A very useful result for generating random numbers is that the fractional part of a sum of independent U(0,1) random variables is also a U(0,l) random variable. In this paper we show that a more general result is true: the fractional part of a sum of n independent random variables, one of which is U(0,l), is also U(0,l). Moreover, we show that the fractional part of a sum of independent near-uniform variables is closer in distribution to a U(0,l) variate than each of the component near-uniform variables. These results are used to characterize the uniform distribution and to give some justification for an algorithm of Wichmann and Hill(1982). In addition, we show how the property of “closeness” carries over to the generation of any random variable.