The most irrational number


Where do the "best" rational approximations come from?

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 a1,a2,a3, ... such that x is the limit of the rational numbers:

c1 = a1

c2 = a1 +  1

c3 = a1 +  1
         a2 + 1

c4 =  a1 +  1
          a2 + 1
               a3 + 1

c5 = etc.

The numbers c1, c2, 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 1/2q2 
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

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© copyright 1999, American Mathematical Society.