Good Math, Bad Math

Looks like I’ve accidentally created a series of articles here about fundamental numbers. I didn’t intend to; originally, I was just trying to write the zero article I’d promised back during the donorschoose drive.

Anyway. Todays number is *e*, aka Euler’s constant, aka the natural log base. *e* is a very odd number, but very fundamental. It shows up constantly, in all sorts of strange places where you wouldn’t expect it.

What is e?
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*e* is a transcendental irrational number. It’s roughly 2.718281828459045. It’s also the base of the natural logarithm. That means that by definition, if ln(x)=y, then *e*y=x. Given my highly warped sense of humor, and my love of bad puns (especially bad *geek* puns) , I like to call *e* the *unnatural natural number*. (It’s natural in the sense that it’s the base of the natural logarithm; but it’s not a natural number according to the usual definition of natural numbers. Hey, I warned you that it was a bad geek pun.)

But that’s not a sufficient answer. We call it the *natural* logarithm. Why is that bizzare irrational number just a bit smaller than 2 3/4 *natural*?

Take the curve y=1/x. The area under the curve from 1 to n is the natural log of n. *e* is the point on the x axis where the area under the curve from 1 is equal to one:

i-943132c94642bfbc16c84895b1b235c5-ln.jpg

It’s also what you get if you you add up the reciprocal of the factorials of every natural number: (1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …)

It’s also what you get if you take the limit: *lim*n → ∞ (1 + 1/n)n.

It’s also what you get if you work out this very strange looking series:


2 + 1/(1+1/(2+2/(3+3/(4+..))))

It’s also the base of a very strange equation: the derivative of *e*x is… *e*x.

And of course, as I mentioned yesterday, it’s the number that makes the most amazing equation in mathematics work: *e*=-1.

Why does it come up so often? It’s really deeply fundamental. It’s tied to the fundamental structure of numbers. It really is a deeply *natural* number; it’s tied into the shape of a circle, to the basic 1/x curve. There are dozens of different ways of defining it, because it’s so deeply embedded in the structure of *everything*.

Wikipedia even points out that if you put $1 into a bank account paying 100% interest compounded continually, at the end of the year, you’ll have exactly *e* dollars. (That’s not too suprising; it’s just another way of stating the integral definition of *e*, but it’s got a nice intuitiveness to it.)

History
——–

*e* has less history to it than the other strange numbers we’ve talked about. It’s a comparatively recent discovery.

The first reference to it indirectly by William Oughtred in the 17th century. Oughtred is the guy who invented the slide rule, which works on logarithmic principles; the moment you start looking an logarithms, you’ll start seeing *e*. He didn’t actually name it, or even really work out its value; but he *did* write the first table of the values of the natural logarithm.

Not too much later, it showed up in the work of Leibniz – not too surprising, given that Liebniz was in the process of working out the basics of differential and integral calculus, and *e* shows up all the time in calculus. But Leibniz didn’t call it *e*, he called it b.

The first person to really try to calculate a value for *e* was Bernoulli, who was for some reason obsessed with the limit equation above, and actually calculated it out.

By the time Leibniz’s calculus was published, *e* was well and truly entrenched, and we haven’t been able to avoid it since.

Why the letter *e*? We don’t really know. It was first used by Euler, but he didn’t say why he chose that. Probably as an abbreviation for “exponential”.

Does *e* have a meaning?
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This is a tricky question. Does *e* mean anything? Or is it just an artifact – a number that’s just a result of the way that numbers work?

That’s more a question for philosophers than mathematicians. But I’m inclined to say that the *number* *e* is an artifact; but the *natural logarithm* is deeply meaningful. The natural logarithm is full of amazing properties – it’s the only logarithm that can be written with a closed form series; it’s got that wonderful interval property with the 1/x curve; it really is a deeply natural thing that expresses very important properties of the basic concepts of numbers. As a logarithm, some number had to be the base; it just happens that it works out to the value *e*. But it’s the logarithm that’s really meaningful; and you can calculate the logarithm *without* knowing the value of *e*.

Comments

  1. #1 venky
    August 2, 2006

    Can you provide a proof that only natural logarithms can be written with closed form series?

    Thanks.

  2. #2 Blake Stacey
    August 2, 2006

    One of my favourite ways of making e intuitive was Feynman’s approach in the “Algebra” chapter of his Lectures on Physics (which I mentioned a few posts ago). The number e arises naturally once you start figuring out good ways to calculate exponents. Say we try to compute powers of a more familiar number, like ten. What is 10x? Well, it’s easy for integer x, but we want to be able to raise 10 to any power we choose.

