The ABC Book of e

Roughly speaking, e is 2.718.

More precisely, e is the essence of existence, the fount of human joy, and (for folks who worry that Pi Day is kinda played out) the perfect constant around which to build your mathematical festivities (e-clairs, anyone?).

Get excited, citizens of math, because Wednesday, February 7th, 2018 is e Day: 2/7/18.

(Well… in America, anyway. Our international pals may wait until Monday, July 2nd.)

In honor of this noble number, I offer an alphabetical celebration:

e is for Euler, one of the most renowned mathematicians of the last millennium. Euler discovered e, although what’s more impressive is where he discovered it: in the public writings of Jacob Bernoulli, who actually discovered it.

1.jpg

 

e is for Exponential, because Euler couldn’t very well name it after himself, could he? That would be immodest. So he named it after a word that happens to start with the same letter. What a hilarious coincidence!

 

e is for Elegance, because of facts like these:

2.jpg[UPDATE: Thanks to those who pointed out the e-gregious error in the pink formula. Previously I had n approaching infinity, which would result in the limit approaching 1.]

 

e is for Economy, because e lies at the heart of compound interest.

Imagine a wildly generous savings account that pays you 100% interest per year. The question is: When do they pay it?

If they give you 100% at the end of the year, you’ll end up with $2. Not bad.

3

But if they give you 50% halfway through the year, and another 50% at the end, then you’ll end up with more: $2.25. (That’s because you earn interest on the first chunk of interest.)

4

And what if they give you 25% each quarter? Then you’ll end up with $2.44, because you earn even more interest on the interest.

5

And what if they give you 10% at each of ten points throughout the year? You’re up to $2.59!

6.jpg

What about 1% at each of 100 points during the year?

Or 0.1% at each of 1000 points during the year?

Or 0.0001% at each of 1 million points during the year?!

As you carve up the year into finer and finer slivers, each carrying a tinier and tinier interest payment, the total value converges. If you could somehow carve the year into infinite pieces, each carrying an infinitesimal payment, then you’d end up with about $2.718.

Or, more precisely: e.

 

e is for Irrational.

(You wish to complain that “irrational” starts with i, not e? Fool: your rationality has no place here.)

Like its colleagues π and √2—not to mention the overwhelming majority of all numbers in existence—the number e cannot be written as a ratio of two integers.

(In fact – like π, but unlike √2 – e goes beyond irrationality to achieve transcendental status, meaning that it isn’t the solution to any polynomial equation.)

7.jpg

 

e is for EEEEEEK! because e offers a simple demonstration of the dangers of gambling.

Suppose there’s a bet you’ll win 1 in 6 times. So you try it 6 times. You ought to win at least once, right?

Nope. There’s a 33.5% chance that you lose ‘em all.

8.jpg

Okay, what about 100 trials of a bet you win 1 in 100 times? Surely your odds are pretty good?

Not really. There’s a 36.6% probability that you won’t win a single one.

Keep going. What about a million trials at a 1-in-a-million bet? A billion trials at a 1-in-a-billion gamble? The further we go, the closer your odds of utter defeat get to roughly 36.79%.

Or, more precisely: 1/e.

 

e is for Eccentric Eggbert, the Egregiously Error-Prone Butler.

9.jpg

When the guests arrive for a party, they all give their fancy hats to Eggbert. But he forgets whose is whose, and winds up giving them all back to random guests. As the party grows ever larger, what’s the probability that nobody gets back the right hat?

1/e.

 

e is for Events. Why? Because as any statistician knows, aggregating many independent events will yield a normal distribution.

Diffusion of molecules. Weights of animals. Inches of rainfall. All of these can be described by the same family of bell-shaped curves.

10.jpg

(The normal distribution is also called a Gaussian, after legendary mathematician Carl Gauss. This makes perfect sense, seeing as the normal distribution was first discovered by de Moivre.)

And what is the formula for such a curve? Well, the simplest example is this:

11.jpg

Throw some π’s and square roots in there, and you’ve got yourself the normal distribution.

 

e is for Exciting Equality, because in calculus, the function ex has a very exciting property: at every point on its graph, the height is equal to the steepness.

12.jpg

Or, in more calculus-y language:

13.jpg

What, are you not excited? (ARE YOU NOT ENTERTAINED???) Then perhaps I should explain. (Don’t worry; it’s easy; e-lementary, even.)

The language by which many scientists understand reality is differential equations. These are statements describing how quantities are changing. Makes sense; we live in a universe of flux.

