Pythagoras' Constant
One of the first theorems people see in their schooling is the Pythagorean theorem.
A special case of this is pictured:
where  .
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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.
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We mention the continued fraction 
which is related to the Pell numbers (analogous to the Golden mean and the Fibonacci numbers).
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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.
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More details and references.
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Copyright © 1995-2001 Steven Finch
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