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## Euler’s Number (the mathematical constant ‘e’)

Last week, in this blogpost here, I tried to explain why the scale height of the atmosphere $H$ is defined as the altitude one has to go up for the pressure to drop by a factor of $1/e \;(\approx 37\%)$ of its value at sea level. After posting that blog I decided I would write a blog to say a little bit more about the mathematical constant e, a very important number in mathematics.

Probably the best known mathematical constant is $\pi$, which is defined as the ratio of the circumference of a circle to its diameter. Pretty much every school child comes across $\pi$; but it is only people who study maths at a more advanced level who come across e, so let me try in this blogpost to explain what e is and why it is so important.

## Euler’s number

e is also know as Euler’s number, named after the Swiss mathematician Leonhard Euler who lived from 1707 to 1783. Euler was one of the great mathematicians, but it was not he who discovered e. The constant was discovered by Jacob Bernoulli (from the same family as the name attached to “the Bernoulli effect” which causes lift in the wings of an aeroplane), but it was Euler who started to use the letter ‘e’ to represent the constant, in 1727 or 1728.

The number itself is defined as the solution to the following sum $\displaystyle e \equiv \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + .... = 2.718281828...$

where $2! \text{ (called "two factorial") } = 2 \times 1$, $3! = 3 \times 2 \times 1, \; 4! = 4 \times 3 \times 2 \times 1$ etc. $0! \equiv 1$ by definition. This is an example of a converging series. It is summed to infinity, but each term is smaller than the one before; but it never ends. So, as you go to more and more terms in the series you only affect numbers which are maybe 100 or even 1,000 places to the right of the decimal point. Calculators usually display numbers to 7 or 8 decimal places, so you would not need to go very far in this series to get the number displayed by your calculator (try it to find out how many!) The mathematical constant e is the sum of 1/n! where n goes from zero to infinity ( $\displaystyle \sum_{n=0}^{\infty} \frac{1}{n!}$). It is equal to 2.718281828…….

Just like with $\pi$, e is a transcendental number (I will explain what that is in more detail in a future blogpost). Briefly, this means that it carries on forever and does not repeat, but it is slightly more complicated than that. Unlike with $\pi$, where people have competitions to remember it to thousands of decimal places (the current world record is 70,000 decimal places achieved by Rajveer Meena on 21 March 2015!!), no one seems that concerned in remembering e.

## Why is e important in mathematics?

Mathematicians often love numbers and formulae for their own sake, sometimes just for their beauty. So, for example, the solution to something like $\displaystyle a = \sum_{n=0}^{\infty} \left( 1 + \frac{1}{n^{3}} \right)$

(which I just made up) may not have any importance mathematically, but still mathematicians may enjoy playing and exploring such a series. But, in the case of $\displaystyle e \equiv \sum_{n=0}^{\infty} \frac{1}{n!}$

the number which comes form this, 2.718281828……., is important mathematically (and physically), and here I will try to explain why.

## Compound interest

I mentioned above that e was discovered by Jacob Bernoulli. His discovery was made in 1683 when he was investigating a question concerning compound interest. Remember, compound interest is when you get a certain percentage interest on the amount you have in e.g. a bank, but the interest is calculated not on the initial amount but on the amount after the previous period’s interest has been added.

Suppose we invest £10 in a bank account which pays an interest of 5% per year, and we leave it there for 3 years. Assuming the interest is added just once a year, after the first year our £10 will have earned £0.50 interest, so we will have £10.50. At the end of the second year the interest earned will be 5% of £10.50 which is £0.53 (rounding up to the nearest penny), so we will now have £11.03 at the start of year 3. The interest at the end of year 3 is going to be 5% of £11.03 which is £0.55, so the total at the end of the 3  years is £11.58.

Doing this for just 3 years manually is quite easy, but if we wanted to do it for e.g. 25 years, there would be a lot of tedious calculation. Also, often the interest is added more than once a year. So, for example, you may have an annual interest rate of 5% but it is added each quarter. You can quickly see that even doing this manually over a 3-year period would be a lot of calculating.

