# Square bracket notation for dimensions and units: usage and conventions

One of the most useful tools in dimensional analysis is the use of square brackets around some physical quantity $q$ to denote its dimension as $$[q].$$ However, the precise meaning of this symbol varies from source to source; there are a few possible interpretations and few strict guidelines. What conventions are there, who uses them, and when am I obliged to follow them?

• Thanks to LaRiFaRi for sparking this post, and providing the BIPM and ISO references, in this question. – Emilio Pisanty Sep 17 '13 at 20:50

I had an extensive look around, and I turned up four conventions. This included a short poll of google, other questions on this and other sites, and multiple standards documents. (I make no claim of exhaustiveness or infallibility, by the way.)

1. Using $[q]$ to denote commensurability as an equivalence relation. That is, if $q$ and $p$ have the same physical dimension $Q$, one might write $$[q]=[p]=[Q],$$ but no bracketed quantity is ever shown equal to an unbracketed symbol. Thus, if $v$ is a speed one might write $[v]=[L]/[T]$ or $[v]=[L/T]$ or $[v]=[L\,T^{-1}]$ or some equivalent construct. You can see $L$ and $T$ as denoting the dimension or just "some length" and "some time". To see how you would work without evaluating braces, here is a proof that the fine structure constant is dimensionless: $$[\alpha]=\left[\frac{e^2/4\pi\epsilon_0}{\hbar c}\right]=\frac{[F\,r^2]}{[E/\omega][r/t]} =\frac{[F r][\omega t]}{[E]}=\frac{[E]}{[E]}= ,$$ so $\alpha$ and $1$ are commensurable. Some examples are this, this, this, or this.
2. Using $[q]$ to denote the dimensions of a quantity. Thus if the physical quantity $q$ has dimension $Q$, one writes $$[q]=Q.$$ A velocity would then be written as $[v]=L\,T^{-1}$ or its equivalents. This seems to be the leading candidate on Google, closely followed by the convention 1. Some examples are this, this, this, this, and this. This is my personal favourite, as I find that it permits the most flexibility without horribly formalizing the whole business (though I will often skip the actual evaluation of the braces, essentially using convention 1).
3. Using $[q]$ to denote the units of a quantity. Here if $q$ can be written as a multiple of some unit $\text q$, you write $$[q]=\text q.$$ This is contingent on what unit system you choose but different units for the same dimension are of course equivalent. When this approach is used, the notation $\{q\}=q/[q]$ is sometimes used to denote the purely numerical value of the quantity. A speed would be written, for example, as $[v]=\text m\,\text s^{-1}$. This use is endorsed by the NIST Guide to the SI, section 7.1, the IUPAC guide Quantities, units and symbols in physical chemistry, the IUPAP guide Symbols, units, nomenclature and fundamental constants in physics, as well as the ISO standard ISO 80000-1:2009, section 3.20. (That document is very paywalled, but chapters 0-3 are available for free preview here.)

Google results seem relatively scarce, with this and this as examples, although that could simply be poor representation. (There is also this document, which uses the notation $[\text W]=[\text V][\text A]$, but I think this is quite uncommon as well as not very useful.)
4. Using $\operatorname{dim}(q)$ to denote the dimensions of a quantity. This is the notation set as standard by the Bureau International des Poids et Mesures in the SI Brochure (8th edition, chapter 1.3, p. 105). This also sets roman sans-serif as the standard for physical dimensions, so $\mathsf{Q}$ would be the dimension of $q$ and you write $$\operatorname{dim}(q)=\mathsf Q.$$ (To typeset roman sans-serif in TeX or MathJax, use \mathsf; note that this is distinct from \operatorname, which is used for $\operatorname{dim}$ and would produce $\operatorname{Q}$ through \operatorname{Q}.) A real-world usage example is thus $\operatorname{dim}(v)=\mathsf L\,\mathsf T^{-1}$ for a velocity.

This use is set as standard by ISO 80000-1:2009, section 3.7, and it is also endorsed by the NIST Guide to the SI, section 7.14. (NIST also reproduces the BIPM text in p. 16 of The International System of Units.) Examples of this online are this, this, this and this; I note, though, that most examples I found are technical, while pedagogical examples tended to use conventions 1 and 2. (This also feels less common, but that's hard to judge.)

