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College Algebra: Rewriting Powers of i


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About this Lesson

  • Type: Video Tutorial
  • Length: 6:00
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 64 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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Recent Reviews

~ amonsalve

excellent work

~ amonsalve

excellent work

Okay, so some people actually might be sort of taken aback and annoyed by the fact that you can take an imaginary number, like i, , and just square it and, all of a sudden, it becomes a real number. Some people, in fact, may say, "That's nonsense." Of course, those people they probably wear really thick shoes. The reality is that while i itself is an imaginary number, it's imaginary because it actually satisfies this property. It's some imaginary number such that its square turns out to be -1, because no real number satisfies that. So, in fact, those people that think, "Oh, this makes no sense" well, anyway, okay.
However, what if we now take i and raise it to the third power? What would that equal? Well, let's figure that out, i^3. Well, one way of thinking about i^3 is saying it's just i^2 times another i. Well, i^2, we know, is -1, so, in fact, I see that this is just -1 i, so it's -i. In fact, i^3 has sort of a simple way of saying it; it's just -i. And why? Well, because i^2 is just -1, and so this just contributes the negative sign.
What about i^4? i^4 would be what? Well, that would be i^2 i^2 - oh, can you here that? There's a plane flying so low that I have to duck. And notice that this is just -1 -1. And that equals 1.
So this is really neat. i = . i^2, we know, equals -1. We see that i^3 is just -i. i^4 is just 1. What would i^5 be? Well, i^5 would just be i^4 i. Well, i^4 = 1, so we just get i again. And what would i^6 be? Now, look over there. i^6 is just going to be i^4 i^2. Well, i^4 is just 1, so we'd just i^2, which is -1. So we would go right back down this list again.
So what I'm saying is the following: If you take a look, here's the list that we just made up. We have that i = , i^2 = -1, i^3 = -i, and i^4 = 1, but now if you want to know what i^5 is, you just go back to the beginning, that's i. i^6 would be here, i^7 would be here. This would be i^8, i^9, i^10 and i^11. So it's just i, i, i, i.
So the point is that you can now find any power of i that you want, and all you've got to do is figure out where you're going to be bubbling out on this chart. This chart has four entries, so basically all you've got to do is take the exponent and divide it by 4 and see what the remainder is. Let me show you with a very specific example.
If we take a look at i^40, what does that equal? Well, that would equal i, and notice that's 4 10 in the exponent. So, using laws of exponents, I could write that as . But i^4 = 1, so this is just 1^10, which equals 1. So, in fact, this is just 1. And the way to see that is that I just have gone through this list ten times, but where do I land? If you imagine having this thing and spinning it, if you do it 40 times, you're going to end right here, because 4 goes into 40 an even number of times, and so there's no remainder.
What about something like this? What about i^223? Well, you're going to spin through this 223 times. Where are you going to end? Well, it depends on how many times 4 goes into 223. So if you do that out, you can take 4 and divide it by 223, and what you would see is 55, and there would be a remainder or 3. So what that means is you would go through here 55 times completely, and then when you're all done, you're going to go 3 more times. So, in fact, it's going to be -i. So what I'm saying is the following: This equals i to the - now, what's that number? We saw it was 4 55 plus the remainder of 3. And if you use the laws of exponents, that equals . And since I'm adding exponents, I multiply the bases and I get that. Well, i^4 is just the 1, so, in fact, that's just 1, and so all I'm left with is I^3, and that's exactly what happened, i^3, and that equals -i. So all you have to do is divide by 4 and see what the remainder is.
So, for example, i^2001, which is the year coming up. What would that be? Well, that's just, if you divide by 4, I'd see 4 500, that gives me 2000, and then plus 1. So the remainder is plus 1. So I'm going to go through here 500 times and where do I end up? I end up right here, at the first one. So this actually will equal i.
So if you have really high powers of i, not a problem, just divide the power by 4, see what the remainder is, and that's where you're going to be on this list. You're going to spin around the quotient number of times, and then the remainder is going to tell you where you end up.
Complex Numbers
Rewriting Powers of i Page [1 of 1]

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