Thankfully, there is a simple formula for calculating the total accumulated value, which is $P(1 + i/n)^{nt}$

where $P$ is the initial amount invested (the ‘principal’), $i$ is the rate of interest, $n$ is how often each year the interest is added (called the ‘compounding frequency’) and $t$ is the time for which we are making the calculation, expressed in years.

If we go back to our example above, and stick to $n=1$, we have $P=10, \; i=0.05, \; t=3$ and so $P(1 + i/n)^{nt} = 10(1+0.05)^{3} = 10(1.05)^{3} = 10(1.157625) = 11.58$

exactly as we calculated manually.

But, you may be wondering, what is the similarity between the formula $P(1 + i/n)^{nt}$

and the formula for $e$ $\displaystyle e \equiv \sum_{n=0}^{\infty} \frac{1}{n!}?$

What Bernoulli noticed was that, if you make $n$ larger and larger (do the compounding daily instead of quarterly or once a year), the sequence approaches a limit. So, for example, in the above example, with $n=1$ we found the amount at the end was £11.58. If we made $n=4$ (compounding quarterly) we would get £11.61 and if we compounded the interest every day ( $n=365$) we would get £11.62. If we compounded every hour (!!!) ( $n=8760$) we would get £11.62, the same answer as if we compound every day, so we have reached the limit.

What Bernoulli did was consider the formula with $P=1, \; i=100\%$ and $t=1$. If we do this for different values of $n$ we get the following curves. The first one is just showing n from 0 to 20, and it is clear that the values are flattening out. The second plot goes from n=0 to 500, and I have just shown on the y-axis values from 2.5 to 2.72. This shows even more clearly that, as n gets larger, the value of $1(1+1/n)^{n}$ tends towards a particular value, and that value turns out to be 2.71828… (which is e).  $y = 1(1+1/n)^{n}$ for n from 0 to 20  $y=1(1+1/n)^{n}$ for n from 0 to 500, but note the y-axis is only displayed between 2.5 and 2.72. The curve is clearly tending towards a value, and that value is $e=2.71828...$

## e in calculus

Even if you don’t know how to do calculus, you have probably heard of it. It was co-invented by Isaac Newton and Gottfried von Leibniz (see my blogpost “Who Invented Calculus” for the fascinating story of the 30-year feud between Newton and Leibniz). Without going into too much detail about all the various things one can do with calculus, one thing is that it gives is the gradient (slope) of a curve at a given point (something which is sometimes called the derivative).

There are an infinite number of different mathematical functions, for example $y=x^{2}$, or $y=x^{3} - 2x + 3$, and we can use differentiation to determine the gradient of these functions for any particular value of $x$. But, the function $y=e^{x}$ is unique; it is the only mathematical function whose derivative is the same as the function. To put this another way, the slope of the curve $y=e^{x}$ is $e^{x}$ for any value of $x$, and there is no other mathematical function whose derivative is the same as the function, only $f(x)=e^{x}$. A plot of $e^{x}$ as a function of $x$. At $x=0, \; e^{x}=1$. As $x \rightarrow -\infty, \; x \rightarrow 0$.

In addition, when we integrate something like $dx/x$, we get a logarithm; but the base of that logarithm is e, not base 10 (our usual base of counting). In fact, we call logarithms in base e natural logarithms. Because the derivation of the variation of pressure with altitude involves integrating $dP/P$, we find that the vertical distribution of pressure is logarithmic, but in base e, $P=P_{0}e^{-z/H}$, where $H$ is the scale height and $P_{0}$ is the pressure at sea level. It is because of the pressure’s exponential dependence on altitude that $H$ is usually expressed as the value for the pressure to drop by a factor of $1/e$.

## The normal (or gaussian) distribution

As one last example of where e pops up in mathematics, it arises in the equation which describes the normal or gaussian distribution. I blogged about that distribution in this blogpost here “What does a 1-sigma, 3-sigma or 5-sigma detection mean?”. The function which describes the normal distribution has the form $f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^{2}/2}$

where e is our friend, Euler’s number. The normal, or Gaussian, distribution $y = \frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$