Arithmetic systems such as those based on IEEE standards currently make no attempt to track the propagation of errors. However a formal error analysis can be complicated and is often confined to the realm of experts in numerical analysis. In recent years there has been a resurgence in interest in automated methods for accurately tracking the propagation of errors. In this article we will describe a model known as significance arithmetic. We will give details of the implementation in Mathematica along with several examples that illustrate the design goals and differences over conventional fixed precision floating point systems.