## 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 a_{1},a_{2},a_{3}, ... such that x is the limit of the rational numbers:

$c_1 = a_1$

$c_2 = a_1 + \frac{1}{a_2}$

$c_3 = a_1 + \frac{1}{a_2 + \frac{1}{a_3}}$

$c_4 = a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \frac{1}{a_4}}}$

$c_5 = \ldots$

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**

*From the decimal expansion.* If we know the number well enough, for example to enough decimal places, we can calculate its continued fraction expansion as we show here with x = = 3.14159265359...
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.

- a
_{1} is the integer part, here 3. This is the largest integer which makes c_{1} less than .
- With a
_{1} = 3, a_{2} is the largest integer which makes c_{2} greater than .

8 is too large because 3 + 1/8 = 3.125... < ,

but 7 makes it since 3 + 1/7 = 3.1428.. > .

So a_{2} = 7, and c_{2} = 3 + 1/7 = 22/7.
- With a
_{1} = 3 and a_{2} = 7, a_{3} is the largest integer which makes c_{3} less than .

16 is too large because 3 + 1/(7+1/16) = 3.14159292... > ,

but 15 works since 3+1/(7+1/15) = 3.1415094... < .

So a_{3} = 15, and c_{3} = 3+1/(7+1/15) = 333/106.
- Continue, alternating "greater" and "less." You should find a
_{4} = 1 and a_{5} = 292. More a's will require more decimal places.

*If the number is a root of a quadratic equation* ax^{2} - bx - c = 0 with b strictly positive, and a = c = 1, the equation yields an algorithm for generating the continued fraction expansion. itself does not work, but x=1+ is a root of x^{2}-2x-1=0. The equation can be rewritten as x=2+1/x. Substituting the right hand side into itself gives x = 2+1/(2+1/x), and then x = 2+1/(2+1/(2+1/x)) etc. Since = (1+) - 1, the procedure yields for the continued fraction expansion:

$1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \ldots}}}}$

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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