Euler's Number

Euler's number, commonly represented as \(e\), is a mathematical constant that is important mainly in calculus but also in complex numbers, probability and other areas. It is irrational, meaning it has no repeating pattern of digits, and it is transcendental, meaning it cannot be expressed algebraically in any way.

Calculating \(e\)

Here are three formulae which produce \(e\). Each contains an input value \(n\), which you can control with the slider. The value produced by the formula becomes more accurate as n rises. The value is shown with correct digits of \(e\) highlighted green and incorrect digits highlighted red.

This is the classic formula used to calculate \(e\). It can be calculated by hand without too much difficulty, as it does not involve the massive numbers created by factorials, but it is very slow to converge.$$e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$$



This formula requires the calculation of massive numbers, due to the use of the factorial function, but is also fast to converge on \(e\). An input value of only n = 97 is required to calculate the 150 decimal places shown below.$$e=\lim_{n\to\infty}\left(\sum_{x=0}^{n}\frac{1}{x!}\right)$$



This infinite continued fraction converges extremely quickly on \(e\), requiring only 41 terms to get 150 decimal places, and is also very simple to calculate. Sadly, I am unaware of a neat way of representing infinite continued fractions. Here, n represents the number of terms, with n = 1 including the 6.$$e=1+\frac{2}{1+\frac{1}{6+\frac{1}{10+\frac{1}{14+\frac{1}{18+...}}}}}$$



Uses for \(e\)

Calculus


One of the ways in which \(e\) is commonly defined, is as the unique positive number \(a\) where the graph of the function \(y=a^x\) has a slope of 1 at \(x=0\).
The function \(f(x)=e^x\) is the unique function that is equal to its own derivative.
The natural logarithm of a number \(k>1\) can be defined as as the area under the curve of \(y=\frac{1}{x}\) (shown below) between \(x=1\) and \(x=k\). It follows that \(e\) is the value of \(k\) for which this area is equal to 1.
graph of 1/x

Complex Numbers


A very well-known application of \(e\) is in Euler's Identity, \(e^{i\pi}+1=0\). This is a special case of Euler's formula, \(e^{ix}=\cos{x}+i\sin{x}\), where \(x=\pi\).

Probability


Consider this problem:
A secretary has a large number \(n\) letters and the same number of addressed envelopes. Each letter needs to go with a specific envelope, but instead, the secretary decides to match each letter with an envelope at random. What is the probability that every letter is matched with the wrong envelope?
The answer is that as \(n\) increases, the probability tends towards \(\frac{1}{e}\).

Interest


Suppose you put £1 into a bank account. This bank pays interest: every 100 days, they will pay you another £1 for every £1 already in your account.
After 100 days, you will have £1 in your account, and so you will receive another £1, bringing your total balance to £2.
Now suppose that instead of paying £1 every 100 days for every £1 you already have, the bank pays you 50p evert 50 days for every £1 you already have. At first, this looks like the same deal, but it's not.
After 50 days, you will receive 50p interest on your £1, bringing your balance to £1.50. However, after another 50 days, you don't just have £1: you have £1.50, so you get 75p interest rather than 50p, giving you a total balance of £2.25.
Now suppose that the bank pays you 20p every 20 days for every £1 you have. After 100 days, this brings your total balance to about £2.49. If instead you received 10p every 10 days, your total balance would be around £2.59.
Clearly, you receive more money the oftener they pay your interest. But what is the limit? What if you received a tiny fraction of a penny every second? Or nanosecond?
The limit is £\(e\).
From here, the general interest formula is derived. If you start of with £\(p\), and you receive \(r\) of your money (for example, \(r=0.1\) would be 10% interest) per unit of time, your total balance after \(t\) units of time will be \(pe^{rt}\).

Stirling's Formula


A more obscure use for \(e\) is in Stirling's formula. This is a formula that allows you to approximate values for factorials rather than calculating the large values directly. The formula is \(n!\approx\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\). The graph below shows Stirling's formula compared with the gamma function (a variant on factorial that exists for non-integer values).
stirling's formula graph