Physics for Mathematicians – Introduction

Examples of discrete quantities are the amount of substance and the electric charge.

The discrete quantities are just counted, so their values are integer numbers. They have a natural unit. Nevertheless, for those that are expressed in very large numbers it may be convenient to choose a conventional unit that is a big multiple of the natural unit, for instance the mole and the coulomb in the SI system of units.

All the continuous physical quantities are derived in some way from the measures of space and time, which is the reason for their continuity. For instance the electric charge is discrete, but the electric current is continuous, because it is the ratio between charge and time and time is continuous.

In order to measure a continuous physical quantity, a unit must be chosen. The unit may be chosen arbitrarily or it may be chosen in such a way as to eliminate universal constants from the formulae that express the relationships between physical quantities.

In either case, the value of a measurement is the result of a division operation between the measured value and the chosen unit, which is a real number, though it is normally approximated by a rational number.

In order to be able to define a division operation on the set of values of a physical quantity that has as a result a scalar, the minimum algebraic structure of that set of values is an Archimedean group.

That means that it must be possible to add and subtract and compare the values of the physical quantity and given two values it is always possible to add one of them with itself multiple times and eventually there will be a multiple greater than the second value (which will determine that the second value lies between two consecutive multiples of the first).

Based on the axioms of Archimedean groups it is possible to devise an algorithm that can multiply a value by a rational number and which can determine that a second value lies between two rational multiples that are as close as desired, producing by passing to the limit a real scalar. Thus any value can be divided by another value chosen as unit.

In practice, all the continuous physical quantities have richer algebraic structures, they are vector spaces over the real numbers, so the division of two collinear vectors is the scalar that multiplies one to give the other.

Nevertheless, the fact that the continuous physical quantities form vector spaces over the real numbers can be demonstrated based only on the supposition that they are Archimedean groups.

So the units of continuous physical quantities are just arbitrarily chosen values of those physical quantities, which are normally vector spaces with one dimension or with more dimensions, while the measured values are just rational approximations of the scalars obtained by division.

This division process is very obvious in the structure of the analog-digital converters used to measure voltages. These ADCs have two inputs, the voltage to be measured and the reference voltage, which is the arbitrarily chosen unit. The ADCs produce a rational number that is the approximate result of the division of the measured voltage by the reference voltage. If the reference voltage is not equal to the conventional unit, i.e. 1 V, the measurement result will be converted by multiplying with an appropriate conversion factor. The division operation can be done in the ADC for example by successive approximation, i.e. by binary search of the two multiples of a fraction of the reference value between which the measured value lies. The fraction of the reference voltage may be generated by a resistive or capacitive divider, while its multiples can be generated by a multiplying DAC.