The Euler number "e" is a special number with many fascinating properties. The symbol e was first used by Leonhard Euler, who studied the number, but did not discover it. The number e is a transcendental number (it goes on for ever, never repeating itself). Rounded to eight decimal places, e can be approximated as 2.71828183.

### Natural Logarithms and Exponentials

A logarithm is a number that has the following property: If y is the base b logarithm of x, written y = log_b(x), then b^x = y. E is often used as the base for logarithms called the natural logarithms. The natural log is often written as ln rather than log_e. Because of the properties of logarithms, ln(e) = 1. Logarithms are the inverse of exponentials, and ln(x) is the inverse of e^x, which is sometimes written exp(x).

### Calculus

E arises very naturally in calculus. The slope of the function e^x is equal to e^x at every point. In other words, the derivative of e^x is equal to e^x: d/dx(e^x) = e^x. E also emerges naturally in a branch of calculus called differential equations, where it arises in the solution to many problems.

### Growth and Decay

The rate at which water flows through a hole near the bottom of a container is proportional to the current water level. As a result, the level of water at any point in time is a mathematical function of the form Ae^(-Bt) called "exponential decay." Each month that a bank account earns interest, the bank adds a small amount of money to the account that is proportional to the current account balance. This leads to "exponential growth," and the future balance at a time t can be approximated by a function such as Ae^(Bt).

### Complex Numbers

Euler created a mathematical identity using e that links real and complex numbers. It was once voted to be the most beautiful mathematical equation: e^(iπ) + 1 = 0.