Let α , β be the linearly independent irrational numbers over Q with α &gt; β &gt; 0 , and A = { n α − m β ∣ n , m are nonnegative integers },m are nonnegative integers}How to prove that A is dense in R ? Is it true?

Question

Let   α  ,  β be the linearly independent irrational numbers over       Q   with   α  &amp;gt;  β  &amp;gt;  0 , and       A    =  {  n  α  −  m  β  ∣  n  ,  m   are nonnegative integers  },m are nonnegative integers}How to prove that       A   is dense in       R   ? Is it true?

Davin Meyer · Accepted Answer

Note that  A  is closed under additioninf   inf  A  =  −  ∞sup  sup  {    x  ∈  A  ∣  x  &amp;gt;  0    }  =  0The first is trivial, the second follows from   β  &amp;gt;  0, and only the third involves the       Q  -linear independence of   α  ,  β.

Let alpha,beta be the linearly independent irrational numbers over Q with alpha>beta>0 , and A={n alpha−m beta∣n,m are nonnegative integers} How to prove that A is dense in R ? Is it true?

Answered question

Answer & Explanation

New Questions in Algebra I