k 
x_{k} 
y_{k} 
r_{k} 
q_{k} 
Error=
x_{k}/y_{k}0.69231 
Remarks 
1 
1 
0 
1 


First set up the table with the 1's and the
0's as shown. r_{1} is the larger of 1 and
the decimal. r_{2} is the smaller. 
2 
0 
1 
0.69231 
1 

Compute
q_{2}=floor(r_{1}/r_{2}).
Continue to do this, using the appropriate q's and
r's until the remainder is zero, or the accuracy is
sufficient.
Compute:
x_{3}=x_{1}q·x_{2}=1, and
y_{3}=y_{3}=y_{1}q·y_{2}=1 
3 
1 
1 
0.30769 
2 
0.30769 
Continue computing the x's and the y's 
4 
2 
3 
0.07693 
3 
0.02564 

5 
7 
10 
0.0769 
1 
0.00769 

6 
9 
13 
0.00003 
2563 
0.000002 
At this point we have a decimal approximation that is
accuracy to five figures. Because we might expect 0.69231
to be accurate within +/0.000005, this may be the
optimal answer. (Actually, this is the fraction I rounded
to produce the starting decimal, and the remainder here
is a rounding error). 
7 
23074 
33329 
0.00001 
3 
0.0000000003 
With greater accuracy, we have a large
denominator. Going this far assumes that the initial
decimal is actually 0.692310000 
8 
69231 
100000 
0 
 
0 
The x's and y's are expressed as a decimal
fraction, the same as 0.69231. This occurs when
gcd=1 