1
|
3
1
o
=}
;
S
3

   
  

4

it

o s R N
I RPN B IR R R

|
: b
Z
3
i
?:
3
i

g

140 THE FASCINATION OF NUMBERS

which, by regrouping, may be converted to:
Pt b A A

and is in fact approximately equivalent to -7, being a non-

terminating expression with a limiting value.

The fallacy here is that both » and y are infinite and
infinity cannot be subjected to normal mathematical pro-
cesses. You cannot mathematically add infinity to infinity
for the result would still be—infinity.

In Chapter 11, it was shown that certain successive opera-
tions upon a three-digit number would always give the
result of 1089. A proof that this must be so was included in
the same chapter. Any attempt to prove it, however, with-
out bearing in mind that a is greater than ¢ and that it is
necessary to employ the strategy of borrowing, would give
the following quite fallacious result.

 

 

Original number 1004 +10b+ ¢
Reverse 100¢ +100+4+ a
Subtract 100(a—c¢) + 0o +(c—a)
Reverse 100(c—a) + o +(a—c¢)
Add 100(a—c+c—a)+ o +(c—a-+a—c)
=100(0) 4+ 0o + (o)
=0

Care is also necessary when dealing with comparative
ages. Thus when Brown is g0 and Smith is 10, Brown is three
times as old as Smith. But in ten years time, when Brown is
40 and Smith is 20, Brown will only be twice as old as Smith.
No time need be wasted in calculating how long Smith and
Brown must live before they both become the same age!

A very old problem was posed thus: An old man in his will
left 17 horses to be divided amongst his three sons in such a
way that they should receive one-half, one-third and one-
ninth of the total respectively. As the number 17 is not
divisible by 2, g or 9, how was the division (in terms of
whole horses) to be effected? The solution to this problem
was provided by the ingenious notion of borrowing another