In this example we see continued fractions used to give rational approximations to irrational numbers.

The following solution was done by Ling Xiang Ning, Allan, Raffles Institution, Singapore.

Using the quadratic formula to solve the equation


x2
=
7x + 1
x2 − 7x − 1
=
0

I find that x is
(7 ±
√
 
53
 
)/2

. The positive solution is approximately 7.140054945

This equation is equivalent to x = 7 + 1/x and hence to the sequence of continued fractions mentioned in the problem. These continued fractions give better and better approximations to the positive root of the quadratic equation and I shall do them one by one.
7+  1
7
=
 50
7
 = 7.142857142
7+  1
7+  1
7
=
 357
50
 = 7.14
7+  1
7+  1
7+  1
7
=
 2549
357
 = 7.140056022
7+  1
7+  1
7+  1
7 + [ 1/7]
=
 18200
2549
 = 7.140054923 ?

To find a rational approximation to

√
 
53
 

we take, as above,
7 +
√
53
2
 ≈  2549
357

which gives

√
 
53
 
≈ 2 (  2549
357
 ) − 7 ≈  2599
357
 .

Similarly, using the equation, x2 = 5x + 1, which has solutions
5±
√
29
2

, we can find a rational approximation to

√
 
29
 

.

The positive root is approximately 5.192582404.

The sequence of continued fractions is:


5+  1
5
=
 26
5
 = 5.2
5+  1
5+  1
5
=
 135
26
 = 5.192307692
5+  1
5+  1
5+  1
5
=
 701
135
 = 5.192592592
5+  1
5+  1
5+  1
5+[ 1/5]
=
 3640
701
 = 5.192582025 ?

As you can see, the sequence of continued fractions gives better and better approximations to the positive root of the quadratic equation.

Using
5±
√
29
2
 ≈  3640
701

gives
 3775
701

as a rational approximation to

√
 
29
 

.