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Old 2017-02-27, 16:46   #6
Dr Sardonicus
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Feb 2017

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Given a real number r, and a small positive value e arbitrarily close to 0,

It depends, of course, on how you are given the real number. A numerical approximation to some prescribed number of decimal or binary digits, would be amenable to conversion to a "simple continued fraction" (an alternative search term). Also pursuant to fivemack's mention of the infinite SCF whose partial quotients are all 1, I mention Roth's Theorem which shows that algebraic numbers are "almost always" hard to approximate very well by rational numbers.

It is generally difficult to know both numerical and SCF representations of a given number. See, however, the following paper.

Last fiddled with by Dr Sardonicus on 2017-02-27 at 16:50
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