IRRATIONALS AND IMAGINARY NUMBERS ITH

The integral part of x, is therefore 2. We extract this from
x; and express the remainder as a fraction:

 

 

I
x1=2 _{.__
Xg
% I
From this Xg=
x1_2
T I I
SO R e
=2 +/2+1—2 4/2—1
! 2 |
and Xy

§1 \/ 2= i \/2 o :
Thus, », and #, are identical, as indeed will be x,, x,, etc.

The integral part of each is 2 and we therefore have a con-

tinued fraction which consists of partial denominators all of

which are 2.
An easier way of writing out a continued fraction is as

follows:

i X

2 =1 —{—5-}—5—{—5—{—5 etc.

The plus signs, printed in line with the divisors instead of in
the usual position, indicate that the expression is a repre-
sentation of a continued fraction. There is, therefore, a great
and fundamental difference between the expressions:

I I I I I
@)E+E+5+5+5

Any positive number may be represented as a continued
fraction. The general procedure is now detailed. We first

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VORI o s e R R