Hi, my name is Bassem Saad. I'm a Math Ph.D. candidate at U.C. Davis, and I'm here today for About.com to show you how to multiply and divide complex numbers.Remember, you can treat the i in a complex number like a variable, with the added property that i squared equals negative one. Then, when you multiply complex numbers, you have to be able to distribute and gather like-terms.So let's multiply these two complex numbers – remember, we're going to use the FOIL method to distribute. So we have five times two is ten; five times i is plus five i; minus three i times two is just minus six i; and finally, we have minus three i times i to give us minus three i squared. But i squared equals negative one, so really we just have minus three times negative one, which gives us a plus three. We gather like-terms to give us the solution of 13 minus i. And now we know if we multiply these two complex numbers, we get 13 minus i.So the trick in dividing two complex numbers is to change the denominator into a real number, and the way to do this is to multiply the top and bottom by the complex conjugate of the denominator. So the complex conjugate times itself is just going to be one squared plus one squared. In the numerator, we again distribute to get three minus three i minus i. And then we have i squared – that gives us a minus one (we put a minus one).Now we just combine all like-terms to give us two in the denominator, and a two minus four i. Now we can distribute this divide two into each term. That gives us one for the real part, and minus two i for the imaginary part. So, three minus i divided by one plus i equals one minus two i. Complex numbers are an important number system for scientists. They must be able to multiply and divide complex numbers in order to use and expand established theory. Even engineers must master complex numbers in their operations to design new products and solutions.Thanks for watching, and to learn more, visit us on the web at About.com.

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