This picture was in my head

Hardy-Littlewood thanks for the memories. I always appreciate it when something in my head shows up as a true idea worked on before, it makes me feel less nutty. But this is where I was, trying to figure out how a bunch of bubbles could compute the value of one. I wonder, let a := 1/(8^1/2) and and let b := 1/(5^1/2), then created two sequences of a^n and b^n, for n from 1 to 16, say. Would they sort in a regular pattern? Each sequence, in its own digit system looks like: .11111  a good approximation to 1. Hmm... Then take, for all three orders of irrationality, the full list, up to 16 bits, of each.  We are quite a bit beyond my brain here.