# How to properly define the physical dimension (unit) of a variable?

Let's say we have an equation like $$v=s/t.$$

I've come across multiple ways to define the units belonging to each variable:

1. $(v/(\mathrm{m}/\mathrm{s}) = (s/\mathrm{m})/(t/(\mathrm{s}))$
2. $v_\mathrm{m/s} = s_\mathrm{m} / t_\mathrm{s}$
3. $[v]=\mathrm{m}/\mathrm{s}, [s]=\mathrm{m}, [t]=\mathrm{s}$
4. $v=s/t|_{[s]=\mathrm{m}, [t]=\mathrm{s}}$
5. $s$ in m, $t$ in s and $v$ in m/s

Please enlighten me if there are more ways and please tell me what is the ''official'' way to define the units properly (by ''official'' I mean somewhere written down, like the SI units system).

• I think that option (3) is good for equations and (5) is good for text. Option (1) is sometimes used in equations where instead of the base SI unit you have some typical scale for that quantity. Like “for M in terms of solar masses, for R in terms of 100 parsec, ...”. – Martin Ueding Feb 16 '17 at 11:30

This is not really about the definition, but rather about the convention of writing.

Personally I only ever encountered "3" and "5", but as long as it is clear what it means, you can write things in any way you want. I'd find "1" and "2" confusing, but perhaps in some context it could make sense.

Also note that the equation you have here is general and does not require "defining" any units. If you put say the distance in km, you just end up with unit km/s for the velocity.

Really, in physics, the only time you need to specify the units for an equation is if you have units that cancel or when plotting data. For instance you might have an equation

$$x=1+\left(\frac{m}{M}\right)^2$$ where $[m]=g$ and $[M]=kg$. and here it is essential to specify units. The same equation written in general terms would look like:

$$x=1+\left(\frac{m}{1000 M}\right)^2$$

• thanks for pointing me to the difference between a definition and a convention, that seems to be indeed more a convention case here. You are also right that my example was not an optimal one. – Alf Feb 16 '17 at 13:59