IRRATIONALS AND IMAGINARY NUMBERS 119

The convergent values for this expression are, in order:

oo e AR B
23 2R P ags g s

The values of x and_y which will satisfy the equation are then
found in the last convergent but one, if the total number of
convergents is odd (as in this case). Thus x=8; and y=p5,
these values being proved as under:

(8 x32) —(51 X 5) =1
The required values are found in a slightly different way

if the total number of convergents is even. For example, to
solve the equation:

13x —48y=1
we set up the fraction 12 and expand it to:
I

 

 

3+1
ey
2+1
4
and the convergents are:
S

If we substitute the integers in the last convergent but one
(i.e. g and 11) for values of » and x, we obtain the result:

(13 x11)—(48 x8)=—1
whereas we require values which will give a positive result
instead of a negative one.

We can, however, find the required values by deducting
the integers in the third convergent from the values of a and &
in the equation

ax—by=1

If we call the integers in the third convergent x, and y, to
distinguish them from the true values of ¥ and », then
x=b—x,=48—11=37
y=a—y,=13— 3=10

premmenmam Y

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