Imagine you have a coin, and I tell you, “Give me this coin, and in a year I’ll give you $100\%$ of the value you put in.” That means in a year, you’ll get another coin.

So after a year, you’ll have $1 + 1$ coins. Now, if I told you I wouldn’t give you the full $100\%$ right away, but would give you $50\%$ after 6 months, and another $50\%$ after another 6 months, how much would you have after a year?

After 6 months, you’ll have

$1 \times 1.5 = 1.5$

And after the next 6 months, you’ll have

$1.5 \times 1.5 = 2.25$

Okay, what if I said 3 months? That means you’ll get $25\%$ every 3 months until a year is up. How much will you have?

After 3 months, you’ll have $1 \times 1.25$

After 6 months, you’ll have $1\times 1.25 \times 1.25$

After 9 months, it will be $1 \times 1.25 \times 1.25 \times 1.25$

After 12 months, it will be $1 \times 1.25 \times 1.25 \times 1.25 \times 1.25$

We can see a pattern. Ultimately, the number will equal $1.25^4$

We can express $1.25$ as $1 + 1.25$, which is $(1 + \frac{1}{4})^4$, and that equals $2.44140625$. Notice that our amount grows every time we calculate our interest over a longer period.

Suppose we want to do the same thing every month, every week, or every day, all the way down to every second and nanosecond, to get the largest possible value?

Our formula will be $(1 + \frac{1}{n})^n$.

If we keep doing this forever and use a very large number—$(1 + \frac{1}{100000})^{100000}$, for example—we’ll notice that our number approaches a specific value: $2.71828$.

It turns out that this number is so important that we’ve given it a special symbol: $e$.

One of the most interesting things about this number is that it has no end. As proof of this, I’d like to introduce you to Euler’s number.

$e$