Pythagoras' Constant

One of the first theorems people see in their schooling is the Pythagorean theorem. A special case of this is pictured:

where .
It has been known for thousands of years that is irrational, that is, cannot be written as a ratio of integers. Here is a proof of the irrationality of based on divisibility properties of the integers. If was rational, then the equation would be solvable in integers p and q, which are assumed to be in lowest terms. Since is even, p itself must be even and so has the form . This leads to and thus q must also be even. But this contradicts the assumption that p and q were in lowest terms.

We mention the continued fraction

which is related to the Pell numbers (analogous to the Golden mean and the Fibonacci numbers).

Plouffe gave a highly accurate approximation of in the Inverse Symbolic Calculator web pages. See also Gourdon and Sebah's essay from Numbers, Constants and Computation.
More details and references (contact Steven Finch).
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