The number e

Most high-school graduates can give the first few digits of pi (3.14159...), and many can even say what this famous number means: you take a circle one meter in diameter, and when you measure the circumference you will get exactly pi meters. Although pi seems to be "about" circles, it turns up in bizarre places in math that, at first glance, seem to have no circle-ness about them whatsoever. This spooky pervasiveness is what makes pi so interesting.

But very few people who haven't particularly studied math can even say approximately how big e is, even though this number, like pi, turns up all over the place in mathematics. I won't keep you in suspense: e is between 2 and 3, rather closer to 3 than to 2; to five digits accuracy it's 2.71828....

Pi and e have many things in common: they are both irrational numbers, which means that they are not exact integer fractions; no matter how big you make the denominator, you cannot capture either pi or e as a fraction. This is equivalent to saying that when written in decimal notation, pi and e both go on forever without repetition: in fact, the digit strings for both look completely random. Both numbers are known to be transcendental, which is a kind of super-irrationality. (Rational numbers are the solutions of first-degree integer polynomial equations; algebraic numbers are the solutions of polynomial equations of any degree; and transcendental numbers fail to be algebraic.)

OK. If pi is about circles, what is e about? It's a little hard to answer, partly because e, like pi, turns up in so many unexpected places. But one possible answer is that e is about exponential growth. The easiest way to think about this is to consider compound interest.

Suppose you have a bank account earning 4% interest per year. (I know, dream on.) If you deposit $100 on your birthday in 2013, and then empty the account on your birthday in 2014, you'll have $104; the bank is paying you four cents on the dollar for the privilege of getting to invest your money for a year. But what if you take your money early? Do you get any extra money? It depends on the exact rules of the account. If it says, "interest compounded yearly", then you get no interest until the bank has had your money for a whole year. But many accounts say "interest compounded quarterly", which means that instead of giving you your four cents on the dollar at the end of the year, they give you one cent on the dollar every three months.

This is a better deal for you than yearly compounding. Not only do you get some gain if you have to take your money out early, but after a year you have more money with quarterly compounding. To see why, we have to run the numbers. You deposit $100 on your birthday. Three months later, the bank pays you one cent on the dollar; this comes out to a whole extra dollar, so now you have $101. Three months after that, the bank again gives you one cent on the dollar. Since you now have $101 dollars, the bank pays you 101 cents, or $1.01. So after six months you have $102.01. Three months after that, the bank again gives you a cent per dollar; neglecting fractional cents this comes to $1.02, so you now have $103.03. Finally, on your birthday in 2014, they pay you $1.03, and you finish the year with $104.06. I know that was a lot of arithmetic to go through, but the bottom line is that quarterly compounding got you an extra six cents over the whole year. What they call the effective interest rate is 4.06% instead of only 4%.

Quarterly compounding is also fairer than annual compounding. Even having your money for one day benefits the bank; the world of high finance is full of things like enormous overnight loans, and the prices of financial assets fluctuate from day to day, so a canny trader is trying to make money constantly, from hour to hour. Maybe it would be fairer for the bank to pay you one 365th of the yearly interest every day? How much benefit would that give you? I just did the math so you don't have to: daily compounding would turn your $100 into $104.08 in a year. You get eight cents extra on a hundred dollars with daily compounding; the advantage over quarterly compounding is two cents on a hundred dollars. Maybe it's not as much as you expected. Even compounding every second gives only a tiny extra advantage over that.

Now we have to make a leap, and without making you sit through several weeks of class you'll have to take my word for it that the leap is justified. With the magic of calculus, we can create an account in which interest is compounded continuously. You would think that this would get you an infinite amount of interest: after all, the interest is paid to you an infinite number of times. But fighting with this is the fact that the amount of interest given with each award has shrunk to an infinitesimal amount. Who wins, the infinity, or the infinitesimal?

