What is e?

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 article gives an overview of the various facets of e. Each section has a link to a more detailed article.

The exponential function

The value of e raised to the power x is an important function that is often used in mathematics. It is called the exponential function, sometimes written as exp(x):

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 that the 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).

For more information, see Formula for e.

The exponential function 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.

For more information, see Derivative of the exponential function.

Maclaurin expansion of the exponential function

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. The reason this arises is that the terms in the Maclaurin expansion of a function are derived by repeated differentiation of the function. Since the exponential function has a very simple derivative, it is perhaps less surprising that its Maclaurin expansion should also be very simple.

For more information, see Maclaurin series of the exponential function.

Euler's identity

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

Before the discovery of this identity, it had generally been assumed that e, pi, and the imaginary unit i were unrelated constants that were applicable in different areas of mathematics - exponentials, geometry, and complex numbers. This formula links the three values in one simple formula.

This identity is derived from Euler's formula:

When θ is equal to π, the cosine term is -1 and the sine term is 0, which leads to Euler's identity:

For more information, see Euler's identity.

See also

• Rules of exponents
• Negative and fractional exponents
• Graphs of exponential functions
• 2 to the power π - exponential function for an irrational value of x
• Formula for e
• Derivative of the exponential function
• Proof that e is irrational