Details

An approximant is defined to be a rational approximation where the numerator is the best choice for the given denominator. Showing the grid gives a helpful visualization of this concept: in each column, the approximant is the dot closest to the target line.

Snapshot 1: the fractions and , together with their mediant, , which is superposed on the target line having slope

Snapshot 2: two fractions whose mediant is 19/7, very close to

Snapshot 3: two fractions, and , both -approximants, whose repeated mediants , also -approximants, can be generated by repeatedly pressing the "" shortcut button

As claimed in the caption, the mediant vector is always between the original vectors and is closer to the target slope than at least one of them. This statement can be significantly generalized, if both fractions are approximants on opposite sides of the target real number. Then their mediant is also an approximant and is closer to the target than one of the original fractions. This allows a sequence of repeated mediants to be generated; for example, keep clicking "" in Snapshot 3 to get:

.

Proofs of these statements and additional material relating to mediants and repeated mediants can be found in [1].

Reference

[1] R. L. Caviness and K. E. Caviness. Making Pi(e) from Scratch, Rapidly and Memorably [Video]. AMATYC Webinar (Sep 27, 2019) www.youtube.com/watch?v=JcBr_v-n6AA.