The significance of Euler's number (2.718...)

Suppose there is a bank that will give you a great [underlying] rate of interest on your deposit: let us say 100% per year. Yes, I know that is an outrageous hypothetical, but humor me.

If you give this "Bank of Euler" $1 on Dec. 31, then on the following Dec. 31 you can withdraw $2.

Great! But, even better, the bank offers a half-year compounding. You can give the bank $1. on Dec. 31. On the following July 1 the bank will calculate your interest so far, and thereafter your interest will compound onto that new figure. This means that on July 1 they gave you half of the annual amount due you. In this case: fifty cents.

So as of July 1 you have $1.50 in the bank. The rest of the year you earn NOT the other 50 cents, but half of the annual rate on that new number. This is $0.75.

On Dec. 31, then, you will have the original $1.00 plus the mid-year $0.50 plus the end of year $0.75 in the bank. This is: $2.25. Better.

By compounding the interest (just this once) we have switched from a single calculation system to a two calculation system. Let us call the number of calculations N. There is a clear gain when we switch from N=1 to N=2.

Still better: the Bank of Euler decides to keep increasing N Compound once a month. N = 12. Once a week. N = 52. Once a day. N = 365. You will find that the total yield, the amount you get on Dec. 31, continues to increase each time, but by a decreasing amount. We are approaching a limit.

So. What if we go with truly continuous compounding of interest? What is the yield as N approaches infinity?

The answer is an irrational number, akin to pi. The digits never repeat and never stop. You will probably get $2.72 as a matter of rounding out. But the true irrational number, and so the amount you will as a matter of principled principal have in the account at the end of that first year of compounding, is $2.71828...

And THAT, deprived of the dollar sign, is e, Euler's number, also known as the base of the natural logarithm.

SO ... the number e is very useful. Most obviously, we calculate the results of continuous compounding with its help even when the numbers involved are not so convenient.

The formula for doing so even resembles the English word "Pert," which is odd and makes it easy to remember. Also reminds some of us of shampoo, as in the image above.

Y = Pert.

How mnemonically convenient is THAT? The principal invested with continuous compounding, times e, to a power defined by the underlying rate and the time the principal is left in there, tells us the YIELD or Y.

If you want the PROOF of PERT, here is one of the many websites that expounds it for you:

https://www.cuemath.com/continuous-compounding-formula/

This means that e is always showing up in financial mathematics. But, more than that, it shows up in every scientific discipline. Y = Pert. is the formula for continual growth and decay everywhere, with Y serving

to mean the total amount of growth/decay, P a beginning for the observed process.

And here you thought math was boring. What could be closer to the antonym of boring than ... PERT!

I want to credit here the best explanation of this subject I've ever read, a long answer in a question at QUORA, by Mihalis Boulasikis.

https://www.quora.com/How-and-why-is-Euler%E2%80%99s-number-e-everywhere

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