For an irrational number α, prove that the set { a + b α : a , b ∈ Z } is dense in R .

Question

For an irrational number   α, prove that the set   {  a  +  b  α  :  a  ,  b  ∈      Z    } is dense in       R  .

Rubi Boyle · Accepted Answer

Let   {  x  } to mean the fractional part of   x, i.e. for x minus the floor of   x. What we need to show is that we can get arbitrarily close to   0 by taking   {  m  α  } for varying integers m. Note that, because α is irrational,   {  m  α  }  ≠  {      m    ′    α  } for   m  ≠      m    ′  .Let&#039;s show that we can get within   1      /    n of   0 for an arbitrary positive integer   n. Divide up the interval   [  0  ,  1  ] into   n closed intervals of length   1      /    n. We have   n  +  1 distinct quantities   0  ,  {  α  }  ,  {  2  α  }  ,  …  ,  {  n  α  }.By the pigeonhole principle, two of these, say   {  i  α  } and   {  j  α  } with  i  &amp;gt;  j, lie in the same closed interval   [  k      /    n  ,  (  k  +  1  )      /    n  ], and so their difference, which is   {  (  i  −  j  )  α  }, is closer than   1      /    n to   0; n  was arbitrary.

For an irrational number α , prove that the set <mo fence="false" stretchy

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