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College Algebra: e


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About this Lesson

  • Type: Video Tutorial
  • Length: 6:47
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 73 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Algebra: Exponential and Logarithmic Functions (36 lessons, $49.50)
College Algebra: The Number e (2 lessons, $2.97)

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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Okay, so now I want to tell you about probably one of the most important numbers in your life. Even if you don't know about it, it's actually in your life, all over the place, and making you healthy and growing. To inspire this number I just want to sort of tell you how one can figure out, roughly speaking, sort of what this number's all about. And maybe you're not going to like this, I don't know, but I hope that you like it, because at least it gives you a sense as to what's going on with this number. This is a very special number. So I'm going to try to define for you a particular number--not a function--just a number. And to do that, let me first just graph the function f(x) = 2^x, that exponential function. We talked about that before. That exponential function sort of has a basic shape that looks like this. So this is just 2^x. This is, of course, (0,1), and then it goes and up and grows and it's asymptotic here. I'm not going to go through the whole thing.
Let me now also graph in the function 3^x. So here's 3^x. Much more dramatic. Now, let's just take a look at what happens at this point (0,1). If you look at the 2^x and you put a line that just nicks it, just grazes it at that point, it's called the tangent line, but who cares what it's called. The important thing is it just nicks the line right at that point. Look at the slope of that line. It doesn't look very steep at all, and in fact, it's not very steep, it's actually less than 1. However, if you look at the slope of the line that just nicks the green function, the 3^x, that's way up here, that's a little bigger than 1, it has a steeper slope. Well, that must mean there's some number, there's some base I can put in there so that when I look at the slope it would actually equal 1. If this is smaller than 1 in slope, and this is steeper in slope and has slope bigger than 1, then there must be some number I can draw in here so that its exponential function, the slope would actually be one.
Let me just draw that in so you can see what it would look like potentially. It would be right between them. And that curve at the point (0,1) would have slope exactly equal to 1. It wouldn't be smaller than 1 like this one, or bigger than 1, like this one, it would be just right. Well, that number, that base, whatever that base is, let me just call it e^x, that number would have to be some number that's between 2 and 3, if you believe what I've told you. I haven't proved anything, but if you believed what I told you that at 2 the slope is actually less than 1, at 3 the slope is bigger than 1, and somewhere in between the slope at that point should equal 1.
Well, that number e is a very special number. Now, it turns out numerically to be equal to 2.71828169, and it goes on forever. We call this number e. This is e. And it's a really important number when you're talking about growth, and in fact, almost all problems that talk about growth, populations expanding, or things decaying and dying, they all usually use this number as the base. It's a very special number. It's sort of like . is a big number for circles, e is a big number for growth and decay.
If you want to estimate it, by the way, because you really can't estimate it from here, the best way to estimate it if you want to see numerically what it equals, is to just take a really big number like 1,000 or something or as big as you want, and compute 1 + 1 over that number, and then raise it to that number power. So put in like 5 in here. (1 + 1/5)^5 or (1 + 1/10)^10 or (1 + 1/100)^100. And it turns out that if you do that, you're going to head towards this number. Try it, you'll see. Make a little chart if you want, and put in bigger and bigger numbers. You'll see something that gets closer and closer to that. It's really cool.
So, in fact, that's the number e, and it appears all over the place. In fact, it appears in this beautiful St. Louis arch. This beautiful curve, in fact, is an exponential function, it's a graph of an exponential function, and that exponential function is--I'll tell you exactly what it is, by the way. Well, at least roughly what it is. f(x) = 693.85-something, that's not bad, but then you have to subtract off, so minus 68.76-something--that's no big deal--but then you've got to multiply this number by this, and look what I've got. I've got e to some power, 0.01..., times x plus e to the -0.01..., times x all over 2. So look how the number e, that special number, 2.71-stuff, actually makes its appearance in this formula that gives you this beautiful arch. Now you're sort of looking, in some sense, at a function that has e in it. So it's really a very important function that appears all over. In fact, like I was saying, when you are talking about populations, you are talking, my friends, about e.
Let me tell you a little story about how things grow. In fact, this story is like a bedtime story, so I'm going to make myself a little bit smaller. Watch this. Ready? Here we go. Okay, that's fine. All right, now here's the story. So first off we have a mommy and a daddy, and they like each other very, very much, so sometimes they even can hold hands. You see, they're holding hands. Well, then what happens after a few times they spend together, all of a sudden a child is born. And watch what happens next. Wow! Another child is born. My God, they're out of control. Stop it! Stop it already! Anyway, you get the idea. And, of course, as these children grow, they will have children of their own--maybe not with each other, hopefully. Anyway, the point is, the more population you have, the greater the rate of population growth. Because the bigger the population, there's more partying going on, and so you actually expect to have more population growth. If you just have a very small population, not a lot of partying going on, you expect smaller growth. Well, that basic idea actually captures the models for growth and also indirectly involves the notion of e.
So we'll take a look at e and we'll take a look at the bridge. The only thing we won't take a look at are the details involved in producing the family. That, I'll leave to another lecture. Good look finding us.
Exponential and Logarithmic Functions
The Number e
e Page [2 of 2]

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