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12

Irrationals, Imaginary Numbers
and Continued Fractions

Whereas the exact nature of many number relationships may
be more easily demonstrated by constructional representa-
tion, the proportionate dimensions resulting from certain
other constructions served only to create confusion in the
early days of Mathematics. The numbers thus produced were
not of the ordinary kind. They were not integral and, indeed,
were subsequently proved to be incapable of expression in
ordinary notation at all.

The equation: a*=62+¢2, when applied to a right-angled
triangle where b =c=1, produces the result:

a=+/2

The inability of the Pythagoreans to represent +4/2 as
an integer was a grave eruption in their philosophical
world.

Another inexpressible quantity which has exercised the
minds of mathematicians for centuries is the ratio between
the diameter and the circumference of a circle. If the
diameter is measured off along the circumference, it is found
that the latter is slightly more than three times as great as the
former. This ratio is represented by the symbol z (the Greek
letter, Pi), and one of the great mathematical pastimes of the
past was the attempt to ‘square the circle’, until it was dis-
covered that & was inexpressible, or transcendental, and that
the objective was therefore impossible.

Euclid proved the impossibility of expressing 4/2 as a
rational number; his reasoning being as follows. If such a

112