One of the most basic and pervasive differential equations is this: A’ = A. In other words: “how fast this quantity is changing depends directly on how big the quantity is.” A bigger quantity changes fast; a small one, slow. Think of a growing population, a nuclear chain reaction, or a burgeoning economy.

The most fundamental solution to this fundamental problem?

ex.

 

e is for Eggheaded Enjoyment. Where its cousin π has gone kind of mainstream, e remains the exclusive estate of the hard-core nerds. And they have a lot of fun with it. Examples:

  • In its 2004 IPO filing, Google announced a goal of $2,718,281,828, or e billion dollars.

14.jpg

  • Google once recruited programmers by posting a billboard that said simply: “{first 10-digit prime found in consecutive digits of e}.com.” (The answer is 7,427,466,391.)

15.jpg

  • Donald Knuth, famed computer scientist, issued a program with version numbers 2, 2.7, 2.71, 2.718, and so on.

16.jpg

 

Finally, e is for the most Exquisite Equation in Existence, Euler’s e-dentity:

17.jpg

Everybody loves this equation. That’s because it’s awesome. Just listen to Keith Devlin:

like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.

The equation is known as Euler’s identity, after the deserving genius Euler. This is altogether fitting and proper, since the formula was most likely discovered by… Jacob Bernoulli’s brother Johann.

POSTSCRIPT: On the name of e and Euler’s identity…

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22 thoughts on “The ABC Book of e

  1. You don’t talk about actually computing e. There are a lot of series that can do this, but I think the Asimov Series is the best:

    e^-1 = 0/1! – 2/3! – 4/5! – 6/7! – …

    Uses each non-negative integer once and only once. All minus signs (except the implicit plus sign before the first term). Efficient: just the four terms above gets you e = 2.71826366016, a relative error of only 1/150,000. As Asimov said in his essay “Exclamation Point!” (reprinted in Asimov on Numbers), you simply can’t ask for anything more beautiful than that.

    1. That’s such a good series! I don’t understand how Asimov wrote like 10^14 books and 10^17 short stories, and still had time to mess around with series like this.

      1. Also 10^15 essays, of which this is one. The man spent his whole life in front of the typewriter (later replaced by a TRS-80), loved to write (even wrote longhand when he was on a cruise, which he did in his later years), and only made two drafts of anything. When asked “What would you do if told you had six months to live?” he replied “Type faster.”

    2. You’ve got the numbers the wrong way round (as should be evident from the first term being zero and all others negative, giving a negative sum while 1/e is positive) and started in the wrong place (skip 0):

      e^{-1} = 1/2! -3/4! -5/6! -7/8! …

      You can derive this easily enough:

      e^{-1} = (-1)^0/0! +sum( (-1)^{2.n+1)/(2.n +1)! +(-1)^(2.n +2)/(2.n +2)!, for n = 0, 1, …, i.e. all natural n)
      = 1/0! -sum( (2.n +2 -1)/(2.n +2)!, for all natural n)
      = 1/0! -sum( (2.n +1)/(2.n +2)! for all natural n)
      = 1/0! -1/2! -3/4! -5/6! -7/8! …

      in which 1 gets repeated, but we can replace 1/0! -1/2! = 1 -1/2 with 1/2 = 1/2! as I gave it initially.

  2. Hi! Great article!
    Please, note that transcendent and Irrational are not synonyms.
    Your illustration may be misleading 😉
    “In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients”
    “In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers”

    1. Yes – this is a fair point. Although, for what it’s worth, the algebraic numbers are a set of measure zero over the reals. With probability 1, any irrational number you pick will be transcendent!

      1. That only works if they’re picking from a uniform distribution over some interval. There are other distributions that give probability less than one of picking a transcendental number. In particular, the probability distribution generated by asking someone “Pick an irrational number” is probably going to include √2 with probability much more than 0.

        Wait, did I just introduce a real-world concern into a mathematics post? Oops.

        1. I think the argument you’re giving is actually a pretty compelling case against the axiom of choice. In any uniform distribution over an uncountable set, there is probability 1 that you pick an element which is noncomputable or equivalently impossible to express. And when the chosen element admits of no finite representation whatsoever, in what sense can you “pick” it?

  3. In your diagram of exp’ = exp, you missed a neat detail worth including: the tangent at x, passing through (x, exp(x)) with gradient exp(x), cuts the x-axis at (x-1, 0), so you can draw a right-triangle with length 1 on the base and the right edge vertical from (x, 0) to (x, exp) and mark that neat base of length 1 to drive home that the slope is the height.

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