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College Algebra: Multiply Complex Numbers


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About this Lesson

  • Type: Video Tutorial
  • Length: 4:43
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 50 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
College Algebra: Basics & Prerequisites (37 lessons, $52.47)
Trigonometry: Complex Numbers & Polar Coordinates (15 lessons, $26.73)
Trigonometry: Complex Numbers (5 lessons, $6.93)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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So adding and subtracting complex numbers is a piece of cake. You just add the real parts, combine the real parts, combine the imaginary parts and you're done.
Now, with multiplying complex numbers, it's actually, in some sense, at least philosophically, we sort of have to binomials, and so a brilliant foiling thing going on. Let me just remind you of one basic fact: if i = , the what does i^2 have to equal? Well, i^2 would be times itself, and so the radical would actually lift and I'm just left with -1. So that's an important fact to keep in mind, that i^2 is just the number -1. It's that number, so that if you square it, you get -1. That what we defined it. So it's worth keeping that in mind, as you'll see in this example, because now to multiply, you just foil away. You're probably saying, "Well, gee, it's too bad he doesn't have a fancy little foil card here." But it's okay, look over there. You can't get any fancier than that.
What do you do? So let's treat this like a foiling thing. So I'm going to multiply 6 2, which is 12. Now my inside terms are quite large. They look like 18i, so I'm going to put that in, 18i. My outside terms are going to give me a -10i. And now what's the last times the last? That's going to be -15, and it's not i, it's i^2. Now remember that i^2 is just -1. So, in fact, that can be cut out and replaced by -1, a factor of -1. So you can see that now I've got two real pieces, namely, this becomes a real. There's no i there anymore. And I've got this piece. And then I've got two imaginary pieces, so I can combine them, like I add complex numbers. So this, a minus a minus, becomes a plus. So this sign flips, because I have an i^2. This becomes plus 15 and 12, which gives me 27. And then I have an 18i - 10i, which gives me a black eye - no, no, just a joke, it gives me 8i. So what's this product? It's some other complex number, and the complex number turns out to be that. So this number times that number equals that number. You see how I did it? I just foiled. And then just remember that i^2 = -1. That's all there is to it.
Let's try one more. Let's square out 3 + 5i. It sort of weird to using foil with numbers. You always think of using foils when you have variables, but, of course, this is a number, but it's a little bit complicated. In fact, it's an imaginary number. I want to square this. That requires me to multiply it by itself, so I've got to foil 3 + 5i with itself. So 3 3 = 9, that's part of the real part. Now I get the imaginary part. So the inside term gives me a 15i. In fact, if you really get good at this, you can start doing this almost together. 15i here, there's another 15i here, that's a total of 30i. And then the last times the last is 25i^2. i^2 is negative, so it's -25. So you could do it that fast. Let's not. So this is going to be plus - the inside terms are 15i, the outside terms are another 15i, and the last times the last is a plus 25i^2. But now I remember that i^2 is just the same thing as -1, so, in fact, this whole thing is the same thing as -25, just change the sign.
So my real parts are over here, my imaginary parts are over here. If I combine those together, let's see, 9 - 25, oh, boy, these are always the hard ones. This is going to be a minus something. It'll be a -16, he says hopefully. And then this is just going to be, 15 and 15 is 30i. So here I have a number and I want to know what number the square is, so I take the number and multiply it by itself and I see that it's this number. So the thing is, this is another complex number and it turns out this is actually the square of this number.
So multiplying, just use the foil method really carefully and remember that i^2 = -1.
Complex Numbers
Multiplying Complex Numbers Page [1 of 1]

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