The constant e
e is one of the most important and fascinating numbers in mathematics. It’s approximately equal to 2.718, and like π, it’s an irrational number, meaning it never ends or repeats, and it quietly shapes huge parts of mathematics, science, and even everyday life.
The origins of e
The constant e was first discovered in the 1600s by Jacob Bernoulli, while studying compound interest: how money grows when interest is added not just to the original amount, but also to the interest that has already built up. He asked a simple but clever question: If you invest £1 at 100% annual interest, and the interest is added more and more frequently, what happens to your total after one year?
The formula for compound interest is:
A = (1 + 1/n)n
where n is the number of times interest is added per year.
If you compound once (n = 1), you get £2.
If you compound twice (n = 2), you get £2.25.
As n increases, the total approaches a limit which ends up being e:
limn→∞ (1 + 1/n)n = e
That limit is the constant e. It represents the maximum growth possible when interest is compounded continuously: every instant rather than just occasionally.
Why e is called ‘natural’
The constant e is known as the natural base, because it describes natural growth and decay better than any other number.
The function f(x) = ex has a remarkable property:
d/dx ex = ex
This means the rate of change (the gradient) of ex is equal to its own value at every point. No other number has this property. In simple terms, e is the perfect balance between growth and rate of growth. It’s what makes things grow naturally.
Just like π can be written as an infinite series, e can be expressed as the sum of infinitely many fractions: e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + … . The ‘!’ symbol means factorial, where 4! = 4 × 3 × 2 × 1 = 24. This series shows how e is built by adding smaller and smaller pieces, and it’s one of the most elegant links between algebra, calculus, and infinite processes.
How e appears in real life
Even though e might seem like a strange idea from pure maths, it shows up all around us in the real world.
Exponential growth and decay
Processes that grow or shrink continuously — like bacteria multiplying, populations changing, or radioactive materials decaying — all follow the same pattern:
N(t) = N0 ekt
Here, N₀ is the starting amount, k is the growth (or decay) rate, and t is time. This shows us how e can affect and predict the growth of populations and so much more.
Compound interest
Banks and financial institutions use e too.
If interest is compounded continuously (meaning it’s added constantly), the total amount after t years is:
A = Pert
where P is the initial investment, r is the interest rate, and t is time in years. This formula comes directly from the limit that defines e, the same one Bernoulli discovered centuries ago.
Probability and statistics
In statistics, e plays a key role in the normal distribution, the famous bell-shaped curve used to model things like test scores, heights, and natural variations:
f(x) = 1/√2πσ2 e-(x – μ)^2/2σ^2
Here, μ (mu) is the mean, and σ (sigma) is the standard deviation. Without e, this elegant and essential curve wouldn’t exist. It’s what makes the shape ‘natural’.
If you take e to the power of πi and add 1, you get 0: eiπ + 1 = 0. This is known as Euler’s Identity, and it’s often called the most beautiful equation in mathematics. It connects five of the most fundamental numbers — e, π, i, 1, and 0 — all in a single, perfectly simple equation. Here’s where e truly shines. It’s a moment where algebra, geometry, and analysis all come together.
The number e may not be as famous as π, but it’s just as essential, and possibly even more powerful. It appears wherever there’s continuous change: in finance, physics, biology, and statistics. It connects growth, decay, and probability to the deeper structure of mathematics itself.
So next time you see the letter e in a formula, remember: behind that single symbol lies one of the greatest discoveries in all of maths: the constant that explains how the world grows, changes, and evolves.