When it comes to mathematical constants, π is probably the most celebrated one. It’s got π-day on March 14th, it’s got hundreds of songs singing its digits, and it’s got a one-line definition that everybody can recite: the ratio of the circumference of a circle to its diameter. But things are not so delightful with the life of e—another equally worthy-of-being-celebrated mathematical constant.
The value of e is 2.718281828459… . It is irrational, meaning that its digits go on forever and ever without repeating. Sounds messy, huh? To many’s surprise, it has a rather shocking and seemingly unfitting name, “the natural base.”
Why does it take such a strange value? And how on earth is it “natural”? You might ask. In this series of articles, we will try to intuitively understand how e is defined, explore e’s “naturalness”, and towards the end, dive into the past and hear about e’s birth and its early life. Buckle up!
One definition of e is
No sweat if you have trouble understanding the meaning of this expression—that’s exactly why we’re here. In this section, we will try to intuitively understand the number e.
Although e is often explained with compound interest, it also arises in biology with cell mitosis in an extremely simplified model. Apparently, cells are not the only ones that are mitotic. In The Big Bang Theory, Sheldon also underwent mitosis after eating too much Thai food…
… of course, in Leonard’s nightmares. One Sheldon is already enough to deal with.
But let’s think about this process: taking Thai food to be the fuel for Sheldon’s mitosis, it takes exactly 1 box of Thai food for one Sheldon to divide into two. At the start, there is only one Sheldon. After one box of Thai food, there are 2 Sheldons. But what happens in between?
For example, at the half-point between 0 and 1, shouldn’t we expect something more than a single Sheldon? After all, Sheldon already ate half a box of Thai food. It’s reasonable to expect the new Sheldon to grow gradually, at a constant rate (1 Sheldon per box of Thai food), out of the old Sheldon. So, in fact, we have something more like the following:
Note that so far we’ve been assuming that mitosis only happens once per box of Thai food. What do you say we make things more interesting? Look at the half blue sheldon at ½. Let’s say that once half a box of Thai food is consumed, even the new Sheldon can undergo mitosis. The situation is becoming increasingly complicated…
Let’s calculate how many Sheldons we have at this point. Over the course of 1 box of Thai food, Sheldon(s) underwent mitosis twice. Once at ½ box, once at 1 box. Since we’ve assumed a constant rate of growth for the new Sheldons, that is 1 per box, we see
# Sheldon at ½ box = 1 + (1 * 1/2) = 1 + 1/2
And since the number of Sheldon at 1 box is growing from 1 + 1/2, rather than 1, we have
# Sheldon at 1 box = (1 + 1/2) + ((1 + 1/2) * 1/2) = (1 + 1/2)(1 + 1/2) = (1 + 1/2)2 = 2.25
We have more Sheldons than doubling! Do I hear Leonard cry?
This is not the end! We can make even more Sheldons. Suppose once ⅓ of a box of Thai food is consumed, the Sheldons can undergo mitosis. And of course, at ⅔ box, they undergo mitosis again.
Let’s calculate how many Sheldons we have this time:
# Sheldon at ⅓ box = 1 + (1 * 1/3) = 1 + 1/3
# Sheldon at ⅔ box = (1 + 1/3) + ((1 + 1/3) * 1/3) = (1 + 1/3)(1 + 1/3) = (1 + 1/3)2
# Sheldon at 1 box = (1 + 1/3)2 + ((1 + 1/3)2 * 1/3) = (1 + 1/3)2(1 + 1/3) = (1 + 1/3)3 = 2.37
In this case, we end up with roughly 2.37 Sheldons. That’s even more than before!
In fact, if we do this properly, it seems likely that with each bite, the Sheldons will undergo mitosis. If we assume one bite is 1/10 of the box, then after eating the whole box, the Sheldons will have undergone mitosis 10 times.
Observe that in our calculation earlier, when Sheldon underwent mitosis 2 times, we had (1 + 1/2)2 Sheldons; and 3 times, (1 + 1/3)3 Sheldons. Can you spot a pattern here? Continuing in this way, after 10 cases of mitosis, we will have (1 + 1/10)10 = 2.5937 Sheldons. If he takes smaller bites, say one bite is 1/100 of the box, there will be (1 + 1/100)100 = 2.7048 Sheldons.
Ah, I definitely hear Leonard cry now—we keep getting more and more Sheldons! If Sheldon just takes really, really, really small bites, can we get infinitely many Sheldons?
Luckily for Leonard, the answer is no. Although the value of (1 + 1/n)n grows bigger as n increases, the value eventually approaches a ceiling, and this is the value of e came. Here’s a plot of (1 + 1/x)x showing exactly this:
And this is how we understand the “lim” in the definition given at the top of the page: no matter how big n gets, the value of (1 + 1/n)n will eventually stay very, very close to e. We can make it as close as we want by just taking larger and larger values of n.
So whilst Leonard will undoubtedly be terrified by the thought of 2.718… Sheldon’s running, at least he knows there can never be any more – no matter how much Thai food Sheldon eats!
All images copyright ‘The Big Bang Theory’/Warner Bros Production.