may not seem much but if we had £10,000 to invest, we would have £2,250
interest instead of £2,000. By compounding every half-year we gain an extra
£250.
But if compounding every half-year means we gain on our savings, the bank
will also gain on any money we owe – so we must be careful! Suppose now that
the year is split into four quarters and 25% is applied to each quarter. Carrying
out a similar calculation, we find that our £1 has grown to £2.44141. Our money
is growing and with our £10,000 it would seem to be advantageous if we could
split up the year and apply the smaller percentage interest rates to the smaller
time intervals.
Will our money increase beyond all bounds and make us a millionaires? If we
keep dividing the year up into smaller and smaller units, as shown in the table,
this ‘limiting process’ shows that the amount appears to be settling down to a
constant number. Of course, the only realistic compounding period is per day
(and this is what banks do). The mathematical message is that this limit, which
mathematicians call e, is the amount £1 grows to if compounding takes place
continuously. Is this a good thing or a bad thing? You know the answer: if you
are saving, ‘yes’; if you owe money, ‘no’. It’s a matter of ‘e-learning’.
The exact value of e
Like π, e is an irrational number so, as with π, we cannot know its exact value.
To 20 decimal places, the value of e is 2.71828182845904523536...
Using only fractions, the best approximation to the value of e is 87/32 if the
top and bottom of the fraction are limited to two-digit numbers. Curiously, if the