Prove:If x is irrational, then n x − [ n x ] are distinct for all n ∈ Z .

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Prove:If   x is irrational, then   n  x  −  [  n  x  ] are distinct for all   n  ∈      Z  .

Pranav Greer · Accepted Answer

Suppose   n  x  −  [  n  x  ]  =  m  x  −  [  m  x  ]. Then  (  n  −  m  )  x  =  [  n  x  ]  −  [  m  x  ]  ∈      Z  But an integer multiple of an irrational number can only be an integer if it is   0, so we must have  n  =  mTherefore it is distinct for all   n.

DIAMMIBENVERMk1 · Accepted Answer

We have to prove that for distinct  i  ,  j  ,  {  i  x  }  ≠  {  j  x  } where   {  x  } denotes the fractional part and   x is irrational.If   {  i  x  }  =  {  i  j  } then   i  x  −  i  j  =  ⌊  i  x  ⌋  −  ⌊  i  j  ⌋ or   x  =            ⌊      i      x      ⌋      −      ⌊      i      j      ⌋              i      −      j        ∈      Q   which is a contradiction.

Prove:If x is irrational, then n

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