## History and applications

### Bernoulli, compound interest and Euler

Although some related ideas appeared in earlier works, arguably the first `discovery' of the number \(e\) was made by Jacob Bernoulli in 1683, in considering compound interest.

Suppose that a bank offers to take your money, and will give you 100% interest on it in a year's time. So if you give the bank $1000 now, you will get $2000 back in a year. (This bank pays well!)

Alternatively, the bank says, it will give you half that interest rate, but it will give you the interest twice as often: so it will give you 50% interest in half a year's time, and then a further 50% on that at the end of the year. Your $1000 will become $1500 after half a year, and then $2250 at the end of the year. By calculating half the interest twice as often, you end up with significantly more at the end.

Now the bank says, it can give you a third of the interest rate, but three times as often. So, 33\(\dfrac{1}{3}\)% interest is paid out three times throughout the year. Your $1000 becomes $1333.33, then $1777.78, then $2370.37 (to the nearest cent) at the end of the year. You end up with more again.

Continuing on in this fashion, the bank might calculate one fourth, one fifth, etc. the interest, four, five, etc. times as often. In general, it might offer you an interest rate of \(\dfrac{100}{n}\)%, calculated \(n\) times a year. You money is multiplied by

\[ 1 + \dfrac{1}{n}, \] \(n\) times throughout the year. That is, your money is multiplied, overall, by \[ \Bigl( 1 + \dfrac{1}{n} \Bigr)^n. \]We saw in the section Sandwiching \(e\) that, as \(n\) approaches infinity (one billionth the interest rate, one billion times a year!), this amount approaches none other than \(e\):

\[ \lim_{n \to \infty} \Bigl( 1 + \dfrac{1}{n} \Bigr)^n = e. \]Thus, in the limit of continuously compounded interest, your $1000 after a year becomes $1000 \(\times e\), approximately $2718.28.

In general, consider an interest rate of \(x\) (so \(x=1\) for our interest rate of 100% above). Again consider an interest rate of \(\dfrac{x}{2}\), calculated twice as often; or an interest rate of \(\dfrac{x}{3}\), calculated three times as often; and, in general, an interest rate of \(\dfrac{x}{n}\), calculated \(n\) times as often. We obtain, in the limit, that our money is multiplied by

\[ \lim_{n \to \infty} \Bigl( 1 + \dfrac{x}{n} \Bigr)^n \] each year. As exercise 21 shows, this limit turns out to be \(e^x\).Although some of the above ideas were discussed by Bernoulli, the notation \(e\) was not used until much later. Its first known appearance is in a letter of Euler from 1731.

Today the notation \(e\) is often associated with Euler's name, and \(e\) is sometimes called Euler's number. (This can be a problem, since there is another mathematical constant \(\gamma \approx 0.5772\) which often goes by the name of Euler's constant.) However, when \(e\) was first used it was not meant to refer to Euler — there is no reason to believe Euler used the letter in honour of his own name.