IRRATIONALS AND IMAGINARY NUMBERS 113

rational number could exist, then it could be represented
alternatively as:

a

b

where the latter fraction is in its lowest terms—that is, all
common factors have been cancelled out. It follows that
either a or b, or both, must be odd; for if both ¢ and 4 are
even, then the fraction cannot be in its lowest terms.

If §= Ve
2
then %:2
and a?=25b?

This means that a2, and therefore a as well, is even. But if
a is even it is a multiple of 2 and may be represented
alternatively as 2m.

Then a?=4m?
but a?=25h2
SO 2h%= gm?
and b?=2m?

so that 42, and therefore 4 as well, is even.

We therefore find that both @ and 4 are even, and this
a
b
original assumption and proves that 4/2 cannot be repre-
sented rationally.

Although irrationals cannot be calculated exactly, it is pos-
sible to obtain approximate values which are quite adequate
for practical requirements. Various mathematicians have
calculated the value of # to remarkable lengths, but such
values as have been produced have no practical use, for after
the first few significant figures, subsequent digits provide a
degree of accuracy which is never called into use.

8

means that - is not in its lowest terms. This contradicts the

E
£
-
3
|
|
A

A