A

YR 10 ALY

B I e T Te e L G T S
P ATt

 

7
Multiplication with a DZfiérence

In addition to the ordinary methods of multiplying or divid-
ing, there are a number of other ways of performing the same
operations and these at first glance appear to have no obvious
connexion with accepted methods at all. Some of the primi-
tive methods appear to be complex in their simplicity, but
the inherent paradox is easily reduced to a pattern when
related to mathematical fundamentals.

7 An ancient method of multiplying together two numbers

larger than 5 enables this to be done by simple operations
with numbers smaller than 5. Thus, if we select the two num-
bers 8 and g, their product can be found by:

(a) Deduct 5 from each, leaving g and 4; add these two
numbers, giving 7.

(b) Deduct from 5 the respective residues in (a), that is g
and 4, leaving 2 and 1; and multiply these two num-
bers together, giving 2.

(¢) Then the answer, consisting of two digits, will have in
its tens column the number in (@), and in its units
column the number in (4), giving 72.

The justification for this method can be seen by substitut-
ing x and y for the two numbers to be multiplied. The stages
are then taken to give the formula:

(¥—5)10+(y—5)10+(5 —%+5) (5—r+5)
=10% —50+410y —50-+100—10X —I0p+xy
=x).
Another early method was to proceed as follows. Form a
quadrilateral divided into small squares, the number of the
72