e is also called Euler's number, named after the mathematician Leonhard Euler, one of the GOATs of mathematics (just like Messi or Ronaldo in football). The number itself comes from a very simple question about growth.
Start with the familiar compound interest formula:
A = P(1 + r/n)nt
Here, P is the starting amount, r is the annual interest rate, n is the number of times interest is added in a year, and t is the number of years.
Suppose the interest rate is 100 percent per year, we start with 1 dollar, and we look at one year. If interest is added once, then n = 1, and the amount becomes 1 + 1 = 2.
If interest is added every six months, then n = 2. The amount becomes (1 + 1/2)2 = 2.25.
If interest is added four times in the year, it becomes (1 + 1/4)4 ≈ 2.441. If interest is added every month, then n = 12, and the amount becomes (1 + 1/12)12 ≈ 2.613.
Continuous growth
Now keep pushing the same idea. What happens if interest is added every second, then every tiny fraction of a second, and then continuously? As n moves toward infinity, the amount approaches:
(1 + 1/n)n → 2.718281828...
That number is e. In this sense, e represents smooth, continuous growth. It is what appears when growth is not happening in a few visible jumps, but is being updated constantly.
Why e keeps appearing
e also has a beautiful mathematical property: the function ex grows at a rate proportional to its own size. More precisely, d(ex)/dx = ex. Its slope is itself.
That is why e appears so often in natural systems. Bacteria growth, the spread of information, and radioactive decay can all be described using equations where the rate of change depends on the current amount. In decay, the direction is negative, but the core idea is the same: the present amount shapes the next instant of change.
That is also why e shows up so much in machine learning. It gives us a natural way to represent smooth curves, growth and decay, probabilities, and transformations whose rates of change behave nicely. It sits behind functions such as the logistic sigmoid and softmax, where raw scores are turned into bounded values or probability-like distributions.
The beauty of e is that it starts from an ordinary question about compounding, and then keeps reappearing anywhere change is continuous and proportional to what is already there.