Math with Bad Drawings

The ABC Book of e

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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.

 

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:

[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.

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.)

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.

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

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.)

 

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.

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.

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.

(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:

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.

Or, in more calculus-y language:

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:

 

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

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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