# Euler’s Number “e”: A Mathematical Marvel

When we think of famous numbers in mathematics, we often recall π (pi) or the imaginary unit *i*. However, another number stands shoulder to shoulder with these in its significance and ubiquity: **Euler’s number, denoted as e**.

The number *e* is often linked to Leonhard Euler, but it was first found by Jacob Bernoulli in 1683. While Bernoulli wasn’t specifically searching for a new mathematical constant, he inadvertently discovered the first approximation of *e* during his studies on continuous compound interest. **This makes the story of e particularly intriguing; it wasn’t the product of abstract mathematical thought, but rather a result of tangible financial studies**.

*e* is approximately equal to 2.71828, though like π, it’s an irrational number, meaning it has an infinite number of decimal places without any repeating pattern.

**e = 2.71828182845**

### Common definitions of *e*

The number *e* is a mathematical constant that is approximately equal to 2.71828. It can be defined in various ways, but here are three of the most common definitions:

**Limit Definition**

The number *e* can be defined using a limit as:

**Sum of an Infinite Series**

*e* can also be expressed as the sum of an infinite series

**Compound Interest Definition**

As Jacob Bernoulli discovered, the number *e* arises in the study of continuously compounded interest. If you invest a sum of 1 unit of currency at an annual interest rate of 100%, and let the interest compound more and more frequently, the amount of money approaches *e* when the interest is compounded continuously:

where *n* is the number of times interest is compounded per year.

All these definitions converge to the same value for *e*, which is approximately 2.71828.

### What makes *e* so special?

**Exponential Growth and Decay**: the natural exponential function *e*^*x* is central to understanding various phenomena that involve growth or decay, such as populations, radioactive decay, and, as Bernoulli discovered, compound interest.

**Calculus**: *e* plays a pivotal role in calculus. The function *e*^*x* is unique because it is its own derivative. This property is foundational to many concepts and calculations in differential equations and other advanced areas of mathematics.

**Complex Numbers**: in the realm of complex numbers, Euler’s formula *e^ix* = cos(*x*) +* i* sin(*x*) is a profound relationship between exponential functions and trigonometric functions. It’s through this formula that we get the beautiful Euler’s identity: *e^iπ* + 1 = 0, which astonishingly connects five of the most important numbers in mathematics: 1,*e*,*i*,*π*,1, and 0.

### Open questions

As with all things in mathematics, there’s always more to explore. Some intriguing questions and topics about *e* include:

- How do other mathematical constants, like
*π*and the golden ratio, relate to*e*? - Why does
*e*appear in so many different areas of science, from biology to quantum physics? - Are there undiscovered applications of
*e*in modern technology or future innovations?

Remember, every answer in mathematics often leads to a dozen more questions. So, the next time you see *e*, give a nod to its mystery and wonder. Who knows what we might discover about it next?