This is insightful, but hard to read.

Essentially:

  1/3 * 4 = 4/3
  4/3 = 1 + 1/3

So, working backwards, our number can be more and more closely approximated by repeatedly adding 1 and dividing by 4 - or in binary, prepending a 1 and shifting two places right: 0.01010101...

Nice. Maybe an easier^3 could be: 1/3 = 0.xyz 4/3 = 1.xyz Meaning (shl,2) leaves the same digits after the decimal point. So it must be 0.010101...