Understanding the Magic of ‘e’: The Mysterious Number Behind Growth

 

When we think about famous numbers, most people know about π (pi). But there's another number, just as important, called e. It may not be as popular outside of math classrooms, but it plays a huge role in many things around us—like money, science, and nature.

So what exactly is e?

What is the Number ‘e’?

The number e is a mathematical constant. It starts as 2.71828 and goes on forever without repeating. Like π, e is an irrational number, which means it can’t be written as a simple fraction.

It is sometimes called Euler’s number, named after the Swiss mathematician Leonhard Euler who studied it deeply.

Even though e looks simple, it shows up in surprising places—like compound interest in banking, how populations grow, how viruses spread, and even in certain areas of physics and statistics.

Where Does ‘e’ Come From?

To understand where e comes from, let’s start with a simple idea: compound interest.

Imagine you put ₹1 in a bank at 100% interest per year.

• If the interest is added just once a year, you’ll have ₹2 at the end of the year.

• If the bank adds interest twice a year, you get a little more:
  ₹1.5 after 6 months, then 100% of ₹1.5, giving ₹2.25 by year-end.

• If interest is added more and more frequently, your money grows faster.

So what if the bank added interest every second of the year? That’s called continuous compounding.

The formula becomes:

$$\left(1 + \frac{1}{n}\right)^n$$

As n (the number of times interest is added) becomes very large, the result gets closer to e.

So, if interest were added infinitely often, your ₹1 would become ₹e, or about ₹2.71828 by the end of the year.

Why is ‘e’ Special?

The number e has some very unique and useful properties:

1. It Grows Like Nothing Else

The function $e^x$ (read as "e to the power of x") is the only one where the rate of growth is the same as its value. That’s why it appears in so many natural growth patterns.

2. It’s at the Heart of Calculus

In calculus, we often work with changing things—like speed, growth, or shrinking. The function $e^x$ is easy to work with because its slope (or derivative) is itself:

$$\frac{d}{dx}{e}^x=e^x$$

No other function does this.

3. It’s Used in Real Life

The formula for exponential growth and decay (like populations, bacteria, or even money) often looks like this:

$$P(t)=P_0{e}^{rt}$$

Where:

• $P(t)$ is the value at time $t$

• $P_0$ is the starting amount

• $r$ is the rate of growth or decay

• $e$ makes the math work perfectly

4. It Connects to Complex Numbers

A truly amazing equation that connects e, π, and i (the square root of -1) is:

$$e^{i\pi} + 1 = 0$$

This is called Euler’s identity, and many say it’s the most beautiful equation in all of mathematics.

e in Everyday Life

You may not see e on your phone screen, but it’s there, hidden in many systems.

• In banking, for calculating continuously compounded interest.

• In science, when studying how populations or chemicals grow or decay.

• In medicine, for understanding the spread of diseases.

• In technology, it helps design computer algorithms that need to model randomness or behavior over time.

Even when scientists model how heat spreads or how atoms behave, e shows up.

Final Thoughts

The number e may seem strange at first. It's not as easy to picture as a circle (like π), and it doesn’t appear in most everyday conversations. But once you look deeper, you’ll find it everywhere — in nature, finance, technology, and even art.

Think of e as the secret rhythm of natural growth. Whether it’s your bank account gaining interest, a tree growing taller, or cells multiplying in your body, e is quietly working behind the scenes.

Understanding e opens the door to many beautiful patterns in the universe. It’s a perfect example of how a single number can tell an endless story.