12

It's everywhere, but no matter who I ask I can never get a straight answer, only "I don't know"s. What is the advantage of using multiplication and negative powers over just using division when writing out units, as in the question title. Granted, in most cases they use superscript rather than how I formatted it in the title, but my phone keyboard would not let me do that.

I don't have a problem reading units written this way, but it just seems inefficient as it uses more characters and makes it more difficult for "regular people" to follow. I'm suspecting it's nothing more than a style choice, possibly because it looks "more scientific".

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    I'm voting to close as off-topic. This isn't a question about academia: it's a question about one of the subjects that academics study. – David Richerby Nov 17 at 22:07
  • @DavidRicherby This is just a question about writing style, exactly as we have questions about styling code or citations. – Massimo Ortolano Nov 18 at 22:04
3

Generally speaking, the choice of one form over another is better done on a case-by-case basis, according to readability.

If I have to specify a speed of 5 meters per second, I'll write v = 5 m/s rather than v = 5 m·s-1, because for most people the first form is more readable than the latter.

However, if I have to label the axis of a quantity with some complex unit, I may choose the form with the exponents. For, instance, if I have to report the spectral density function Sv(f) of some voltage noise, whose unit is V2/Hz, I may label the vertical axis (according to the style I described here) either

Sv(f)/(V2/Hz) (with parentheses to avoid any ambiguity)

or

Sv(f)/V2Hz-1 (for compactness).

In this case, I'd probably choose the second form because is probably more readable even in the case of more complex units. Strictly speaking, also this second form without parentheses is ambiguous because multiplication and division are associative from left to right, but frequently people understand that unit as a single block.

Remark. If you use LaTeX to write papers I cannot but recommend to use the package siunitx to typeset the units, because it allows you to switch between the two forms with just one option (per-mode=symbol or per-mode=reciprocal). Moreover, it automatically adds a small space between the units.

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    Hold up, isn't $Hz^{-1}$ just $s$? Because isn't a Hz simply 1/s? – vikarjramun Nov 17 at 18:25
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    @vikarjramun Yes it is, but certain units are commonly written in certain preferred, more recognizable ways. – Massimo Ortolano Nov 17 at 18:28
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    You can't write "5 ms^-1" because "ms" means millisecond. You would have to write "5 m·s^-1" instead. – Nayuki Nov 17 at 20:54
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    @Nayuki Unfortunately there's no way here to render the small space, but using the dot between units is generally discouraged. See also my comment above to Federico. – Massimo Ortolano Nov 17 at 21:07
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    Who said middle dot is discouraged? I see documents like this and it seems fine. physics.nist.gov/cuu/Units/checklist.html – Nayuki Nov 17 at 21:09
25

I think the main advantage is that an ambiguity of a/bc is avoided. For example, g/m^2s may be read as g m^{-2} s^{-1} or g m^{-2} s depending on whether multiplication takes priority over division. As far as I know, there is no consistency in opinions about it, so to avoid confusion a slightly uglier notation with negative powers may be worth accepting.

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    Though in programming, the pretty much universal convention is a / b * c ≡ (a / b) * c. This is exactly analogous to ab + c ≡ (ab) + c. But of course, these language don't allow writing b * c as b c (let alone bc). – leftaroundabout Nov 17 at 17:13
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    In that case you can write $gs/m^2$ or $g/m^2/s$. – R.. Nov 17 at 17:30
  • @leftaroundabout I agree about conventions in programming, however, my maths students often miscode $\frac{a}{b c}$ as a/b*c which makes me think they've used to a different convention somewhere. – Dmitry Savostyanov Nov 17 at 19:13
  • @leftaroundabout: but few will say that 1/2pi is equal to pi/2. – Martin Argerami Nov 17 at 19:48
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    But, if you're attempting to eliminate ambiguity, then a/bc can be trivially written as a/(bc), or g/(m^2s) as in your example, which adds two characters to the representation, whereas g m^{-2} s^{-1} and g m^{-2} s adds 7 characters and 3 characters respectively (not counting the space characters added to your examples). Arguably, those also remove the ambiguity between interpreting the original as g/((m^2)s) and g/(m^(2s)), but that could be resolved by writing it as g/(sm^2). – Makyen Nov 17 at 20:40
1

Dimitry's answer is probably the most complete, but for me, the issue is around ambiguity in fractions. For example I would never write a/bc, because it is ambiguous, I would right (in LaTeX) \frac{a}{bc} or ab^{-1}c^{-1}. The former I find much neater for an equation out in the open, while the later looks a bit better if included in-line text.

1

Personal preference, but having the numbers be explicit helps me with checking the dimensions of the final answer (it's easier to see which terms cancel, especially if the exponents are non integers as I don't need to keep subtracting 1 as 1/x^0.5 is actually x^-0.5)

1

Another significant advantage of using negative powers to me is that checking for correctness of units becomes easier. In physics especially, any physically meaningful formula or identity should have units which match up on both sides, so if the units not match up for an equation, this can be an effective check that the equation can't possibly be correct. This is a common way to check one's work, and can even be taken further: see Dimensional Analysis.

Now, working with units is much easier with negative exponents. Consider the equation distance = velocity × time. The (SI) units of the LHS are (m), that of velocity is (m/s), and that of time is (s). It is easier to see that m·s^(-1)s=m than that (m/s)s=m to many people. Of course the difference is slight because the formula is simple in this case, but once one starts working with more complicated equations dimensional analysis becomes much easier once you abandon the forward slash notation.

0

Perhaps its not an academic reason, but more of a practical reason combined with tradition/preference for the earlier format learned?

Historically, typesetters may not have been able to render what is now easy with various markup notations.

39.3396 m/s might be accidentally read as "mis" or "mls" or "mi5" where (m.s^-1) is unambiguous.


A horizontal fraction bar will also break up the flow of a paragraph of printed text, so this presentation keeps the units to be one "line of text" high.


"m/s" when sent by morse code is

   --     -..-.    ...

Since the forward slash is a relatively rare character, could be accidentally transcribed as one of these:

   --     -..     -.    ...
    m      d      n      s

   --     -.     .-.    ...
    m      n      r      s

   --     -   ..   -   .    ...
    m    you get the idea...

  $39.3396 \frac{m}{s}$

should render unambiguously like this (if mathtex were supported here)

enter image description here

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