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When we encounter "real world problems" in math, one can chose different levels of detail with regard to units: from leaving them out completely up to using them everywhere.

I'd argue that both extremes have their merits and inconsistent usage of units (using them in some places but not always) is wrong and leads to confusion.

To me, the most important aspect is that units are great in telling you if a calculation makes sense - and more often than not, a calculation is right if and only if the units are right. For example, due to the units alone it is evident that distance has to be divided by time to get speed.

On the other hand, units can get hard, for example if you're solving for time.

Is there scientific research or results that is pro/contra using units in math?

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    $\begingroup$ I wonder which inconsistencies you see with both extremes. This might sidetrack your question, but it puzzles me since I see no inconsistencies in either. $\endgroup$ – Michael Bächtold Jun 5 at 14:00
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    $\begingroup$ What level are interested in educating towards. Constructing scientific laws as unitless is hugely important, as area scaling laws. In QFT, the units have been understood as a grading on the algebra of observables and the fact that all exponentiated quantities must be unit-less has been used to great effect to restrict the space of possible theories. I don't know highschool level source for any of this though. $\endgroup$ – Nate Bade Jun 5 at 14:35
  • $\begingroup$ @MichaelBächtold each extreme is "fine", I meant using units only in some places (e.g. at the beginning and the end of the calculation but omitting them in intermediate steps). $\endgroup$ – Jasper Jun 5 at 17:50
  • $\begingroup$ @NateBade I've added the target audience in the title and as a tag. I'm not (primarily) looking at QFT stuff :) $\endgroup$ – Jasper Jun 5 at 17:52
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    $\begingroup$ Is there scientific research or results that is pro/contra using units in math? What do you mean by scientific research? Do you mean educational research on whether one approach or the other produces better educational results? On the other hand, units can get hard, for example if you're solving for time. Why would this be hard? m/(m/s)=s...? and more often than not, a calculation is right if and only if the units are right If the units are wrong, the calculation is wrong. It's not an if-and-only-if. Units won't catch unitless factors like 2 and $\pi$. $\endgroup$ – Ben Crowell Jun 7 at 13:41
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This is not exactly what you seek, but at least you can find educational research articles under the key phrase dimensional analysis. Below [1] says it's useful, [2] questions that usefulness.

[1] Hrayr Ohanyan. "The Application of the Method of Dimensional Analysis When Solving Problems." American Journal of Educational Research. Vol. 4, No. 1, 2016, pp 5-7. Journal link.

[2] Canagaratna, Sebastian G. "Is dimensional analysis the best we have to offer?" Journal of Chemical Education 70, no. 1 (1993): 40. Journal link.

And this is interesting: Buckingham's $\pi$ theorem. It states that, if there are $n$ physical variables in a $k$-dimensional equation, then the equation can be rewritten in $n-k$ dimensionless parameters. For example, if we have an equation in the $n=3$ variables $\{$distance $d$, velocity $v$, and time $t$$\}$, $k=2$ ($t$ & $d$), and the $n-k=1$ dimensionless parameter is $\pi_1 = t v / d$.

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  • $\begingroup$ The second linked article does not seem to question the use of units in the sense of the OP. It's a critique of something they call "dimensional analysis" versus the "method of equations." Both methods use units, but it's not clear (to me) what the difference is. $\endgroup$ – Michael Bächtold Jun 6 at 10:38
  • $\begingroup$ @MichaelBächtold: Yes, I realize these papers are about dimensional analysis rather than the OP's focus on using units. $\endgroup$ – Joseph O'Rourke Jun 6 at 13:53

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