e - the Unnatural Natural Number

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

Thanks.

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.

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?

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.

Does e have a meaning?

Try page 15 of this.

Bronze Dog, isn't the punchline "I'm d/dy"?

Probably. I didn't do well in calculus.

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.

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?

Craig -

Mark's formula is correct. 0!=1.

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

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.

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?

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.

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

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 ;)

Of course, "j" is short for "jimaginary".

(-:

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.

Of course, "j" is short for "jimaginary".

Or even: "je n'existe pas" (I don't exist)
:-D

Thanks for your post!!

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

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?

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.

@MiguelB:

In lieu of "je n'existe pas," how about "j'imagine" = "I imagine"?

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.

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.

@Blake:

You win :-)

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) = eiθ

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 + Ï):

ei(θ + Ï) = eiθeiÏ,

and pull out Euler's formula, thus:

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

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