The number e can be defined in a variety of ways:

Method 1
In high school, the number e is often introduced in the context of continuous compound interest.
This can be understood as follows. Imagine that a bank offers 100 percent interest on deposits, with interest compounded annually. After one year, a deposit of one euro would increase to two euros.
If interest is now compounded every half-year, then at the end of the year, a deposit of one euro will have grown to (1+1/2)2.

By analogy, we see that if the year is divided into n equal segments, then through n compoundings over the course of the year, a deposit of one euro will have grown to the value of (1+1/n)n euros. The question then arises whether by taking n larger and larger we would have an increasingly large bank balance at the end of the year.

Surprisingly, the numbers (1+1/n)n do not grow without bound, but approach a limiting value, namely the number

2.71828182845905….

This number is so important that it has been given its own special symbol, the letter e, in honor of the Swiss mathematician Leonhard Euler.

Method 2:
The number e arises in a natural way in growth processes. Imagine a population of cells or organisms or anything else in which the rate of growth of the population is proportional to the existing population. In particular, if at time t there are f(t) individuals, then the growth in the population over the next s time intervals will be a•s •f(t) (here a is some constant that represents, say, the population’s fertility). We may write this as follows:
[f(t+s)-f(t)]/s=a•f(t).

And as s approaches 0, we have the formula f'(t)=a•f(t), where f' denotes the derivative of f.

It suffices now to consider the special case a = 1, since the general situation differs only by a scaling factor.

We may consider, then, the following problem: find a function f for which the relationship f'= f holds (that is, find a function whose derivative has the same value as the function itself) and for which f(0) = 1 holds as well. The latter condition is simply a practical normalization.

One can then prove that there exists a unique function with these properties. It is called the exponential function, and its value at a number x is denoted by exp(x). The number e is then the value of this function at the point x = 1. We then have that exp(x) = ex for all real numbers x.

Method 3
For concrete calculations, it is often useful to know that exp(x) can be written as an infinite sum:

exp(x) = 1 + x +x2/2! + x3/3! + ...;

recall that n! ("n factorial") stands for the product 1•2•...•n.

With the substitution x = 1, we obtain

e = 1 + 1+ 1/2! + 1/3! + ...

Since the denominators in this sum become very large very quickly, it is possible to approximate e very well using only a few terms.

Method 4
Those who understand what is meant by the area under a curve can define e as the value of x for which the area under the curve 1/x between 1 and x has the value 1. That this method yields the same value for e as in the previous methods follows from elementary rules of integration (the indefinite integral of the function 1/x is the logarithm function, whose value at x = e is 1).