Unit Symbols for Equations

This question falls in the general category of another question from 5.5 years ago with the addition of the explicit request from the reviewer.

A comment from a reviewer requested that units of all physical quantities must be clarified (e.g., N, N/m, etc...). eg, F = m * a. Assuming based on the review, it would be m is mass in kg and a is acceleration in m/s^2 and F is force in N.

Now I see a few issues and wanted advise:

1. The equations are not empirical, hence it can be any unit for mass and acceleration.
2. If I use units in terms of fundamental dimensions (as per the link) I do not satisfy the requirements of the reviewer.
3. Personally, I have not come across any data historically that has units associated to equations in a manuscript.

Hence, I am seeking advise as to, either:

1. How to reply politely to the reviewer.
2. An IEEE or any other journal guide to author that explicitly states how to make such connection for the units in the equations.

Manuscript Field : Civil Engineering / Computational

UPDATE: Going with the suggestion below (and accepted answer) we wrote to the editor about it, and clarified that we will use units in fundamental dimensions for all equations. The editor and reviewer were both satisficed and the paper was accepted. Looking back, I just think it was a misunderstanding of the how the comment was written plus the fact that one equation parameter had a unit missing.

Thank you for the help and suggestions.

• To be clear, now you have something like "F = m * a, where m is the mass and a is the acceleration" and the reviewer would prefer either "F = m * a, where m is the mass given in kg, and a is the acceleration given in m/s"? Or would perhaps a statement like "All equations are expressed in the SI system of units" suffice? Aug 5, 2022 at 18:26
• @Anyon Based on the comment, he is looking for the actual units. Meaning "F = m * a, where m is the mass given in kg, and a is the acceleration given in m/s" Aug 5, 2022 at 19:03
• Does the equation happen to have exponential or logarithmic terms? Aug 6, 2022 at 3:16
• The question you linked is not "unanswered" -- it just doesn't have an accepted answer. Aug 6, 2022 at 3:19
• I don't think the reviewer wants you to add units in an equation such as `F = m * a`. Most likely it's a misunderstanding. Maybe you forgot units somewhere when giving values, or maybe you have a dimensionless value somewhere and the reviewer didn't notice that it was dimensionless and thought that you forgot units for this value. Double-check your paper, and explicitly state when a value is dimensionless.
– Stef
Aug 6, 2022 at 12:12

Before considering a reply, I would first try to work out if the reviewer has misunderstood something about your paper/calculations which have resulted in them asking this question. If this is possible, then addressing this source of confusion could answer the reviewer's comment.

Otherwise, you could respond with stating any consistent system of units can be used. You could also specify the units in a general way such as: where m is in units of mass, a is in units of acceleration, and F is in units of force. I can imagine there could be situations where this may not be obvious. For example people sometimes use "moment of inertia" and "second moment of area" (m^4) interchangeably but this could be confused with the "mass moment of inertia" (kg m^2) which is also sometimes called just "moment of inertia".

If there are any specific values given in the paper, perhaps in an example, then make sure these have units.

In my opinion it's not reasonable to ask for specific units to be provided for expressions that are dimensionally consistent and I wouldn't do so.

I'm not aware of any style guide which addresses this specific case.

• Correct, it was raised as a question for one equation. And that equation has been explicitly clarified and addressed in the reply to reviews. We hope that such explanation is what he was looking for. Aug 5, 2022 at 23:18

It might be worth pointing to items 3.21 and 3.22 of the ISO 80000-1:2013 standard. It's clear to me from those two items that the referee is wrong to ask for this, although I can't guarantee that it will be clear to the referee.

• Interesting ISO read, never knew of its existence. Thank you. Agreed, I dont think it would be clear to the referee, in addition I really do not want to end up starting a debate. Aug 5, 2022 at 23:16

Some engineering papers assume a specific background set of units, and then omit all units from numeric constants. Thus (for example) one might say that an object falling with negligible air resistance obeys the law

with the implicit assumption that t is measured in seconds, h in meters (down), and you don't worry about checking that all the units cancel out when you substitute for a specific moment. This convention is particularly useful if the equations include a large number of empirical constants, and your audience is sufficiently geographically restricted to use only one unit system. (From your item (1), maybe you already know this?)

From in your question I think you do not want to use this sort of convention, but your manuscript does not sufficiently clarify that you do not want to do so. If so, a sentence like

All constants in this paper have units marked explicitly.

or

All constants in this paper without marked units are dimensionless.

ought to suffice.

• Nitpick: ∆h = (1/2) * 9.8 t^2. (Unless you're on a planet where gravity is twice that of Earth.) Aug 6, 2022 at 18:08
• @MichaelSeifert: It's been a long time since I did kinematics.... (fixed) Aug 6, 2022 at 21:41