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114 THE FASCINATION OF NUMBERS

Archimedes attacked the problem of calculating the value
of = by the method of approximating to a circle by means of
alternately inscribed and circumscribed polygons. This, of
course, can only give an approximate value because, no
matter how near a polygon may be made to resemble a
circle, it can never actually become a circle. The polygons
may therefore be said to converge on the circle without
actually approaching it, and the unattainable (i.e. the circle)
may be said to constitute the limit of such a convergence.

This concept of a convergence to a limiting value is of
immense importance in theories concerning the irrational
numbers and the infinite nature of their composition.

One method of calculating approximate values is by the
use of continued fractions. The square root of 2 may be repre-
sented by this means as follows:

V2=I1+4I
2+1
241
241 ... etc.

 

In order to represent +/2 in this way, we first extract the
integral part, 1, and express the remainder as a fraction:

 

Va=14—
X1
where x, is greater than 1. From this,
\/2—~1=—I—
X1
I
SO T/;:’fi
But
I
-\/2_1.=\/2+1, because (4/2—1)(4/241)=1
therefore
Xy =14/241.