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).

1. The number e is undoubtedly one of the most important numbers in both pure and applied mathematics. This has to do primarily with its significance in growth processes. Properties of the exponential function—including those in the domain of complex numbers—were studied extensively by Euler, and as we mentioned above, the number e is so named in his honor.
2. Like the number ?, the number e belongs to the most complicated class in the number hierarchy. That it is irrational can be shown without much difficulty from elementary properties of the exponential function. It is much more difficult to show that e is transcendental as well. This latter fact was proved in 1873 by Hermite.
3. In contrast to ?, there is no fan club for the number e. Nor is there much prestige to be won by setting the world record for the number of decimal digits of e. This is because the above formula is so simple that one needs only elementary knowledge of computer programming to determine as many decimal digits as desired.

4. The formula

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

implies the remarkable fact that the function ex can be defined not only for real numbers x. This formula has meaning in all areas of mathematics in which sum, product, and convergence are defined. This is of importance first for the extension of the real numbers to the complex domain, but this construction also plays an important role in many high-level relationships. (For example, one can calculate ex when x is a square matrix.) The extension of the exponential function to the complex domain plays an important role in the explication of one of the most interesting formulas of mathematics.

(Ehrhard Behrends, Freie Universität Berlin)