Construction

Start with 0/1 and 1/0 (imaginary infinity). Mediant of a/b and c/d = (a+c)/(b+d). Insert mediants recursively.

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Path encoding

Each rational corresponds to unique path L/R. Path = continued fraction expansion.

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Best approximations

Truncated path gives best rational approximation with denominator bound. Foundation of Diophantine approximation.

Applications

Best rational approximations. Farey sequences. Number theory research.