06 e
e is the new kid on the block when compared with its only rival π. While π is more august
and has a grand past dating back to the Babylonians, e is not so weighed down by the
barnacles of history. The constant e is youthful and vibrant and is ever present when
‘growth’ is involved. Whether it’s populations, money or other physical quantities,
growth invariably involves e.
e is the number whose approximate value is 2.71828. So why is that so
special? It isn’t a number picked out at random, but is one of the great
mathematical constants. It came to light in the early 17th century when several
mathematicians put their energies into clarifying the idea of a logarithm, the
brilliant invention that allowed the multiplication of large numbers to be
converted into addition.
But the story really begins with some 17th-century e-commerce. Jacob
Bernoulli was one of the illustrious Bernoullis of Switzerland, a family which
made it their business to supply a dynasty of mathematicians to the world. Jacob
set to work in 1683 with the problem of compound interest.
Money, money, money
Suppose we consider a 1-year time period, an interest rate of a whopping
100%, and an initial deposit (called a ‘principal’ sum) of £1. Of course we rarely
get 100% on our money but this figure suits our purpose and the concept can be
adapted to realistic interests rates like 6% and 7%. Likewise, if we have greater
principal sums like £10,000 we can multiply everything we do by 10,000.
At the end of the year at 100% interest, we will have the principal and the
amount of interest earned which in this case is also £1. So we shall have the
princely sum of £2. Now we suppose that the interest rate is halved to 50% but
is applied for each half-year separately. For the first half-year we gain an interest
of 50 pence and our principal has grown to £1.50 by the end of the first half-
year. So, by the end of the full year we would have this amount and the 75
pence interest on this sum. Our £1 has grown to £2.25 by the end of the year! By
compounding the interest each half-year we have made an extra 25 pence. It