June 2000 15+ Challenges
Tough Nut

Solution to 
Good Approximations

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.
A:\Sep00\jun00_tn397.doc   30 July, 2000