# Calculating Euler’s Constant (e)

Euler’s Number, written as $e$, is probably the second most famous mathematical constant after Pi. But what is Euler’s Number, and how do we calculate it?  In fact, why has e become so famous, and why does it deserve a place on our calculators and in the mathematical constant hall of fame?

### What is Euler’s Number (e) and where did it come from?

Euler’s number has a value of 2.718… , however just like Pi, it is an irrational number, meaning that it cannot be written as a fraction and that it has a decimal expansion which will continue forever without repetition. Euler’s number e has become famous for two main reasons: firstly, it is used in lots of important real-life situations, and secondly it has many interesting mathematical properties. This makes $e$ a fascinating and useful number for scientists, engineers, and mathematicians alike.

### Euler’s Number e and Compound Interest

Euler’s Number $e$ was first discovered by Jacob Bernoulli in the 17th Century when he studied the problem of Compound Interest.

Imagine you have £1 and that you get interest twice a year at a rate of 50%.

At the end of the year you would end up with £1 $\times \left( 1 + \frac { 1 } { 2 } \right) \times \left( 1 + \frac { 1 } { 2 } \right) =$£ $1 \times \left( 1 + \frac { 1 } { 2 } \right) ^ { 2 }$ = £2.25

Now imagine you have £1 and you get interest 12 times per year, or every month at a rate of $\frac { 1 } { 12 }$ (8.3%)

At the end of the year you would end up with £1 $\times \left( 1 + \frac { 1 } { 12 } \right) ^ { 12 } =$ £2.61303529

Now imagine you have £1 and you get interest 365 times per year, or every day at a rate of $\frac { 1 } { 365}$ (0.2739….%)

At the end of the year you would end up with £1 $\times \left( 1 + \frac { 1 } { 365 } \right) ^ { 365 } =$ £2.714567482

Jacob Bernoulli asked an important question: what would happen if you received interest so often that you received it continuously?

In fact, what is the value of $\left( 1 + \frac { 1 } { n } \right) ^ { n }$ as n tends towards infinity?

You might already have guessed the answer, just by looking at our example where n=365, which is already getting quite close to $e$.  This brings us to the most well known way of calculating $e$:

### Calculating the value of Euler’s Number e as a limit: $\large e =\mathop {\lim }\limits_{n \to \infty } \left( {1 + \frac{1}{n}} \right)^n$

(Keep putting in bigger and bigger values of $n$, until you get really close to the true value of $e$.) Jacob Bernoulli

Unfortunately, Jacob Bernoulli didn’t have a computer at his disposal and was only able to say that the value was between 2 and 3.  Some years later Leonhard Euler, one of the greatest mathematicians in history managed to calculate the value of $e$, correct to 18 decimal places. Euler had also discovered the following:

### Calculating the value of Euler’s Number e using an infinite series: $\large e = 1 + \frac { 1 } { 1 ! } + \frac { 1 } { 2 ! } + \frac { 1 } { 3 ! } + \frac { 1 } { 4 ! } + \frac { 1 } { 5 ! } + \cdots$

(In case you are wondering, 5! means $5 \times 4 \times 3 \times 2 \times 1$ and is the factorial function)

The more terms you calculate, the closer you will get to the true value of $e$. You will only arrive at the exact value of $e$ if you carry on adding up the sequence forever.

Nobody knows exactly how Euler calculated $e$ to 18 decimal places, however the best guess is that he used the sequence above. It was also Euler who named the constant ‘ $e$’. Surprisingly, historians are fairly certain that he didn’t name it after himself, but that it was a pure coincidence that he chose the first letter of his surname.

### Continued Fractions and e

Euler was also able to represent $e$ in the form of a “continued fraction”. There are lots of different ways to represent e as an infinite continued fraction. Here is one of them:

Calculating the value of Euler’s Number e as a continued fraction: $\Large e=1+\dfrac {2}{1+\dfrac {1}{6+\dfrac {1}{10+\dfrac {1}{14+\dfrac {1}{18\dfrac {1}{22+\dfrac {1}{26+\ddots }}}}}}}$

### Other ways to calculate e

The three main ways of calculating $e$ have been listed above. There are however many other lesser known representations of $e$ such as: $\Large e = \mathop {\lim }\limits_{n \to \infty }\left( \frac { n } { \sqrt [ n ] { n ! } } \right)$

If you visit the Wolfram Mathworld page on e, you can browse through a huge collection of different ways of calculating, some of which are very complicated indeed. This same page also lists a collection of mnemonics to help you remember the digits of $e$. A favourite has to be:

“It enables a numskull to memorize a quantity of numerals” (10 digits)

Count the letters in each word and you will have: 2.718281828

### Where is Euler’s Number e used in the real world? Compound Interest is not the only practical use for $e$. In fact, Euler’s number $e$, the function $y = e^ { x }$, and the natural logarithm with base $e \enspace( \ln x )$ appear a lot in real-life processes. The main reason for this is that the exponential function $y = e ^ { x }$ can be used to describe growth and decay.

Examples include:

• How populations grow
• How temperature changes as materials heat up or cool down

### Unique mathematical property of $y = k e ^ { x }$

The function $y = k e ^ { x }$ has a special mathematical property which has important consequences for calculations involving $e$, making the mathematics involved work much more easily than with many other functions. It is one of the reasons that $e$ is used so frequently to model the real world.

The function $y = k e ^ { x }$ is the only function where it is equal to its derivative ( $k$ stands for any number, and this just means that the property also holds for multiples of $y = e ^ { x }$). When you differentiate $y = e ^ { x }$, it remains unchanged: $\large \frac { d } { d x } \left( e ^ { x } \right) = e ^ { x }$. This also means that when you integrate $y = e ^ { x }$ it will remain unchanged apart from the constant of integration. This unique property simplifies many calculations involving $y = e ^ { x }$

### Don’t forget about $e ^ { i \pi } = - 1$

No discussion about Euler’s Number e would be complete without mentioning one of the most famous equations in mathematics called Euler’s Equation: $\Large e ^ { i \pi } = - 1$

(If you aren’t sure what stands for – it is equal to the square root of minus 1 and is called an imaginary number.)

Euler’s Equation shows that both $\pi$ and $e$ are connected to one another. This is really surprising, given that $\pi$ comes from looking at the properties of a circle, and $e$ arises from situations which have nothing to do with circles such as compound interest. Euler’s Equation shows that $e$ is more than just a useful number which can be used by scientists to model the real world – it is a fascinating number in its own mathematical right.

Image Credits:

Leonhard_Euler by Jakob Emanuel Handmann [Public domain], via Wikimedia Commons