# Why are all basic physical measurements ratios?

Why is it that "all basic physical measurements [...] are ratios"? This question comes from the book Mathematical Physics by Donald H. Menzel. Here is the explanation given:

1. The significance of an observation. Any direct observation of a physical nature ordinarily results in a number expressing the magnitude of the measured quantity. The simpler measures are those of lengths, of masses, or of times. More complicated ones may be of velocities, of energies, or of angular momenta. The number, by itself, does not indicate what is being observed; its magnitude depends on the upon the type of measuring scale employed. We may express lengths in centimeters, miles, or light years; we may define masses in grams, tons, or in units of solar mass. All basic measurements are, therefore, ratios.

Consider the definition of a 1 metre according to IUPAC

The metre is the length of path travelled by light in vacuum during a time interval of 1/299 792 458 of a second

So 2 metre would be twice of this. Three would be thee times of the above value.

Another example

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the round state of the caesium-133 atom

So two seconds would be twice the above duration. Three would be thrice the duration.

In a way all the measurement units are ratios of the basic definitions. See the complete list here(page 2) http://iupac.org/publications/analytical_compendium/Cha01sec41.pdf

• This makes sense, but what is he saying about centimeters, miles, light years, etc.? – Randy Randerson Oct 24 '13 at 5:38
• 100 centimetres is 1 metre. So 1 centimetre would be $\frac {1}{100}$ of a metre and so on. – GTX OC Oct 24 '13 at 6:01
• But a unitless measurement doesn't have to be a ratio. For example, I can count that I have five apples in a bag. – user4552 Oct 24 '13 at 15:49
• But first you must define what "one apple" would be. What is this 'one apple' that you speak off? Once you answer that, 5 apples would be well five times the "apple" that you defined. – GTX OC Oct 25 '13 at 12:34

The magnitude of the measured quantity depends on the type of measuring scale employed (i.e. the system of units used). Let $m$ and $M$ be magnitudes of a measured quantity, and let $u$ and $U$ be the corresponding units (definite magnitudes of the physical quantity). Then, we have $$mu = MU \implies m = \frac{MU}{u}.$$ Therefore, all basic physical measurements are ratios. To illustrate this result, consider two measurements of the mass of the Sun: $$1.9891 \times 10^{30} \textrm{ kg} = 1 \textrm{ solar mass} \implies 1.981 \times 10^{30} = \frac{\textrm{solar mass}}{\textrm{kg}}.$$ Thus, our measurement for the mass of the Sun (in kilograms) is a ratio.