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QIR L R e e T Y

78 THE FASCINATION OF NUMBERS

factors for convenience: they should be understood to be
there. If 99 is represented by 28+42%+2-41 and is divided
by 2 the result will be 25+2%+1, ignoring the remainder (as
in the second line). Eventually, as we divide by 2, each
power of 2 will be reduced to 1, and it is the lines in which
each power is so reduced that contain, in the right columns,
the terms which must be added. Thus each factor is reduced
to 1 as follows:

Factor Term in which reduced
2" 7th

. 28 6th
2 ond
I Ist

It is clear that the 7th, 6th, 2nd and 1st terms will be odd
in the left column since each of these is a multiple of 2, plus 1.
And it is also clear that the relative terms in the right column
are those which have to be added. Omitting irrelevant terms
we have:

99 =
49 4
3 64
I 128

Add: 198 (=2x99)

The reason for this is that as each left-hand factor is reduced
to 1, the right-hand term is the result of multiplying the
original number by the factor reduced. Thus the factor 2
is reduced to 1 in the 2nd term and the right-hand figure has
been multiplied by 2. Similarly, 2¢ is reduced to 1 in the
seventh term and the right-hand figure has been multiplied
by 28 So, if we add the 1st, 2nd, 6th and 7th terms in the
right column we are in fact adding:

(1 x2)4(2 x2)+(2%x2)+(2%x2)
or 2(14+2+2542% =2 x99