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When I studied physics (both in high-school and university), in all of the worked examples, variables would be used to denote physical quantities, and thus include dimensional units.

For example, with $F$ being a physical variable denoting force, the variable itself could be calculated in units of Newtons: $$ F=2\,\text{kg} * 5\,\text{m/s${}^2$}=10\,\text{N} $$ But I've recently started tutoring an engineering student, and saw in the worked examples from his lectures the use of variables as plain numbers, with the unit being outside the variable.

So modifying the same example, $F$ would now be a pure number that is multiplied by Newtons: $$ F\,\text{N} = 2\,\text{kg} * 5\,\text{m/s${}^2$} =10\,\text{N} \Rightarrow F=10 $$

Is there an educational benefit to one notational approach vs. the other? Has there been research done on this? Is this more relevant to particular levels of education or particular fields?

innisfree
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yoniLavi
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    I can't see how this has any benefits, pedagogical or otherwise. For me, it's just confusing and cluttered, and elevates (arbitrary) units into vital pieces of an equation. – innisfree Oct 08 '15 at 00:09
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    Should I write e.g. $E,\text{J} = m,\text{kg} \times (c ,\text{m/s})^2$? Completely awful! – innisfree Oct 08 '15 at 00:11
  • I don't like the 2nd way of writing out an equation. It leads to clutter and confusion because you have letter symbols representing both quantities and units. –  Oct 08 '15 at 00:16
  • The example is a seriously bad (or at least confusing) idea. The first quantity you encounter is FN. Is that a variable? The product of two variables? A variable and a unit marker? Sure, it becomes clear very quickly, but it's a bad way way to teach. – WhatRoughBeast Oct 08 '15 at 00:37
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    Since I see that you are a programmer, imagine a language that required you to write a variables type out with it every time you wanted to use it: float tk = int 5* (float tf - float 32) / int 9 + float 273.15. Yuck. – dmckee --- ex-moderator kitten Oct 08 '15 at 02:05
  • To follow up on the comments, I just wanted to remark that to me the second way (with the unit outside the variable) is confusing too. I was just wondering if maybe I'm missing something. – yoniLavi Oct 08 '15 at 02:13
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    Note that in astronomy it is very common to divide by a dimensional quantity, usually one chosen based on typical scales in the problem rather than just 1 of the appropriate unit. As a random example, see all the formulas in this answer of mine. In this way one can immediately see the order of magnitude of the left side of the equation assuming "typical" numbers are plugged in on the right. –  Oct 08 '15 at 18:14

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Mathematicians just know pure numbers (or non-scalar entities, besides operators themselves) and don't like physical units, so I guess a math teacher designing an exercise involving a physical example might be tempted by anything that through units away so as to play with plain number.

While physicists consider than plain numbers mean nothing, excepted in the case of dimensionless quantities such as proportions. What has a unit is not only the final result but really each term in an equation. And since you cannot add inhomogeneous units (different unit or different power), this constrains the small subset of physically consistent equations among all the possible math equation (e.g. exp(x), log(x), sin(x) is physically consistent only if x is a dimensionless expression ). This allows physicists to immediately detect many errors. But it can also help even young math students to simplify fractions ( (m/s) / (m/s) = (m/m).(s/s) but can't be (m/s)*(m/s) ), and to understand why perimeter corresponds to + and area correspond to * (since unit is L+L=m vs L*L=m²).

So like colleagues, I just can be horrified by this new notation, and see mostly problems with using it. ( Contrarily, I urge math teachers to use more units in their examples and exercises. )

Fabrice NEYRET
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