# Old square bracket notation for units

As discussed in this answer https://physics.stackexchange.com/a/77691/667 there are several common conventions for the notation $[q]$ of a physical quantity $q$.

However, I often see people to put the square brackets around the (SI-) unit itself. For example $[\mathrm{kg}]$ (for kilogram). Most time the people using this are over 50 years old and often engineers. So, I got the impression that this convention was common some years ago.

So, what are the origins and reasons for this old convention?

• I don't follow the reason for your comment "though from a modern point of view it doesn't seem to make sense". In particular, what do you mean by "modern" with respect to not making sense. As for people who are over 50, including me (my age is in open interval $69.551 < \textrm{my age} < 70$, I have always thought the $[]$ bracket notation as used in dimensional analysis was useful (not sure though that "making sense or not making sense" is applicable). Commented Apr 20, 2017 at 17:48
• @K7PEH: Ok I guess I should formulate it a bit weaker. For me it is clear that the bracket notation as discussed in the link above, where the bracked is around the physical quantity makes perfectly sense for dimensional analysis. However (and this is the weaker version of my comment) I don't see why writing the brackets around the unit itself makes any sense, for example $[\mathrm{kg}]$. For example people use it like this: $E [\mathrm{J}] = \frac{1}{2} m [\mathrm{kg}] \cdot v^2 [\mathrm{\frac{m^2}{s^2}}]$. Commented Apr 20, 2017 at 19:45
• In my own experience with dimensional analysis, I would not write the equation has you have done. Instead it would be written like $[E] = [M][L]²[T]^{-2}$. Where, E is dimension for Energy, M is Mass, and L is length, and T is time. Note that constants are not included the the $1/2$ is left out. If you have not yet done so, Google Buckingham Pi Theorem to see how such dimensional analysis is used. Commented Apr 21, 2017 at 3:22
• @K7PEH Ok, so you use one of the modern conventions (as discussed in the link), but that's not what I am asking about. Commented Apr 21, 2017 at 5:44
• I couldn't find the use you describe in books or publications, do you have a reference? It might just be taking SI symbology too literally:"The value of a physical quantity Q can be expressed as the product of a numerical value {Q} and a unit [Q], Q={Q} [Q]". Of course, braces and brackets are supposed to be replaced by numbers and unit names, not enclose them, but the phrasing is highly misleading. Commented Apr 21, 2017 at 21:38

The German DIN Norm 461 from 1973 explicitly says, that units must not be put in brackets and further relates to DIN 1313. The first version of DIN 461 is from 1923.

Die Einheit darf keinesfalls in Klammern gesetzt werden (siehe DIN1313)

According to DIN, the square brackets are used as an operator to extract the unit from a physical quantity and curly brackets are used to extract the number:

m = 5 kg

{m} = 5

[m] = kg

In that context [kg] does not make sense.

Therefore, the idea of not using units in (square) brackets is quite old.

I just had a look at some papers from the 1980s. Both notations occurred there. Therefore, the idea to use units inside brackets is quite old, too.

Would be interesting to know which of the notations is older. Has the square bracket notation around units already been used before 1923?

It is a habit of text books and handbooks (of engineering branches as well as of chemistry and even physics) to indicate the unit of numerical values given in tables by column headings like "m [kg]" or "m in kg". The mathematically correct way is "m/kg" since if the mass is m = 3 kg, then the numerical value is clearly 3 = m/kg. But even when students are told this, I see often the aformentioned habit show through. This might be the reason for further [ab]use of square brackets. (It is not distorting. We know what is meant.)

• I am personally involved in the accelerator physics community, and I would say that square bracket notation for units is the most common, closely followed by parentheses, in both internal documents/presentations, and in proper peer-reviewed papers. The particle physicists however tend to write /kg in their publications.
– a20
Commented Mar 27, 2019 at 8:03
• I really don't see how "m/kg" deserves to be called 'mathematically correct'. This is because the example offered as a logical justification (" if the mass m is 3 kg then the numerical value is clearly 3 = m/kg") actually seems logically defective. It treats the expression 'kg' as if it were a coefficient, capable of being used symbolically as a multiplier or divisor, instead of the descriptive qualifier that it actually is. Commented Dec 12, 2021 at 0:16
• @terry-s: It is simply an abbreviation of "1 kg," which is obviously a quantity that you can divide by. Since 1 is the multiplicative identity, 1×kg = kg, right? Commented Jan 12, 2022 at 20:10
• @Kevin well with respect it's not an obvious abbreviation, and putting it forward as an abbreviation also doesn't address the point that the logic here is about descriptive qualification, essentially not about multiplication or division. Commented Jan 14, 2022 at 1:31

The / sign before the unit (e.g. mass/kg) would indeed be the most correct mathematical way for expressing a physical quantity in theory. However, I guess it falls short of being in wide usage because many units already contain a slash in their common representation (e.g. m/s, J/K, etc, assuming the equivalent power notation such as m.s^-1 is seen unpractical for simplest units). An extra slash would simply make the notation awkward (or confusing in worst cases: in m/kg and m/s, the m obviously doesn't have the same meaning, and even the same nature; of course, in that quite artificial case, the issue is not with the notation per se, but with the choice of a conflicting symbol). Brackets or parentheses are less prone to such shortcomings.

Let's let m be the mass of some star. We can the write the equation:

m = 2.0 ((30))[kg]

The equation is saying that the mass , a physical attribute of some star, is equivalent to a rather huge number, 2 000 000 000 000 000 000 000 000 000 000, of units which are abbreviated to [kg].

The square brackets are interpreted to mean "the unit" and "kg" is the SI symbol for the unit of mass, "kilogram".

The square brackets are indicating a constant attribute. The mass of a kilogram is constant. The mass of a star is a variable that varies depending on which star you consider. But the function of a unit such as [kg] is that every [kg] is the same quantity of mass as every other [kg].

Algebraically the equation above can be converted to

m/[kg] = 2.0 ((30)).

This equation is saying the mass of this star divided by the mass of the SI unit mass, [kg], results in a huge number. This agrees with the original equation. There is no contradiction.

If astronomers found it convenient to use this particular star as an astronomical unit of mass, they could define [m*]:

[m*] = 2.0 ((30))[kg].

Dividing both sides by the mass of the astronomer's unit star we have the equation;

1 = (2.0 ((30)) [kg])/ [m*].

This we can use to convert from a physical quantity expressed in [m*] to the same quantity expressed in the SI standard [kg].

For example the mass of a massive black hole, mbh, measured to be say:

mbh = 1.54 ((8)) [m*]

could be converted to:

mbh = 3.1 ((38)) [kg]

by multiplying {1.54 ((8)) [m*]} times { 2 ((30)) [kg]/ [m*]}.

The right hand bracketed expression is equivalent to one.

Note that mbh could be rapidly changing in time, but [m*] is not.

And that mbh could be used for the mass of any size of black hole where [m*] is tied through its definition to the SI standard kilogram.

Without the square brackets" m* " would not be considered a unit of mass because it looks like a variable quantity.

The letter "m" has many meanings, the bracketed "[m]" means the SI standard [meter], a unit of length. The brackets prevent confusion.

I see below this box that I should have used MathJax for the equations. My apologies.... my bad.

Wow! Looks like MathJax is rewriting my equations. Mostly just removing blank spaces.