People have a strong preference for fairness. For many, fairness means equal reward and punishments for equal efforts and offences. However, this belief does not specify the units in which equality should be expressed. We show that people generally fail to take the interchangeability of units into account when judging and assigning fair punishments and reward. Therefore, judgments about and distributions of resources are strongly influenced by arbitrary decisions about which unit to express them in. For example, if points represent different monetary values for different recipients, people attempt to distribute money equally if money is salient but attempt to distribute points equally if points are salient. Because beliefs about fairness are a fundamental principle in many domains, the implications of these findings are broad. Essentially any distribution of outcomes can be made to appear more or less fair by changing the units these outcomes are expressed in.