r_{0} = 3, |
r_{1} = 3.1 = 31/10, |
r_{2} = 3.14 = 314/1000, |
r_{3} = 3.141 = 3141/10000, |
... |
give a sequence of better and better approximations to . We can measure the quality of these approximations by noticing that - r_{k} < 1/10^{k} : the error is less than one over the denominator of the fraction.
Similarly = 1.41421... can be apprroximated by the sequence of rational numbers
1, |
14/10, |
141/1000, |
1414/10000, |
... |
with the same accuracy as the approximations to .
When we allow other denominators than powers of 10, the picture becomes different. For example 22/7 is a well known rational approximation to . The error in the approximation is 0.00126. Another rational approximation to is 355/113; this time the error is 0.000000266.
The best rational approximation to with a denominator less than 10 is 7/5; the error is .0142. The best with a denominator less than 200 is 239/169, with error .0000124. This is much less satisfactory than the rational approximations to .
Observations like these have led mathematicians to set up a hierarchy among irrational numbers, according to how difficult they are to approximate with rationals. It is in this sense that one irrational is more irrational than another. To make the criterion precise, we start from the following fact:
Hurwitz' Theorem: Every number has infinitely many rational approximations p/q, where the approximation p/q has error less than 1/q^{2}.
The criterion can then be stated in terms of: how much less than 1/q^{2}? to compare and in this regard, some of the best rational approximations can be displayed in a table. Here we will reckon the error E in terms of Hurwitz' bound M = 1/q^{2} by tabulating the quotient E/M.
: | p/q | E = error | M = 1/q^{2} | E/M |
22/7 | .00126 | .0091 | 0.13 | |
355/113 | .000000266 | .0000350 | 0.007 | |
: | p/q | E = error | M = 1/q^{2} | E/M |
7/5 | .0142 | .0179 | 0.79 | |
239/169 | .0000124 | .0000156 | 0.79 |
These tables suggest that admits much better rational approximations than . In fact no rational approximation to ever gets an E/M ratio as small as .13, let alone .007, and is really harder to approximate with rationals than . In this precise sense is a more irrational number than .
Corrected 11/12/13
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