary.

So let's review FOIL. Remember FOIL is the acronym that we use to remember the order in which we multiply terms when multiplying two binomials. And FOIL stands for first, outside, inside, last.

So for example, if we want to multiply the two binomials, 3x minus 2x and x minus 4, we can use FOIL. So we're going to start by multiplying our first two terms, 3x and x. Multiplied together that's going to give us 3x squared.

Then we'll multiply our outside terms. So 3x times negative 4 is going to give me a negative 12x. Then we're going to multiply our inside terms. Negative 2 times x will give me negative 2x. And finally, multiplying our last two terms, negative 2 times negative 4 will give me a positive 8.

So I can combine these two middle terms, negative 12x minus 2x. And a negative 2x is going to give me negative 14x. And I can bring down my other two terms. So I've found that these two binomials multiplied together is equal to this.

So now let's do an example multiplying two complex numbers together. Again, because complex numbers contain two terms, they're binomials. So when we're multiplying two complex numbers together, we can use FOIL to multiply our terms.

So I'm going to start by multiplying my first two terms together. Negative 5 times 3 will give me negative 15. Multiplying my outside two terms together, negative 5 times negative 4i is going to give me a positive 20i.

Multiplying my inside two terms together, 2i and 3. That's going to give me positive 6i. And finally, multiplying my last two terms together, 2i and negative 4i is going to give me a negative 8i squared.

Now as usual I can combine my inside two terms. 20i plus 6i will give me 26i. Bring down my other two terms. However, I can also simplify this last term, negative 8i squared, because we know that i is equal to the square root of negative 1. And we know that i squared is equal to negative 1. So I can substitute negative 1 into my expression for i squared.

So this is going to become negative 8 times negative 1. Negative 8 times negative 1 is going to give me a positive 8. And now that this is just a constant term, I can combine it with negative 15. Negative 15 plus 8 is going to give me negative 7 plus 26i for my final answer.

Let's go over our key points from today. Complex numbers consist of a real part and an imaginary part. The square root of negative 1 is imaginary because no real number squared results in a negative number. Because complex numbers are binomials, use FOIL when multiplying. And simplify expressions with an i squared term by substituting negative 1 for i squared, multiplying by the coefficient, and writing as a real number with the opposite sign.

So I hope that these key points and examples helped you understand a little bit more about multiplying complex numbers. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Imaginary Number