Mathematics
Euler's Number (e)
Quick fact
If you invest $1 at 100% annual interest compounded continuously for one year, you get exactly $2.71828...—that is, $e$.
Why this is interesting
You’ve probably heard of π, but there’s another magical number—about 2.71828—that describes how things grow continuously. Why does this number show up everywhere from bank accounts to virus spread?
Read the full explanation
Understanding Euler's Number (e)
Imagine you deposit $1 in a bank that offers 100% annual interest. If interest is paid once at the end of the year, you get $2. If it's compounded semi-annually (50% each half), you get $2.25. Compounding monthly yields about $2.61. As you increase the number of compounding periods (daily, hourly, every second), the total approaches a limit: $2.71828... That limit is e. It represents the maximum you can get from 100% growth when growth is applied continuously—every instant. Think of it as the 'natural' rate of growth when there's no pause between compounding events.
A deeper explanation
Mathematically, e is defined as the limit of (1 + 1/n)^n as n approaches infinity. Its deepest property is that the function f(x) = e^x is its own derivative: the slope of the curve at any point equals its height. This self-referential quality makes e the natural base for calculus. Because of this, e appears in differential equations describing population growth, radioactive decay, cooling, and electrical circuits. The natural logarithm (ln) is the inverse of e^x, so it's the tool to solve for time in continuous growth problems. Euler's number isn't just a constant—it's the DNA of continuous change.