We say a real number $y$ preserves normality in base $b$ under addition if for every $x$ which is normal in base $b$ the number $x+y$ is also normal in base $b$. The rational numbers and any number with only finitely many non-zero digits have this property. G. Rauzy defined an entropy-like quantity called noise and showed a real number preserves normality in base $b$ under addition if and only if it has noise zero. Using this characterization we find the complexity of numbers with noise $s$ which gives the complexity of the set of normality preserving numbers as well as another proof of previous complexity results for the set of normal numbers.