P i e e s e s

FALLACIES 7 Ry
But, as y=x, then x+y=x-+x
SO ox=2x; afid" 9=}
Similarly x24xy—2y2=x2—xy
Factorizing:  (x—») (x+2y) =x(x—))
therefore (x+2y)=x [Dividing by (x—»)]
But, again, y=x, so gxX=x
and g=1

The fallacy is not difficult to find. In each of the above
examples each equation, after factorization, was divided by
(x —). But if x =y, then x —y=0. When we divide, say, 2 by
nought, the answer, so far as can be stated briefly, is infinity;
the answer is certainly not 2. Similarly, when we divide
x(x—y) by (x—») the answer cannot be x when (x—y)=o.
In point of fact, therefore, when we divide by (x—») we are
claiming to do the impossible.

There are also a number of equations for x which hold
when x =1, but not when ¥ is greater than 1.

Example: xt-+-x8=2x
If x is equal to 1, then this is obviously correct, but if x=2,
then it is not. This, of course, is because the number 1
multiplied by itself any number of times never varies; it
always remains 1. This is the only number which acts in
this way.

The use of the minus sign can also result in fallacious
results, as here:

{g) x{5ng) =aj
and (+5) x(+5) =25
50 (—6) wirmg) =t L5 ()
or (r8) Yo by S
and, taking the square root:

=8 T

so that —5 —§ =0
or O sk b ey

The fallacy here is in taking the square root of (—5)* and
(+45)2 Both these expressions are really equivalent to 25,