The fraction part of xR is {x}=x-x, where is the Gauss floor function. Note that 0={x}1. let R\Q.

The fraction part of x∈R is {x}=x-⌊x⌋, where ⌊ ⌋ is the Gauss floor function. Note that 0<={x}<1. let β∈RQ. show that

(1) for any m,n∈Z, if m≠n then {mβ}≠{nβ}

(2)for any k∈N, there exist m,n∈Z , such that 0<mβ+n<1/k

(3)use (2) to conclude that the set {mβ+n: m,n∈Z} is dense in R