    One way to make progress is to ask, what is 10x when x is small, much less than 1. The answer has to be a little bit greater than 1, because (10x)1/x equals 10; when you multiply 10x by itself many times, you bring yourself back to 10, meaning that the number you start with can only be a smidgen greater than one.

    Working out the details involves extracting ten successive square roots of ten (using, e.g., Newton’s method). The answer comes out to this: for small x,

    10x = 1 + 2.305x.

    Now, that’s a horribly arbitrary-looking number, 2.305. Is there a way to make the formula appear more “natural”? That is, for what number will raising that number to a small power x just give 1 + x? The answer, of course, is e: this is where the limit formula MarkCC gave above comes from.

  3. #3 Chris Nelson
    August 2, 2006

    Thanks. May I suggest — for those of us who think graphically — that showing how y=1/x and the integral from 1 to n are related would help a great deal?

  4. #4 Bronze Dog
    August 2, 2006

    Let me see if I remember this correctly:

    On a bus, there was a psychopath shouting, “I’ll differentiate you! I’ll integrate you!” All the functions onboard scrambled for the door as soon as the bus came to a stop, afraid of this person.

    One function, however, was unintimidated and walked right up to the threatening individual. “I’m e^x.”

    “I’m d/dx,” was the reply.

  5. #5 David Corfield
    August 2, 2006

    Does e have a meaning?

    Try page 15 of this.

  6. #6 dave glasser
    August 2, 2006

    Bronze Dog, isn’t the punchline “I’m d/dy”?

  7. #7 Bronze Dog
    August 2, 2006

    Probably. I didn’t do well in calculus.

  8. #8 BenE
    August 2, 2006

    I feel that the most intuitive meaning of e comes from the fact that e^x is equal to its derivative. Just knowing this, makes me expect to see it appear in all sorts of areas. I mean surely some phenomenon will have this simple f(x)=f’(x) property.

  9. #9 Craig Stuntz
    August 2, 2006

    I think there’s a typo here:

    It’s also what you get if you you add up the reciprocal of the factorials of every natural number: (1/0! + 1/2! + 1/3! + 1/4! + …)

    Shouldn’t that zero be a one?

  10. #10 John Johnson
    August 2, 2006

    Craig -

    Mark’s formula is correct. 0!=1.

  11. #11 John Johnson
    August 2, 2006

    Actually, I was half right. He left off the 1/1!.

  12. #12 KeithB
    August 2, 2006

    I like e because of Euler’s equation and the fact that it is so easy to memorize to 10 signifigant digits. My teacher in high school said to remember it as “2.7 Andrew Jackson Andrew Jackson”, but that really wasn’t much help for historically ignorant high school students.

  13. #13 Kurt
    August 2, 2006

    Just a nitpick here. I’ve heard e refered to as “Euler’s constant” before, but isn’t that term more properly reserved for this value?

  14. #14 MiguelB
    August 2, 2006

    Another interesting property of e, from my field:

    x(t)=e^{jwt} is the only signal that has the property that, when filtered by a linear system, produces an identical output (up to a multiplicative constant).

    In other words, if F() is a linear system,

    k*x(t) = F(x(t))

    This result is one of the foundations of Fourier analysis.

  15. #15 Chris Nelson
    August 2, 2006

    Where, of course, j is the Fourier way of writing i. Right? ;-)

  16. #16 MiguelB
    August 2, 2006

    Yes, j=i in my post. We EEs tend to write j instead of i, as was noted in one comment on Mark’s article about it. We already use i for current.

    Sorry about that, I’m so used to j that I don’t really think about it anymore ;)

  17. #17 Blake Stacey
    August 2, 2006

    Of course, “j” is short for “jimaginary”.

    (-:

  18. #18 Zombie
    August 2, 2006

    In this post, I have too much fun with Taylor series.

    You can start with the question of integrating 1/x. The normal rule for integrating powers doesn’t work for x^-1. So you just define an unknown function, call it ln, to be this integral – ln(x) = integral[1,x] dt/t. Then d/dx ln(x) = 1/x, that’s pretty straightforward.

    What’s the inverse? Well, dy/dx = 1/x, dx/dy = x (that’s not formal, but I summarize). So you know all the derivatives of the inverse in terms of the inverse. The definition of ln does give you ln(1)=0 so exp(0)=1, which allows you to construct the Taylor series for exp(x) expanded about zero: sum x^n/x!.

