# What is e?

By Martin McBride, 2022-08-05
Tags: exponential power root
Categories: exponentials

You probably know of the mathematical quantity e. It is known as Euler's constant and is the base of natural logarithms. It is an irrational number, meaning that it cannot be represented exactly as a fraction, but instead has an infinite number of digits that never repeat. Its value is approximately 2.718281828459045.

The value e crops up quite a lot in mathematics, in ways that might appear to be unrelated. This series of articles attempts to tie those various facets of e together and give some insight into why they are all true of one special number.

## Formula for the value of e

One of the standard equations for the value of e is this one:

This tells us that we can calculate an approximation to the value of e by calculating the formula with any chosen value of n. It also tells us the that larger the value of n we choose, the better the approximation will be. For example, if we choose a value of 2, the result will be 2.25, which isn't particularly close to the true value of e:

If we repeat the calculation with a value of 100 we get a slightly closer value of 2.704813829:

With a sufficiently large value of n we can calculate e to as many decimal places as we wish (although, later on, we will see a better way to calculate e).

## e to the power x is its own derivative

Next, we will look at one of the most important and useful properties of e, expressed in this equation:

This applies to the exponential function with a base equal to the special value e. It tells us that for any value x, the slope of the curve at that point is equal to the value of the curve at that point.

In fact, the exponential curve, with a base e, is the only function that has that property. (That isn't quite true, the function f(x) = 0 also has that property, because the value of the function and the slope of the function are both 0 everywhere, but that function isn't very interesting or useful).

This is an extremely useful property because there are many situations where the rate of change of a system is proportional to the value. Examples include radioactive decay, and population growth.

You might wonder why the exponential function has this property, and in particular why it only happens when we choose this strange number e as the base.

## Maclaurin expansion of e to the power x

Next, we have this formula for calculating the exponential function as an infinite series:

Again it seems very strange that a function of the irrational number e should have such a simple series. We will cover this in a future article.

## Euler's identity

Finally, e is famously part of Euler's identity:

This formula links the value e, pi, and the imaginary unit i in one simple formula. We will cover this in a future article.