|

k3
S
5
:
g
2
#

A R L O S H B

B e TR g T T T e i
1 Lo it L DL R R R gl R L L

126 THE FASCINATION OF NUMBERS

The fact that the number 1089 will always result from the
above operations may easily be understood if it be remem-
bered that any number ‘ab¢’ is really a number of the form
100a+-10b +¢. Every number of three digits is essentially of
this form and the process may therefore be represented as
follows, stage by stage, irrespective of the numerical values of
the digits a, b and .

Original number

100a + 10b+¢
Reverse 100¢ + 10b-+ta

 

Subtract 100(a—c—1) + 9o +(10+c—a)
Reverse 100(10+c¢—a)+ 9o +(a—c—1)

Add -IOO —1-410) +180 +(10—1)

 

=1089

Since the first number is greater than its reversal, it follows
that the digit a is greater than ¢, so that in order to subtract
the former from the latter, it is necessary to borrow 1o from
the tens column. As, however, there are no tens from which
to borrow (because 100 —106=0), we have to borrow 100
from the hundreds column. Of this 100, we carry 10 to the
units column to assist the subtraction of a from ¢, and the
balance of go is then left in the tens column.

A similar proposition is to write down a sum of pounds,
shillings and pence, so that the number of pounds is less than
12 but exceeds the number of pence by two or more, and
then to treat this in exactly the same way as the number
‘abc’ in the previous example.

Original £8 15 6
Reverse

Subtract
Reverse

Add