R TR T R

R RS S R R

Fv R R s R

T ekt o AT 1 e e U R
PATYECT AEET SV IO fuaeti s

 

 

 

16 THE FASCINATION OF NUMBERS

fully grasped the fact that a decimal number is only another
representation of a fractional number. Confusion arises, for
example, because the multiplication of decimal numbers
always results in an answer less than either of them, whereas
the multiplication of integral numbers always results in an
answer greater than either of them (except where the multi-
plier is negative).

Thus: 5 X 5=25
But 5 X 5="25

That this must be so is clearer when it is appreciated that
-5 % -5 is the same as 4 X 4 and is therefore equal to }. For
although the expression # x4 means ‘multiply a half by a
half’, it also means ‘calculate a half of a half’ and this must
obviously be less than a half.

An extension of the same principle also explains the super-
ficially remarkable behaviour of two numbers, one slightly
larger than 1 and the other slightly less than 1, when they
are successively raised to higher powers of themselves. The
numbers 1-1 and -9, for example, both approximate closely
to the number 1, and it might seem, on first considerations
that (1-1)" should always be relatively close in value to
(-9)". There is, for instance, only a small differéence between
the values of (1-1)2 and (-9)?, the values being 1-21 and -81
respectively; but the difference between (1-1)°° and (-9)*°°
is immense, the former number being nearly 14,000, while

 

the latter is less than :
10,000

Thus, as 1-1 is progressively raised to higher powers it
becomes increasingly larger whilst g, under the same treat-
ment, becomes progressively smaller.