Continued Fraction Geometry

Every real number has a unique CF expansion; Farey sequences and the Stern-Brocot tree encode rational approximations

Number

Value (0 to 1):

Depth: 8

Convergents p_k/q_k:

Farey Sequence F_n

Order n: 7

|p/q - p'/q'| = 1/(qq') for adjacent Farey fractions
Mediant: (p+p')/(q+q') — heart of Stern-Brocot

φ = [1;1,1,1,...] — worst approximable
π = [3;7,15,1,292,...] — 355/113 is excellent
e = [2;1,2,1,1,4,1,1,6,...]
Liouville: aₙ grow fast → very approximable