Functions, not dynamical equations, are the definitive mathematical objects in equilibrium thermodynamics. However, more than one function is often described as “the” equation of state for any one physical system. Usually these so named equations only capture incomplete physical content in the relationships between thermodynamic variables, while other equations, no less worthy of the name equation of state, go inconsistently by other names. While this approach to terminology can be bewildering to newcomers, it also obscures crucial properties of thermodynamic systems generally. We introduce specific principal equations of state and their complements for ideal gases, photons, and neutrinos that have the complete thermodynamic content from which all other forms can be easily deduced. In addition to effortlessly clarifying many smaller classical issues, they also make properties like the second law of thermodynamics and local thermodynamic equilibrium completely visual