Is the sum of two irrational numbers almost always irrational?

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ebizsavvy1txn · Accepted Answer

Let   N  P be the set of pairs whose sum is rational. I think its easier to prove   λ  (  N  P  ∩  B  (  x  )  )  =  0. In fact since   N  P  ∩  B  (  x  )  ⊂  N  P, we just prove   λ  (  N  P  )  =  0 and we are done. Let  N      P    x    =  {  (  x  ,  y  )      |    x  +  y  ∈      Q    }Notice that the restriction addition to this set is translation by   x which is measure preserving, hence the inverse image of the rational numbers has measure zero. However, there is a weak form of Fubini&#039;s theorem that says that if a subset of a product measure space ( which             R        2   is) has the property that its intersection with each slice has measure zero then the set has measure zero. Hence   λ  (  N  P  )  =  0.To bring this back to the specific question you are asking,   N  P  ∪  P  =            R        2  , so for any open ball   B  (  x  )   λ  (  P  ∩  B  (  x  )  )  =  1. On the other hand the set of points whose coordinates are irrational is the complement of a set of measure zero, so   λ  (  R  ∩  B  (  x  )  )  =  1. Hence you are taking the limit of   1      /    1.

Is the sum of two irrational numbers almost always irrational?

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