# Show that all numbers can be written as $x=k\alpha+x'$, given...

The question is: Suppose $\alpha>0$. Prove that any real number $x$ can be written uniquely in the form $x=k\alpha+x'$, where $k$ is an integer and $0\leq x'\lt\alpha$.

How do I approach this problem? Perhaps construct a set for all $x$ that cannot be expressed in this way and show that it is empty?

• The word "numbers" should be identified a little more clearly I think...are you referring to integers? If it is then you are looking for en.wikipedia.org/wiki/Euclidean_division Oct 24, 2013 at 18:15