Euler’s Number

OldTrout, Going Postal
The blue curve is the function eˣ and the red line is the tangent to this curve at (0, 1). The dotted curve is the function 2ˣ and the dashed curve is the function 4ˣ

“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).

OldTrout, Going Postal
Compounding at 20%

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.

© OldTrout 2018