[Basic question] What is the 'physical' explanation of a division by a fraction?

[OmCheeto]

This reminds me of the time my sister tried teaching me about negative numbers when I was 4 years old:

Om's Sis; "If you have 3 pies, and take away 4 pies, that leaves you minus one pie."
Om; "That's impossible. You can't take away more pies than actually exist. Negative numbers are stupid."

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The irony of getting an infraction in a post about fractions, did not elude me, nor dissuade me from making my post. I stand by my silliness.

ps. I like Mark44's answer the best, as it was the first correct answer, AFAICT. Everything following a correct answer, is just arguing for no good reason, IMHO.

 

[cdux]

Yeah she should have said you lose 3 pies and you owe one. But you have to remember that you owe one, it's important!

 

[Jeff Cook]

@ Halls of Ivy,

Yes, you are correct, those are instructions into how equations are to be evaluated. In fact, I retract much of what I wrote above in that it seems upon further investigating this, there is no authority over this matter. But also note, that this isn't a Physics take alone, and I too want Pluto back! :)

The truth is, as far as I have found since I first wrote the above, is that there is some authority of precedence over the other operators, just not (yet) for multiplication over division. I find this truly odd, but some of you will find it satisfying. Below is a discussion on this very topic I found online between a professor and another:

"...it would be easier and perhaps more consistent to give
multiplication precedence over division everywhere; but of course
there is no authority to decree this, so the more prudent approach is
probably just to recognize that there really isn't any universal rule.
I ran across the same AMS reference that you found while trying to see
if any societies had made official statements on the rules of
operations in general; the fact that they took note of this one rule
alone demonstrates only that this is the one rule on which there is
not universal agreement at the present time, but it probably is
growing in acceptance.

I've been continuing to research the history of Order of Operations,
and one of the references in our FAQ now includes a mention of
something I had also discovered, that the multiplication-division rule
has never really been fully accepted:

Earliest Uses of Symbols of Operation - Jeff Miller
http://jeff560.tripod.com/operation.html"

I suppose what's in order for those of us (myself included) who believe otherwise would need to find a hard fast mathematical example that presents a formal paradox (as I expressed above "should" occur) on a fundamental level...outside of evaluation of an equation. I cannot do that here, as it would represent independent research. But I find it difficult to accept that evaluation of an equation for all other operators can carry physical meaning, but multiplication over division cannot. I have heard the arguments above that it probably shouldn't be such an odd condition to consider, but it seems too at odds with uniformities in mathematics that hold so well in other cases.

Maybe mathematicians have not been too terribly concerned to prove multiplication should have universal precedence over division in order warrant any investigation into any such a proof, as I would feel more comfortable accepting that multiplication has most always taken prior to division in that such paradoxes would rarely be found in this regard. Maybe no such proof is possible too. Maybe you all are right. But since there is no authority stating otherwise, none of the counter statements to mine are proof to the opposite either. It's simply not proven either way.

Then, to summarize a response to the original post: one CAN look at a fraction divided by one half still as the number of parts that add up to the whole if you are seeking physical meaning of an expression...but then again, you don't need to either. ;)

Jeff

 

[Mark44]

* "Physical Review Style and Notation Guide". American Physical Society. Section IV–E–2. Retrieved 5 August 2012.

* the third edition of Mechanics by Landau and Lifgarbagez contains expressions such as hPz/2π (p. 22), and the first volume of the Feynman Lectures contains expressions such as 1/2√N (p. 6–8). In both books these expressions are written with the convention that the solidus is evaluated last.

I would bet that the expressions you quote appear like this:
$$ \frac{hPz}{2\pi}$$
and this:
$$ \frac{1}{2\sqrt{N}}$$

and NOT like hPz/2π and 1/2√N.

In the first forms above, the formatting makes it clear that the multiplications in the numerator and denominator are to be performed before the division, so they are not redefining the order of operations.

 

[lavinia]

A good physical model of addition,multiplication and division is the creation of points with a straight edge and compass. Form this point of view division by two is constructing the midpoint of a segment. Multiplication by two is constructing a segment whose midpoint the point you start with. These constructions are easily seen to reverse each other.