I also find it important to add that few academic journals impose standards in this area. As a working physicist in academia, the style guidance of one's chosen journal is often the only style standard one is really obliged to follow. The style manuals of the American Physical Society, the Institute of Physics, Reviews of Modern Physics, Nature Physics and several Elsevier journals have no mention of which convention should be used in their publications.

As was made clear in Should we necessarily express the dimensions of a physical quantity within square brackets?, the choice of what the symbol $[q]$ means is entirely a matter of convention. The most important thing is that your usage is consistent. Do not jump conventions within a document. If your work is closely allied to other resources (e.g. textbooks) that use a particular convention, it is best to stick to that, to avoid confusing your students. If you are presenting an exam, use the notations used in your course to avoid confusing your examiner, or - at the very least - define all non-standard notation you use.

So, what convention should you use? There is really no requirement to use any one of the above (and you can even make up your own notation, as long as you define it appropriately and don't overdo it). This is really less of an issue than it looks, as there's actually rather rarely a need to use this notation in print except in pedagogic settings. (That's not to say that professional physicists don't use it in practice: we do use it, often, in everyday life, but it's mostly informal work used on the side to keep calculations straight or as exploratory scaling arguments when starting work on a problem, for example.)

If your work is a commercial report, or similar document, and it could potentially have legal repercussions, then you should check whether there is a legal standard you should be using, which will then probably be conventions 3 and 4. Academically, you are typically free to choose the conventions you find most convenient as long as you use them properly and you avoid conflicts with other allied resources. If you are publishing in a journal or as part of a bigger work, you should check if they provide style guidance on this, though as I said journals rarely take stances on this. (You should really be reading the style guidance anyway as part of your submission process, though.) For your informal work, you should use whatever you're most comfortable with!

Finally, if you have questions about the typesetting of these notations in LaTeX, you should go to How should I typeset the physical dimensions of quantities? on TeX.SE.

• @Emilio I am having problems to understand the first point. If $q$ and $p$ have the same dimension, it is the same as point 2. If $[q]=Q$ and $[p]=Q$ than $[q]=[p]$. But in the examples for point one, you actually write $[q]=[Q]$. I hope, you understand what I mean. I am getting a bit confused on all my "dimensions-research" :-) – LaRiFaRi Sep 18 '13 at 7:53
• To the examples in 1: The second one is of type 2. $[q]=[p]^\alpha [r]^\beta \dots$. The third and fourth is of type "The dimension of $q$ is $[Q]$". The first one is strange and like the one, you explained further. $[q]=[Q]$ which I find very confusing. – LaRiFaRi Sep 18 '13 at 8:05
• Am I the only person in the world who writes $[t]=\text{s}$? Maybe this is just an artifact of the way I teach the topic, but I teach my students to write out an expression like $m\cdot a$ in terms of the SI base units like $\text{kg}\cdot\text{m}/\text{s}^2$. If you think of the algebra of units as a vector space, then this corresponds to fixing a basis. Of course "a gentleperson never fixes a basis," but for freshmen, abstraction is hard and anything concrete is easier. – user4552 Sep 20 '13 at 15:51
• @BenCrowell Sure, that's convention 3. It's not that common online but my feeling is people do use it. It's also the recommended standard in as much as there is one. – Emilio Pisanty Sep 20 '13 at 16:23
• @BenCrowell: I am also using convention 3. – student Apr 19 '17 at 20:27

It may also be worth mentioning another shorthand convention that appears in the field theory literature and some textbooks (e.g. A. Zee, Quantum Field Theory in a Nutshell, Chapter III.2). This is essentially a modified version of the second convention on Emilio's list. If one is working in the "natural" units of the problem then often everything can be expressed in terms of a single dimensionful quantity. For example, particle physicists usually set $\hbar = 1$ and $c = 1$ so that all quantities carry the dimension $\mathrm{mass}^x$ for some power $x$. In that case, one can simply write $$[q] = x.$$ To give an example, with $\hbar = c = 1$ a length $l$ would be assigned a mass dimension $[l] = -1$.