S S

L

W

sl it

3
-
|
]
$
i
S
5
S
E.
%
=
i

 

122 THE FASCINATION OF NUMBERS

and at the end of the year:

-02\12
f(i+33)
If these expressions are simplified, we arrive at a different
value at the year end for each of the different methods
employed.

By calculating interest still more frequently (e.g. weekly),
we find that the value at the end of the year is again higher
than if we calculate yearly or monthly. It does not, however,
increase indefinitely. There is a value beyond which it will
not increase, or in other words, there is a limit to its possible
value. If the loan had been at 100 per cent. interest this limit
would have been [£2-718 approximately and its connexion
with the number ¢is at once apparent. In the present example
the limit is £2-718%? or L1-021.

The number ¢ may be represented in a number of different
ways:

 

 

e=2-1

I+1

242
3+3
4+4 ...

I I I I
i ey +

TSR CRUREBC RSt

U +[I L1 g }
W 1+2414+144...
=the limit of the sequence: (11)2, (11)3, (11)4, (11)5...
In addition to irrational numbers of the type previously
mentioned, there are other numbers, the values of which
we cannot even calculate approximately. These are called

itmaginary numbers and the most important of these is i,
equivalent to the square root of minus 1.

i=vV —1