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# Multiplying complex numbers

Sal multiplies (1-3i) by (2+5i). Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

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.