To make matters simpler mathematically, imagine that the interest rate is a whopping 100% per annum. Again you deposit $100 on your birthday. With annual compounding, you walk away next year with $200. With quarterly compounding, you would get $244.14. With daily compounding, $271.45. And with continuous compounding ... will it be infinite? No. The answer is perhaps counterintuitive, but with continuous compounding you walk away with $271.83 minus a couple of tenths of a cent.

We have set up a situation where we have a quantity whose growth rate is equal to the quantity itself. After one time unit has passed, such a growing quantity will always be found to have increased by exactly the same ratio: 2.71828... to 1. This ratio is the number e. It is one greater than the effective interest rate of a bank account that pays 100% per year, compounded continuously. (The effective interest rate is 171.828...%, but you get back your original investment too, so your money has grown by a factor of e.)

Continuous exponential growth is a situation that comes up frequently in mathematics, and every time it does, you can expect e to turn up in the analysis. (The effective interest rate of an account paying 4% per annum, compounded continuously, is e raised to the power of 4%, minus 1.)

Computing e is surprisingly easy. The classic procedure is to start with 1, add 1, add 1/2, add 1/6, add 1/24, and so on: the denominators of those fractions is calculated by starting with 1 and multiplying by 2, 3, 4, and so on, so the next fraction to add is 1/120, because 24 times 5 is 120. This procedure gives a very close approximation to e after very few steps. It is intriguing that nobody has been able to find a procedure that is so inexpensive for calculating pi; e seems to be intrinsically easier to calculate than pi, though no one has succeeded in proving this; it's an open mathematical question and if you could settle it you would be famous in the mathematical world.

I mentioned that e comes up in surprising places, so I'd like to conclude with a little story to illustrate the weird places that e tends to hide. The story is true in outline, though I never knew most of the details, so I will have to tell it vaguely. There was a television program on which a man who was skeptical of astrology issued a challenge to an astrologer. The astrologer would be presented with a panel of twelve volunteers, each of which was born under a different sun sign, so the panel would contain one Aries, one Taurus, one Gemini, and so on around to Pisces. The astrologer claimed that the birth sign affected the personality in ways that a trained astrologer could detect; he thought that given time to interview the panel, he could correctly assign each panelist to their proper sign. This was a very strong claim, and the skeptic agreed that if the astrologer could do that without chicanery, the skeptic would have to admit that there was something to astrology. (The claim was strong because the odds of getting all twelve signs right by blind chance is a smidge better than 1 in 480 million.)

The astrologer was given all the time he wanted to interview the panel, and finally wrote down his guesses on cards which were handed face down to the panelists. The panelists were instructed to turn over their cards and stand up if the astrologer's prediction was correct. With appropriate dramatic flourish, the panelists turned over their cards and every one remained seated. The astrologer had failed as badly as it was possible to fail, assigning every single panelist to the wrong sign.

That's the end of the story proper, but any moderately curious person might think, "Hey, getting every single one wrong is sort of unlikely too, isn't it? Wouldn't one expect the astrologer to have gotten one or two right, just by luck?"

If there were only two people on the panel, the odds of getting both signs wrong would be 1 in 2. If there were three people on the panel, the odds would be 1 in 3. If there were four people, the odds would be 3 in 8, or about 1 in 2.667. As the number of people goes up, the odds settle very quickly to about 1 in 2.718; some mathematical sleight-of-hand reveals that the limiting odds are exactly 1 in e. (For twelve panelists, the odds differ from this ideal by only a few parts per billion.)

So yes, it's a bit unlikely to get them all wrong, but no supernatural explanation is needed for something that would happen about one time in three by chance. (If an astrologer could get them all wrong for, say, ten successive panels, then I would have to agree that maybe there was something weird going on.) But I think you will agree that it is odd to see e pop up in this question about probability. And that's just the beginning, but the deeper ubiquity of e (and the utterly gobsmackingly beautiful deep connection between e and pi) are way beyond my scope.