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

[cdux]

For example, dividing by 2, means we cut something in two.

But dividing by 0.5, can only be explained with multiplying something by 2.

So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?

 

[Number Nine]

A number divided by 0.5 is equal to however many time 0.5 goes into that number. There's nothing more to it than that, and there's no reason it should have any physical significance.

 

[cdux]

That's a good answer with a 'physical meaning':) "I fill that number with 1/2s".

 

[Mark44]

For example, dividing by 2, means we cut something in two.

But dividing by 0.5, can only be explained with multiplying something by 2.

So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?

I have $37. How many quarters does that represent?


In case you're not familiar with US money a quarter is one-fourth of a dollar.

My question could be represented as 37/(1/4)

 

[cdux]

That too. It's impressive how many ways there are to represent that concept in a physical form.

 

[symbolipoint]

Division has another interpretation, simpler than "cut into" so many "equal pieces". Division means repeated subtraction of a quantity until no more of that quantity can be subtracted. Work with that idea and see what you find!

 

[Jeff Cook]

"dividing by 2, means we cut something in two."

Not necessarily...the deeper meaning of a fraction to me is how many halves does it take to add up to a whole, as wholes are created before they are destroyed.

J

 

[Jeff Cook]

Or by the order of operations, the whole is a product first. Then it can be divided. How does one crack and egg before it is created?

 

[cdux]

Not necessarily.

I don't mind other interpretations but can you deny that it's correct to say it cuts something in half? It obviously cuts something in half. It's a valid interpretation.

 

[Jeff Cook]

It's an expression so it could express just about anything you want, the past, the present or the future or something else. But the origins of the concept of division must follow after multiplication...don't you think? Else one runs into paradoxes and maybe even 2+2=5 if you continue forward. Order of operations, got to follow them always.

 

[Jeff Cook]

Take 2 right triangles back to back to form an isosolese. How many triangles do you have? 3? The two rights and the isosolese, the sum of the two? Take one away from 3, how many do you have? 1? 3-1=1? No...cause the whole is equal to the sum of its parts. Gotta follow that else all math is useless.

 

[rcgldr]

One physical example would be the amount of linear force applied by a tire with a given torque and radius. The formula is force = torque / radius. If the same torque was applied to a tire with 1/2 the radius, the linear force applied by the tire would be doubled.

 

[HallsofIvy]

For example, dividing by 2, means we cut something in two.

But dividing by 0.5, can only be explained with multiplying something by 2.

So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?

You need to understand that there exist an infinite number of physical applications for any mathematical operation. There is no such thing a single "physical explanation".

 

[symbolipoint]

When you divide by a number, that number by which you divide is a DENOMINATOR. You may often stop focusing on a physical meaning and at least temporarily, focus on the mathematical meaning.

Example: 10 divided by 0.5,
Meaning, quotient =\frac{10}{0.5}.
If you work the number properties correctly, you will find it is equivalent to 20.

 

[Mark44]

Or by the order of operations, the whole is a product first. Then it can be divided.

Huh? How does it follow from the order of operations that a whole is first a product?

How does one crack and egg before it is created?

 

[HallsofIvy]

Or by the order of operations, the whole is a product first. Then it can be divided.

Huh? How does it follow from the order of operations that a whole is first a product?

Jeff Cook may be thinking that by "PEMDAS" multiplication comes before division. He is wrong of course. In fact, the order of multiplication and division is arbitrary- it just happens that "M" is ahead of "D". "PEDMAS", which is, in fact, used by some, would work as well. And even so, the order of operations is purely customary, having no relation to the "meaning" of any operation.

How does one crack and egg before it is created?

We could as well ask "How does one put things together before they have been separated?".

 

[Jeff Cook]

The American Physical Society states that multiplication is of higher precedence than division with a slash, as do texts by Feynman, Landou, Lifgarbagez and others. The original question asked about the "physical meaning", and the physical world does require precedence of multiplication in a number of physical cases--not arbitrary at all. Problems arise in terms of vectors, in that if a function takes a scalar into values of imaginaries, one must deduce that it was already a vector on the complex plane (as i doesn't come out of thin air...it carries physical meaning that i was multiplied by x ahead of time). By not considering that x was already multiplied by i, paradoxes do arise (Grandfather Paradox is one that seems fitting for this discussion: the grandfather must have produced his son, for his grandfather to return back in time to kill him...such situations can only be resolved by understanding that the product arises ahead of the quotient even in cases of time reversal...if those may be so). In the same, one cannot equally ask, "how does one put things together before they have been separated?" because we only define a whole as equal to the sum of its parts. If it has no parts, then it is not a whole in the first place...and you cannot separate the whole if it doesn't exist, which is why you cannot divide by zero. There is no meaningful solution in doing so.

 

[Mark44]

The American Physical Society states that multiplication is of higher precedence than division with a slash

Please cite a specific publication by this organization.

, as do texts by Feynman, Landou, Lifgarbagez and others.

Please cite a specific publication by any of these authors.

The original question asked about the "physical meaning", and the physical world does require precedence of multiplication in a number of physical cases--not arbitrary at all. Problems arise in terms of vectors, in that if a function takes a scalar into values of imaginaries, one must deduce that it was already a vector on the complex plane (as i doesn't come out of thin air...it carries physical meaning that i was multiplied by x ahead of time).

This doesn't make sense to me. If the domain of a function is real numbers, and the range is complex numbers, just because a complex number is produced doesn't mean that the input had to also be complex.

