Multiplying complex numbers
We're asked to multiply the complex number 1 minus 3i times the complex number 2 plus 5i. And the general idea here is you can multiply these complex numbers like you would have multiplied any traditional binomial. You just have to remember that this isn't a variable. This is the imaginary unit i, or it's just i. But we could do that in two ways. We could just do the distributive property twice, which I like a little bit more, just because you're doing it from a fundamental principle. It's nothing new. Or you could use FOIL, which you also used when you first multiplied binomials. And I'll do it both ways. So this is just a number, 1 minus 3i. And so we can distribute it over the two numbers inside of this expression. So when we're multiplying it times this entire expression, we can multiply 1 minus 3i times 2 and 1 minus 3i times 5i. So let's do that. So this can be rewritten as 1 minus 3i times 2-- I'll write the 2 out front-- plus 1 minus 3i times 5i. All I did is the distributive property here. All I said is, look, if I have a times b plus c, this is the same thing as ab plus ac. I just distributed the a on the b or the c. I distributed the 1 minus 3i on the 2 and the 5i. And then I can do it again. I have a 2 now times 1 minus 3i. I can distribute it. 2 times 1 is 2. 2 times negative 3i is negative 6i. And over here, I'll do it again. 5i times 1-- so it's plus. 5i times 1 is 5i. And then 5i times negative 3i-- so let's be careful here-- 5 times negative 3 is negative 15. And then I have an i times an i. Let me do this over here. 5i times negative 3i-- this is the same thing as 5 times negative 3 times i times i. So the 5 times negative 3 is negative 15. And then we have i times i, which is i squared. Now, we know what i squared is. By definition, i squared is negative 1. i squared, by definition, is negative 1. So you have negative 15 times negative 1. Well that's the same thing as positive 15. So this can be rewritten as 2 minus 6i plus 5i. Negative 15 times negative 1 is positive 15. Now we can add the real parts. We have a 2, and we have a positive 15. So 2 plus 15. And we can add the imaginary parts. We have a negative 6. So we have a negative 6, or a negative 6i, I should say. And then we have plus 5i. And 2 plus 15 is 17. And if I have negative six of something plus five of that something, what do I have? Or if I have five of that something and I take six of that something away, then I have negative one of that something. Negative 6i plus 5i is negative 1i, or I could just say minus i. So in this way, I just multiplied these two expressions or these two complex numbers, really. I multiplied them just using the distributive property twice. You could also do it using FOIL. And I'll do that right now really fast. It is a little bit faster. But it's a little bit mechanical. So you might forget why you're doing it in the first place. But at the end of the day, you are doing the same thing here. You're essentially multiplying every term of this first number or every part of this first number times every part of the second number. And FOIL just makes sure that we're doing it. And let me just write FOIL out here, which I'm not a huge fan of, but I'll do it just in case that's the way you're learning it. So FOIL says, let's multiply the first numbers. So that's going to be the 1 times the 2. That is the F in FOIL. Then it says, let's multiply the outer numbers times each other. So that's 1 times 5i, so plus 1 times 5i. This is the O in FOIL, the outer numbers. Then we do the inner numbers, negative 3i times 2. So this is negative 3i times 2. Those are the inner numbers. And then we do the last numbers, negative 3i times 5i. These are the last numbers. So that's all that FOIL is telling us. It's just making sure we're multiplying every part of this number times every part of that number. And then when we simplify it, 1 times 2 is 2. 1 times 5i is 5i. Negative 3i times 2 is negative 6i. And negative 3i times 5i-- well, we already figured out what that was. Negative 3i times 5i turns out to be 15. Negative 3 times 5 is negative 15. But i times i is negative 1. Negative 15 times negative 1 is positive 15. Add the real parts, 2 plus 15. You get 17. Add the imaginary parts. You have 5i minus 6i. You get negative i. And once again, you get the exact same answer.