    That gives you a couple of the above for free, one of which is the exp(x)=1+x for small x, and also gives you the series for exp(1)=sum(1/n!).

    The Taylor series also allows you to demonstrate exp(ix) = cos x + i sin x, given Taylor series for sin and cos, and you get the constants formula setting x=pi.

    To find a simpler formula for exp(x) (which so far is still just the inverse of a strange function defined by an inconvenient integral), try differentiating d/dx exp(x)^p = p exp(x)^p, and so (d/dx)^n exp(x)^p = p^n exp(x)^p. That works in a Taylor series too, and you get the series sum (px)^n/n!, which is just the Taylor series for exp(px).

    Plug in x = 1 and that leaves you with exp(p) = exp(1)^p, where exp(1) = e.

  19. #19 MiguelB
    August 2, 2006

    Of course, “j” is short for “jimaginary”.

    Or even: “je n’existe pas” (I don’t exist) :-D

  20. #20 Gabriella
    August 2, 2006

    Thanks for your post!!

  21. #21 Brandon
    August 3, 2006

    e also finds itself the limit of a primorial function. See http://mathworld.wolfram.com/Primorial.html next to (3)

  22. #22 QuantumQuery
    August 3, 2006

    pi and e are naturally existing objects as well as mathematical objects.

    pi is circles and e is change of state (growth,decay,movement etc).

    Euler equation is not amazing just a special case; it describes motion around a unit circle in the complex plane.

    i is just an operator that links e and pi since cannot be linked normally as both are transcendental.

    Phi is interesting too, do that next?

  23. #23 Paul "the tree" Carpenter
    August 3, 2006

    An exponceptional blog (my puns are just disgusting), I’ve never quite got my head around why e comes up in real-life situations like finance, items that just happen to depreciate in value following aproximately e^{-14x} are so well, strange.
    If you carry on with fundamental numbers then do take a look at phi, even though it doesn’t seem to do anything. That’s probably part of phi’s awesomeness, the way it doesn’t serve much purpose apart from being pretty.

  24. #24 Blake Stacey
    August 3, 2006

    @MiguelB:

    In lieu of “je n’existe pas,” how about “j’imagine” = “I imagine”?

  25. #25 Pi Guy
    August 3, 2006

    Very cool. As you can see, I am partial to pi but would be very interested in a discussion about phi.

    Addressing Paul C’s comment above, there are quite a number of real-world applications for phi such as the volume of shell sections, the frequency of tree brances, et al. Mario Livio’s book, “The Golden Ratio: The Story of Phi” is an entertaining, readable book on the subject. As Paul mentions, the fact that it appears so frequently and yet does not seem to be fundamental in the same way as i/j, e, pi, zero and one are (re: Euler’s Eq – those five numbers are all intimately intertwined if x = pi) makes it all the more interesting.

  26. #26 QuantumQuery
    August 3, 2006

    Hi Pi Guy

    0 and 1 are only fundamental in mathematics (multiplacative and additive entities) not in nature (unless you allow 0 to mean nothingness and 1 to mean oneness).

    You can rewrite Eulers formula so as not to include 0 or 1.(i to power i = e to power of -pi over 2 but that does not mean that 2 is fundamental, naturally or mathematically).

    So if pi is circles and e is change what is phi in nature?

    In mathematics it is only irrational not trascendental.

  27. #27 MiguelB
    August 3, 2006

    @Blake:

    You win :-)

  28. #28 Canuckistani
    August 4, 2006

    Here’s a neat feature of Euler’s formula:

    f(θ) = cos(θ) + i sin(θ)
    f’(θ) = -sin(θ) + i cos(θ)
    f’(θ) = i(cos(θ) + i sin(θ))
    f’(θ) = if(θ)

    Solving the differential equation, you get

    f(θ) = e

  29. #29 Blake Stacey
    August 4, 2006

    One of my favourite tricks with Euler’s formula is using it to get the sum and difference formulas for sine and cosine. I saw these in high school trig; our textbook proved them geometrically somehow, I forget the details. Neither I nor anybody else could remember them, and I’m sure none of us could produce a derivation upon demand.

    At some point, I discovered that I just had to expand ei(θ + φ):

    ei(θ + φ) = eiθeiφ,

    and pull out Euler’s formula, thus:

    ei(θ + φ) = (cos(θ) + isin(θ))(cos(φ) + isin(φ)).

    Multiply everything out, collect the real and the imaginary terms, and you’re done!

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