Understanding Euler's number

December 5, 2025

Euler’s number (e ~= 2.718), possible definitions:

a measurement of the natural rate of growth
the number where continuous growth at a rate of 100% for 1 unit of time gives you exactly e times the starting amount
e is what you get when growth compounds as smoothly as possible

Based on the last definition in that list, I took a stab at defining e as “analog growth as opposed to digital or sampled growth”. (See the analog vs sampled section below.)

The compound interest formula

A = P(1 + \frac{r}{n})^{nt}

Where:

A = final amount
P = initial principal balance
r = interest rate
n = number of times interest is applied per time period
t = number of time periods elapsed

Here’s a demonstration using the above formula that shows how e is what you get when you push compound growth to its absolute limit:

In [15]: P = 1
In [16]: r = 1
In [17]: n = 12

In [18]: P * (1 + (r/n))**n
Out[18]: 2.613035290224676

In [19]: n = 24

In [20]: P * (1 + (r/n))**n
Out[20]: 2.663731258068599

In [21]: n = 365

In [22]: P * (1 + (r/n))**n
Out[22]: 2.7145674820219727

In [23]: n = 3650

In [24]: P * (1 + (r/n))**n
Out[24]: 2.717909554576972

In [25]: n = 100000

In [26]: P * (1 + (r/n))**n
Out[26]: 2.7182682371922975 # approaching e

The mathematical definition of e

e = \lim(n \rightarrow \inf)(1 + \frac{1}{n})^n

Euler’s number as a representation of analog growth (speculative)

In the Python demo above, each value of n is like a different sampling rate for growth:

n = 12 is like sampling growth 12 times a year
n = 365 is like sampling growth daily
as n approaches infinity, you get continuous (analog) growth that converges to e

With this conception, the compound interest formula (1 + 1/n)^n is like a digital sampling of continuous growth. (maybe?)

Where natural growth occurs

population growth: bacteria grow continuously
radioactive decay: atom decay happens smoothly over time

The key is that these processes aren’t scheduled in the way that interest payments are.

Euler’s number in calculus

From Claude: when you see e^x in an equation, read it as “starting from 1, what do you get after x units of continuous growth?” When x = 1, you get e, when x = 2 you get e^2

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