The "best" rational approximations, as well as most of the theory of rational approximation, arise from continued fraction expansions.
A continued fraction expansion for a positive number x is a sequence of positive integers a_{1},a_{2},a_{3}, ... such that x is the limit of the rational numbers:
c_{1} = a_{1} c_{2} = a_{1} + 1 --- a_{2} c_{3} = a_{1} + 1 ------- a_{2} + 1 --- a_{3} c_{4} = a_{1} + 1 ------- a_{2} + 1 ------- a_{3} + 1 --- a_{4} c_{5} = etc.The numbers c_{1}, c_{2}, etc are called the convergents of x. They are important in this context because the best rational approximations to an irrational number are always found among its convergents.
Any rational number p/q which approximates x to within 1/2q^{2} must be one of the convergents of x.
In terms of our table, this means that any rational approximation with E/M < 1.118.. must be a convergent.
Calculating a continued fraction expansion
The point is to make c_{1}, c_{2}, c_{3}, etc. alternately smaller and larger than x by choosing the largest possible a_{1}, a_{2}, a_{3}, etc.
1 + 1 ------- 2 + 1 ------- 2 + 1 ------- 2 + 1 ------- 2 + etc.
and the convergents for are c_{1}= 1, c_{2}= 3/2, c_{3} = 7/5, c_{4} = 17/12, c_{5} = 41/29, c_{6} = 99/70, c_{7} = 239/169, ...
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