[MikeBattaglia]

He points out that the YouTube video "uses mediants" to compute rational approximations to real numbers. He then says that the continued fraction expansion is better. However, the mediant-based method is totally identical to the continued fraction expansion method. The two are related via the Stern-Brocot tree (https://en.m.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree).

The fractions that one gets from iterating the mediant process are exactly the "semiconvergents" one gets from the continued fraction process. Truncating the continued fraction gives the "convergents"; these are special mediants which are right before switching direction as one zig zags on the tree.