User blog comment:Vel!/Happy Pi Day/@comment-5150073-20150315122531/@comment-1605058-20150316114421

Wyth, what you have proven is that every irrational number has infinitely many 1's and infinitely many 0's in its binary expansion. Normality is far stronger property than this. It says us that these 0's and 1's are evenly distributed, as are pairs of consecutive digits, blocks of three consecutive digits etc.