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College Algebra: Writing Complex Numbers


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About this Lesson

  • Type: Video Tutorial
  • Length: 4:42
  • 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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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

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I don't know, do you ever thing about things that are imaginary, like unicorns or leprechauns, or the Tooth Fairy, things that are really - let's face it, I'm thinking about things that are imaginary. That's right. This is nice. We're in a happy place now. But really, sometimes things that are imaginary are required, because they're just not real. Let me show you something that just ain't real. In fact, you may remember, when you're taking square roots, if I said to you, "What's the ?" what you've got to say there is you've got to say, "Okay, no problem. I've got to find some number that, when I multiply it by itself, it's going to equal -9." Now, wait a minute, if I take a number and multiply it by its self, whether it's positive or negative, when I multiply it by itself, it becomes positive, unless it's zero, in which case then it's zero. There's no way that I'm going to be able to take two numbers, multiply them together and get -9. So this is just not going to happen, this is just not real. So therefore, it must be imaginary. It's pretend time.
Now, if we were to pretend, let's just pretend for a second that this really was some sort of wacko number. Well, what would we say about it? So now, let's just have a little fantasy. I could write this, for example, as -1 times 9. And then I could use properties of exponents, because remember a square root is just an exponent of , and say that's just . Now, , I happen to know there's no imagine required for that. That's just 3, cut and dry. So let me write the 3 out in front. And then I've got . So it's that's sort of the imaginary part. It's not a real number. And, in fact, we have a name for that, we call it i, i for imaginary, just like my imaginary friend.
So if I call that i, and usually we use a fancy - we don't use a capital I because we don't want to make a big stink about it, we just use a lowercase i. And that stands for - it's not a real number, it's an imaginary number, . So that's what this symbol means. By the way, this is a little trivia fact you can impress your friends with if you want, that if you're an engineering person, do you know what they use? They use little j. They have the thing sort of dangle this way, but who cares? Anyway, we'll use i, because that's the way it's supposed to be. The engineering people don't know what they're doing.
Anyway, I'd write this as 3i, and that means it's 3 times this imaginary number. Okay, so when you're armed with that, now you can actually start writing all sorts of numbers, because what we can do is write what are called complex numbers. There it is. And what a complex number is, is a number that has i's in it, that's all. So, for example, something like this, if I took 2 + , I could write that in the following way: well, I'll just keep the 2 there, but we just saw is actually the same thing as 3i, because the = 3 and = i. So, in fact, an imaginary number is any number that looks like this, some real number plus a real number times i. So this is called a complex number. This number here is called the real part. This is the real part, because it's just real. Now, this part actually is the part that's the imaginary part, so this 3 is called the imaginary part. So the real part of this number is 2, the imaginary part of this number is 3. If I look at this number, how about 3 + , I could write that as 3 plus - and while the = 5, but then I have , which is i. So this is a complex number, 3 + 5i, and the real part is 3 and the imaginary part is 5.
So, now you can actually start dealing with imaginary numbers. And guess what? That means it must be imaginary arithmetic. Cool! We'll see what's up next.
Complex Numbers
Introducing and Writing Complex Numbers Page [1 of 1]

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