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$ M AR RTERE L i iistal BRURES

 

136 THE FASCINATION OF NUMBERS

be able to plough the same two fields in one-twentieth of the
time, for the simple reason that there may not be room in the
two fields for so many ploughs. Even where the toil is of a
manual nature the argument is the same. Three men will be
able to make a concrete path in three hours, but ninety men
would not, as a consequence, be able to make the same path
in ¢ths of an hour (i.e. 6 minutes). Even allowing for the
shovel-leaners, tea-makers and card-players of this large
labour force, the remainder of the path-makers would still
be so numerous that they would constantly be falling over
each other.

In Algebra and other branches of Mathematics, we use
certain letters and other symbols to replace unknown quanti-
ties and relationships. If these can be placed into the form of
an equation containing only one unknown quantity, then
that quantity may be evaluated simply by working out the
equation. Similarly, the values of two unknown quantities
may be calculated from simultaneous equations. But the use
of letters is intended to simplify these calculations; if we are
not careful, they serve only to confuse. Indeed, certain
equations may be expressed in an algebraic form which
appears to be correct but which, in fact, may contain hidden
fallacies leading to fantastic results.

The best known examples of such results are those involved
in ‘proving’ that a number greater than nought is neverthe-
less equal to nought or to any other desired number. This
spoofing is a simple matter and may be accomplished in a
number of different ways.

The following is one demonstration, in the case where two
unknowns x and y are found to be equal to each other—and,
therefore, their squares are also equal.

Thus: X ==y
Therefore x3=xy=9y3
and x2—yi=x%—xy
Factorizing: (x—9) (x+2) =x(x—y)
SO X+y=x