The values of pairs of related physical quantities are shown to arise in the judgement of one person, the first cogniser, in the union of observation and concept. An expression of each value is observed and, using an original method, its numerical value is mapped, by a mathematical function deemed appropriate, to an explanatory rational number that both indicates the significance of the physical quantity and establishes its value. If the physical quantity is considered to be significant the explanatory number will take a rational value of low denomination. If the quantity is considered to be of great significance the explanatory number will take a special integer or half-integer value. Having been explained with sound judgement, the value of the physical quantity is what anyone will experience on observation. By the use of two mapping functions, the numerical values of significant quantities deemed to be paired are mapped to closely coincident special integers or half-integers. Mathematical relationships established between the values of quantities reflect the thinking of the observer. Fundamental particle and atomic quantities are judged in Planck units. Astronomical and cosmological quantities are judged in SI units.