e is an irrational real number that has interesting properties, much like that number pi that seems to be so popular.

How do we define e?

For the purposes of this writeup, we'll say that e is an irrational number that follows the definition that d/dx(e^x)=e^x.

What does this definition of e mean and imply?

Returning to arithmetic and functions, this definition depends on the nature of an exponent. By raising e to a certain exponent x, written e^x, one is simply multiplying e by itself x times. (See writeups on exponents for more details on this)

The statement above refers to the graph of e^x. At any point on the graph, because it is a continuous and differentiable function of x, there can be a line drawn tangent to the graph, so that the line touches the graph at that point only. Think of a ball resting on flat ground. The ball is tangent to the ground at the point of contact. According to this definition of e, the slope of a line tangent to any point on the graph of f(x)=e^x is equal to the value of that value of f(x). It's crazy, but it works.

Another interesting thing about e

The alternate definition of e is the base of the natural lograthm, ln(x), which is defined as the integral ∫₁ 1/x dx. This is an interestingly simple definition that has many applications in calculus.

So how do we calculate e?

This number, like π, has a simple definition, but cannot be represented as a simple rational or the value of an easily evaluable elementary function.

The answer lies within a concept called Taylor series, and more specifically, Maclaurin series (knowledge of some calculus required). A Taylor series polynomial attempts to reconstruct the behavior of a function around a certain point. A Maclaurin series polynomial is a Taylor series polynomial with its "seed point" being chosen as x=0.

An informal definition of Maclaurin series is that you continually differentiate the function to attempt to reproduce the behavior of each xⁿ term with n from 0 to ∞.

       ∞
     ____
     \   `  f[i](0)   i
P(x)= )     ------- * x
     /___,     i!
      i=0

Where f[n](x) is the n-th derivative of the function f(x), defined as f(x)=e^x. Note the term i! in the denominator. This is called the factorial operation, and is necessary in the denominator to counteract the factorial expansion of values in the numerator. Any number of derivatives of this function will still be f(x) because of its definition (see above for definition). Also, f(0) is 1 because any number (except 0) raised to the 0 power is 1. Also, we are looking for P(1), so we can replace x with 1.

       ∞
     ____
     \   `   1
P(1)= )     -- * 1
     /___,  i!
      i=0

    ∞
  ____
  \   `   1
e= )     --
  /___,  i!
   i=0

And so we have the definition of a series that can calculate the value of e, quite possibly the most important irrational number in math, science, and engineering, rivaled only by π and half the square root of two.