BetterExplained » Q: Why is e special? (2.718…, not 2, 3.7 or another number?)

A common question is why e (2.71828...) is so special. Why not 2, 3.7 or some other number as the base of growth?

First off, e was discovered, not chosen. Think of the speed of light, c. It wasn't originally decided to be 299,792,458 m/s -- we did experiments and realized under ideal, universal conditions (a vacuum), this was the fastest light could move¹.

Let's look at growth and ask under ideal, universal conditions, what's the fastest something can possibly grow? Ideal, universal assumptions would be:

Growing by the unit rate (100%)
Growing for the unit time (1 time period)
Growing perfectly, without any delay (continuous)

Turning these assumptions into a formula, we get:

If we actually use the formula (using large values of n for more accuracy) we get e = 2.718281828459...

Objection: But $13.74^x$ can model exponential growth just like $e^x$ can!

Sure. But what assumptions did you make to get 13.74? They probably weren't "unit rate, unit time, perfectly compounded". (You can pick k as the speed of light through Kool Aid too -- but why?)

Arguably, $2^n$ is also universal ("the discrete e"), because you have zero compounding (n is an integer like 0, 1, 2, 3). Instead of perfectly continuous, it's perfectly non-continuous (discrete), and we take growth step-by-step.

So, I'd say either $2^n$ (in discrete systems) or $e^x$ (in continuous systems) are "universal".

Objection: But things can grow faster than $e^x$, which is just $2.71828^x$ -- what about $13.74^x$?

What is it with you and 13.74? Yes, you can beat $e^x$ in an exponential footrace, if you use a rate more than 100%. $13.74^x$ is really $[e^{\ln(13.74)}]^x$. Because ln(13.74) ~ 2.6, you are assuming a 260% continuous interest rate, more than the 100% $e^x$ uses. (Alternatively, you can grow for 260% of the unit time period that $e^x$ uses.)

Funny enough, in 1983 c was decreed to be 299,792,458 m/s by redefining the length of a meter. Similarly, you could decide that e is a clean "10" in your base-e number system. ↩

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