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A geometrical approach to measure irrationality. (English) Zbl 1432.11093

Summary: We present a geometric way of describing the irrationality of a number using the area of a circular sector \(A(r)\). We establish a connection between this and the continued fraction expansion of the number, and prove bounds for \(A(r)\) as \(r\rightarrow\infty\) by describing the asymptotic behavior of the ratios of the denominators of the convergents.

MSC:

11J82 Measures of irrationality and of transcendence
11A55 Continued fractions
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References:

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