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The Most Irrational Number


Rational approximation of irrational numbers

The decimal expansion of an irrational number gives a familiar sequence of rational approximations to that number. For example since pi = 3.14159... the rational numbers

r0 = 3,
r1 = 3.1 = 31/10,
r2 = 3.14 = 314/100,
r3 = 3.141 = 3141/1000,
...

give a sequence of better and better approximations to pi. We can measure the quality of these approximations by noticing that pi - rk < 1/10k : the error is less than one over the denominator of the fraction.

Similarly sqr2 = 1.41421... can be approximated by the sequence of rational numbers

1,
14/10,
141/100,
1414/1000,
...

with the same accuracy as the approximations to pi.

When we allow other denominators than powers of 10, the picture becomes different. For example 22/7 is a well known rational approximation to pi. The error in the approximation is 0.00126. Another rational approximation to pi is 355/113; this time the error is 0.000000266.

The best rational approximation to sqr2 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 pi.

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/sqr5q2.

The criterion can then be stated in terms of: how much less than 1/sqr5q2? to compare pi and sqr2 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/sqr5q2 by tabulating the quotient E/M.

pi: p/q E = error M = 1/sqr5q2 E/M
  22/7 .00126 .0091 0.13
  355/113 .000000266 .0000350 0.007
sqr2: p/q E = error M = 1/sqr5q2 E/M
  7/5 .0142 .0179 0.79
  239/169 .0000124 .0000156 0.79

These tables suggest that pi admits much better rational approximations than sqr2. In fact no rational approximation to sqr2 ever gets an E/M ratio as small as .13, let alone .007, and sqr2 is really harder to approximate with rationals than pi. In this precise sense sqr2 is a more irrational number than pi .


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