Lets say we have irrational numbers α 1 , . . . , α n in the interval ( 0 , 1 ). Represent each αi as a binary expansion 0. a i 1 a i 2 . . . where each a a i j ∈ { 0 , 1 }. Define the "dovetail" of the α i to be the number with binary expansion 0. a 1 1 a 2 1 a 3 1 . . . a n 1 a 1 2 a 2 2 a 3 2 . . . a n 2 a 1 3 a 2 3 a 3 3 . . .So that the dovetail is irrational?

Question

Lets say we have irrational numbers       α    1    ,  .  .  .  ,      α    n   in the interval   (  0  ,  1  ). Represent each αi as a binary expansion   0.      a    i    1        a    i    2    .  .  . where each a      a    i    j    ∈  {  0  ,  1  }. Define the &quot;dovetail&quot; of the       α    i   to be the number with binary expansion   0.      a    1    1        a    2    1        a    3    1    .  .  .      a    n    1        a    1    2        a    2    2        a    3    2    .  .  .      a    n    2        a    1    3        a    2    3        a    3    3    .  .  .So that the dovetail is irrational?

isscacabby17 · Accepted Answer

It&#039;s never rational. This follows from the fact that a real number is rational if and only if its binary expansion is eventually periodic (possibly with endless   0s).