# Euler’s Number

“To express $$e$$ remember to memorise a sentence to memorise.”

$$e \approx 2.7 \,1828 \,1828$$

This mathematical constant is a rate of growth.  It was discovered by Jacob Bernouilli $$(1683)$$ whilst studying compound interest.  Euler referred to the constant as $$e$$ and it is now generally known as Euler’s number.

Imagine a bank that pays you $$100$$% interest per year.  How often it pays you the interest makes a difference to your bank balance in terms of the interest on the interest; this is compounding.

You have an initial investment of $$£1$$ and you are paid interest at the above rate on an annual basis;  you would have $$£2$$ at the end of the year.

$$1 + 1 = 2$$

If you were to be paid twice in the year, then after six months you would have $$£1.50$$ and at the end of the year you would have $$£2.25$$.

$$1 + 0.5 + 0.75 = 2.25$$

You have been paid interest on the interest already earned.

If you were to be paid quarterly, then your investment grows from $$£1$$ to $$£2.44$$.

$$1 + 0.25 + 0.3125 + 0.3906\,25 + 0.4882\,8125 = 2.4414\,0625$$

Monthly interest gives :

$$\displaystyle \left(1 + \frac {1}{12}\right)^{12} = 2.6130\,3529$$

Weekly interest gives :

$$\displaystyle \left (1 + \frac {1}{52} \right)^{52} = 2.6925\,9695$$

Daily interest gives :

$$\displaystyle \left (1 + \frac {1}{365} \right)^{365} = 2.7145\,6748$$

In monetary terms we have arrived at the limit of 2.72 (by choosing to round up).

Mathematically, the limit as $$n$$ approaches infinity is given by :

$$\displaystyle \lim_{n\to \infty} \left (1 +\frac {1}{n} \right)^n = e$$

We can also express $$e$$ as a sum:

$$\displaystyle e = \sum_ {n = 0}^ \infty \frac {1}{n!} = \frac {1}{0!} + \frac {1}{1!} + \frac {1}{2!} + \frac {1}{3!} …$$

where the sequence of sums begins $$1, 2, 5/2, 8/3, 65/24, 163/60, …$$.

For interest calculations, use :

$$\displaystyle P’ = P\left (1 + \frac {r}{n} \right) ^{nt}$$

where $$P’$$ is principal plus interest, $$P$$ is the initial principal, $$r$$ is the interest rate, $$n$$ is the compounding period and $$t$$ is time.

Euler proved that $$e$$ is irrational by showing that its simple continued fraction expansion is infinite.

Hermite proved ($$1873$$) that $$e$$ is transcendental, that is, it is not a solution to any non-constant polynomial equation with rational coefficients.