Coherent units
Part of my units pages
A system of measurement is coherent when it only contains one unit of distance, one unit of force, one unit of pressure, and so on for each distinct dimension.
The SI system is coherent, since it only has one unit of length (the metre), one unit of time (the second), and so on. In contrast, an “imperial” system containing more than one unit for the same dimension, say both feet and inches for distance, is not coherent.
Redundant names are not redundant units
The SI unit of pressure is the Pascal. However, we know from physics that pressure can be measured as a force divided by an area: P = F/A
The SI unit of force is the Newton, and the unit for area is the square metre, so “Newtons per square metre” is also an SI unit of pressure. It appears like SI is not coherent, but in fact the Pascal is just a shorter name for the “Newton per square metre”; that’s how the Pascal is defined.
Likewise, the Newton itself is a shorthand. We can express it as a combination of other units using Newton’s second law of motion: F = ma
This tells us that force (SI unit: Newtons) is the same as a mass (SI unit: kilogram) multiplied by an acceleration (SI unit: metres per square second). Hence the Newton is short for the “kilogram metre per square second”.
Such derived combinations are coherent as long as they involve no scaling. The metric system is hence not coherent, since it defines scaled units like the litre (0.001 cubic metres), the hour (3600 seconds) and the tonne (1000 kilograms). Likewise, in imperial systems the foot is not merely another name for a dozen inches; but even if it were, that would still be incoherent, since it’s a scaled unit.
Numerical terminology does not create redundant units
“10 sec” is not a unit of time distinct from the sec, it’s just ten of the latter. Likewise, a “dozen Volts” is not a unit of potential-difference distinct from the Volt, it’s just twelve of the latter. Similarly, a “kilometre” is not a unit of distance distinct from the metre, it’s just a thousand of the latter.
Conversion factors are proportionality constants
SI defines some “base units” (metre, kilogram, second, Ampere, Kelvin, mole, candela) and derives the rest using known scientific laws, like the Pascal and Newton shown above.
Such laws are actually statements of proportionality, e.g. F ∝ ma tells us that doubling the mass will double the force (for the same acceleration), but it doesn’t let us calculate one side from the other; since that depends on the system of units being used, which is circular if we’re using these laws to define our units!
To turn these into equations, we need to introduce an arbitrary scaling factor (called a “constant of proportionality”) like F = fma
SI units are those whose constants of proportionality are all 1, resulting in those simple equations like F = ma
Once we know a few physical laws, it’s then straightforward to find the relationships between SI units. For example, the Watt is usually defined as a Joule per second; but for electrical devices it can also be calculated as the current (SI unit: Ampere) multiplied by the potential-difference (SI unit: Volt). Those are equivalent to each other, and also to many other SI units:
1 Watt = 1 Joule per second
= 1 Hertz Joule
= 1 Hertz Newton metre
= 1 Hertz Pascal cubic metre
= 1 Hertz square Coulomb per Farad
= 1 Newton Coulomb per Tesla Farad metre
= 1 Newton Joule per Tesla Coulomb metre
= 1 Newton Volt per Tesla metre
= 1 Amp Volt
= 1 Watt