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ARRANANGNI

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138 THE FASCINATION OF NUMBERS

but the square root of 25 is neither +5 nor —5 alone. The
correct root is plus or minus 5.

The minus sign is also featured in the following. Let x be
the quantity which satisfies the equation ¢®*= —1. By squar-
ing both sides we obtain ¢**=1. But a° always equals 1, no
matter what value a represents. Thus, a®=a®, so that 2x=o,
and x=o.

But, again &=

and, if ¥ equals nought, then a° may also be shown as a~.

Therefore a*=+1
But, from our data: a®=—1
SO +1=—1I
It has been noted that a®=1, irrespective of the value of a,
and that, therefore, 19=20=g0

but it does not follow from this that:

Similarly g: 1; except where a=b, for then (a—b) =o, so

a—b o
that e has no meaning in fact.

) o
In the same way, the expression y—- appears to be mean-
x

ingless since it is impossible to divide nought by x. The

expression is not, however, entirely devoid of meaning. It
may be re-expressed as:

Xy =0

whence we know that either x or » must be equal to nought.
This fact is a fundamental one and the solution of quadratic
equations depends largely upon it. Thus, where

(x—1) (*+1)=0
we can state that either (x—r1) or (v+1) is equal to nought
and that, therefore:

X=-+1 or —1I