Science

Wednesday Is E Day. Can We Make It As Fun As Pi Day?

Feb. 7, 2018, is the calendar date of the mathematical constant 2.718, and we have several ways to celebrate.

Euler's Number 2.718, Euler Day, 2/7/18!
Animation by Lisa Larson-Walker. Images by Wikimedia CC.

Wednesday is E Day. That’s because the date, Feb. 7, 2018, is reminiscent of the mathematical constant e, which starts out 2.718281 (and goes on forever). As with Pi Day 2015, the day, month, and year correspond to the first few digits of this constant, which means it’s time to party. (People who use the day/month/year convention can save this article and celebrate on July 2.)

How do you celebrate E Day? An eggplant-edamame enchilada would be one way—perhaps Pi Day has a bit of an edge on the culinary front. Instead, I think the first thing you should do is invest a bit of money in a savings account, because the first place students usually encounter the number e is in the formula for continuously compounded interest. That formula is A=Pert, where A is the amount of money in the account, P is the principal amount invested, r is the interest rate, and t is the length of time the money has been there.

If your high school interest rates lessons are a bit foggy, here’s a refresher. If you open an account with, say, $1,000 in it and a 10 percent annual interest rate (it’s nice to dream), the amount of money you have at the end of the year actually depends on how often the interest is compounded. If you compound it just once over the course of the year, you’ll end up with $1,100. If you compound it once after six months, the interest you earn in the first half of the year then goes on to earn more interest in the second half of the year. After six months, you earn $50 in interest. In the next six months, that total amount of $1,050 earns $52.50 in interest for a grand total of $1,102.50 at the end of the year. An extra $2.50 might not sound like much, but I’m sure Paul Ryan will be very excited for you. If you compounded three times in the year instead of twice, you’d end up with $1,103.37. The more frequently you compound interest, the more money you end up with, though at some point you’re talking minute fractions of a penny.

Compound interest is the first unambiguous use of the number e that we learned about. (It had secretly made an appearance decades before in tables of natural logarithms, but no one had explicitly looked at it as a number in and of itself.) In 1683, Jacob Bernoulli was studying compound interest. He calculated that the formula for compounding interest a certain number of times over the course of the year, let’s say n times, will be A=P(1+r/n)n. Bernoulli found that if you wanted to compound interest continuously—that is you want every morsel of interest itself to earn interest immediately—you would take the limit of P(1+r/n)n as n went to infinity. The number e shows up when you set r to the value 1, so it is the limit of the expression (1+1/n)n as n goes to infinity. (As a side note, banks often tell you an annual percent yield that takes into account the way they compound interest rather than an annual percent rate.)

Bernoulli himself didn’t figure out e’s value beyond noting that it was between 2 and 3. But a few decades later, Leonhard Euler named it e and found that it was the limit of another expression, the infinite sum 1+1/1+1/2+1/6+1/24+1/120+… . That helped him compute e to 18 decimal places and show that, like π, it is irrational, that is, it can’t be written as a fraction. Though Euler didn’t actually choose the letter e to name it after himself, we now call it the Euler constant. (Sadly, Euler waited until April 15 to be born instead of making an appearance on E Day in 1707.)

The number e is good for a lot more than bank accounts. If you took calculus, you might remember your sighs of relief when you were asked to find the derivative of the function y=ex. The derivative is a measure of how quickly a function is changing at a particular point. Any exponential function has a derivative that is a multiple of itself. For example, the derivative of the function y=2x is approximately (0.69)2x. The derivative of the function y=4x is approximately (1.39)4x. Only when the base is e do you get to discard that clunky number at the beginning: The derivative of ex is just ex. To know how quickly this function is increasing at any given point, you just have to look at the value of the function at that point.

For example, here is a graph of the function y=ex. The slope of the graph at any point (x,ex) is ex:

A graph of the function y=e^x.

That property of the function ex is why some people like to describe e as the natural base for growth, as James Grime explains in this Numberphile video:

Like π, the constant e seems to show up in all sorts of places in math. Both numbers are part of what is often described as the “most beautiful equation,” which says eπi+1=0. Here, e is indeed the base of the natural logarithm and π is the circle constant we know and love. The number i is the imaginary unit, a square root of -1.

I must admit I have my qualms about proclaiming eπi+1=0 to be the most beautiful equation. People are entranced by the fact that it contains five of the most important numbers—0, 1, i, e, and π—but you can easily stuff 0 and 1 into any equation you want by adding 0 or multiplying by 1. A more natural way to write this expression, in my opinion, is eπi=-1. Somehow that doesn’t capture people’s imaginations the same way, even though it is a more compact way of writing the exact same thing. But beyond my cosmetic preference for this writing, the equation eπi=-1 also depends in a deep way on how we define an awful lot of things, and I think the magic is in all of those definitions, rather than one example.

But back to e—is there anything this great number can’t do? It finds its way into population equations, the famous bell curve, calculating probabilities of success in repeated experiments, even making important life decisions about hiring, dating, or what public toilet to use at a music festival. That leaves E Day celebrations wide open. Anything you want to do in life that involves change or growth probably has an e hiding in the background somewhere. So perhaps instead of activities like opening a savings account or going on a date, E Day can be a day for reassessing or renewing those resolutions that might have faded since Jan. 1. Change and growth, here we come!

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