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70.42 The density of Pythagorean rationals

70.42 The density of Pythagorean rationals If a, b, c are positive integers which satisfy a + b = c, then the number a/b is called a Pythagorean rational. These numbers are 'dense' in the set of nonnegative real numbers in the sense that, if x and y are different positive numbers, then there is a Pythagorean rational between them. After a discussion with L. H. Lange about his proof of this result in [1] I began looking for a picture that would yield a proof and I came up with the following argument.