IRRATIONALS AND IMAGINARY NUMBERS I

we proceed, the approximation becomes closer to the actual
value at each stage, and the different values thus obtained
are called the convergents of the fraction. Successive con-
vergents of the continued fraction for 221 are:

Ist convergent: 1
2nd e 1
grd &
4th Tos
sth 35
6th h 122
7th » 1502213

 

These convergent values are calculated direct from the con-
tinued fraction, taking one further step for each convergent.
After the first two convergents have been calculated, subse-
quent convergents may be calculated from the two preceding
convergents. This may be shown by substituting letters for
the respective integers. For example:

Ist convergent: ———

1D

2nd i
: $ I

grd term in fraction: —=
2

Then the third convergent may be shown as related to the
values a, b, ¢, d, ¢ and f as follows:
(¢ X f)+(axe)

grd convergent=,——_-"—-

dx f)+(bxe)
1x26)+(1x1) 27

" (2x26)+(1X1) 53