By not considering that x was already multiplied by i, paradoxes do arise (Grandfather Paradox is one that seems fitting for this discussion: the grandfather must have produced his son, for his grandfather to return back in time to kill him...such situations can only be resolved by understanding that the product arises ahead of the quotient even in cases of time reversal...if those may be so).

This is really a stretch, giving a situation that has never occurred as an example that is somehow related to your thesis.

In the same, one cannot equally ask, "how does one put things together before they have been separated?" because we only define a whole as equal to the sum of its parts. If it has no parts, then it is not a whole in the first place.

This also makes no sense to me. Things do not need to consist of parts to be considered to exist. The early Greeks theorized that things were made up of atoms, which were indivisible. We know now that atoms consist of subparticles: protons, neutrons, and electrons. AFAIK, and I am not a physicist, electrons are indivisible -- they are "whole."

..and you cannot separate the whole if it doesn't exist, which is why you cannot divide by zero.

That isn't why. The reason is that division and multiplication are inverse operations. If a/b = r, then it must also be true that b*r = a. If b happens to be zero, and a is nonzero, there is no number r such 0 * r = a.

There is no meaningful solution in doing so.

 

[1MileCrash]

I never understood where the idea of multiplication preceding division came from. Division is exactly multiplication by multiplicative inverse of the number we are "dividing by."

To claim that multiplication precedes division is to claim that certain multiplication precedes other types of multiplication which is nonsense because multiplication is commutative, it could never have any effect on the problem.

 

[Jeff Cook]

@ CDUX,

Your post was straight forward and clear, and as easy to understand as "what is the physical meaning of subtraction?" If 4 parts amount to a whole, if you take away 2 parts, you get still 2 parts. If you take away 1/2, you get 3 1/2. Similarly goes for division in my answer to your reasonable question: the numbers in the denominator are the number of parts that add up to the whole. If you divide 4 by 2, then you have 2 parts in the denominator that still add up to the remaining whole. Same goes in the end for rational numbers, real numbers complex numbers or any other kind of number. Whenever you look at a fraction now, you can say with certainty what the denominator means and never run into any contradictions otherwise: it is the number of parts that add up to the whole. Failing to do so however, may (in certain cases) amount to paradoxes if you continue to take the problem farther. There really is no serious debate otherwise about such matters; they are backed by axioms back to Euclid's Elements and farther. All these matters can be found in standard mathematical literature, but I do hope you will continue to post questions as such to this forum and hope you will be able to avoid confusion in the process. My best.

Jeff

 

[micromass]

The American Physical Society states that multiplication is of higher precedence than division with a slash, as do texts by Feynman, Landou, Lifgarbagez and others.

I'll ask again: please cite specifically where they say that. If you can't, then please retract this statement. Otherwise, this is considered misinformation.

 

[pwsnafu]

If it has no parts, then it is not a whole in the first place...and you cannot separate the whole if it doesn't exist, which is why you cannot divide by zero. There is no meaningful solution in doing so.

Except we do have mathematics where division by zero is allowed. See wheel theory.

 

[1MileCrash]

Except we do have mathematics where division by zero is allowed. See wheel theory.

...

Is that not an entirely different algebraic system? How does that apply to what we are talking about (standard algebra)?

 

[micromass]

...

Is that not an entirely different algebraic system? How does that apply to what we are talking about (standard algebra)?

I think the point is that standard algebra can be extended to a wheel (by adjoining some elements). So division by 0 can be defined by adding elements to the real numbers. Of course, in standard algebra (= real numbers), division by 0 is not defined.

 

[pwsnafu]

Is that not an entirely different algebraic system? How does that apply to what we are talking about (standard algebra)?

Jeff makes statements like this

Same goes in the end for rational numbers, real numbers complex numbers or any other kind of number. Whenever you look at a fraction now, you can say with certainty what the denominator means and never run into any contradictions otherwise: it is the number of parts that add up to the whole.

I keep getting the impression from Jeff's comments that he thinks that his opinions are universal truths. Wheels are an explicit counter-example, and as micromass points out, every commutative ring (includes integers, rational, reals and complexes) has an extension to a wheel, so if you wanted to do division with zero with the integers, you can enlargen your space to accommodate it.

I personally don't have much of a problem with Jeff's comments if we stick with Q (as per the thread title), my beef is that, well, he doesn't.

 

[boit]

One Kenyan shilling is a hundreth of a Euro.
1KSh1= 1/100 Euro
So you give me 1 euro, how many shillings are that?
1/(1/100) or
1/(.01).
Thats KSh100.
So division by a fraction is just looking at matters from the other guy's point of view :)

 

[HallsofIvy]

The American Physical Society states that multiplication is of higher precedence than division with a slash, as do texts by Feynman, Landou, Lifgarbagez and others.

If the American Physical Society is going to legislate precedence of arithmetic operations, can the American Mathematical Society decree that Pluto really is a planet?

 

[Jeff Cook]

* "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.

 

[HallsofIvy]

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

Okay, that is a guide to how to format equations for publication in that particular journal. It has nothing to do with precedence rules for mathematics.

* 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.

In other words, "1/2√N" is to be interpreted as 1/(2√N), not (1/2)√N. Again that is about formating of equations, NOT precedence rule for mathematics.

 

[micromass]

Okay, that is a guide to how to format equations for publication in that particular journal. It has nothing to do with precedence rules for mathematics.


In other words, "1/2√N" is to be interpreted as 1/(2√N), not (1/2)√N. Again that is about formating of equations, NOT precedence rule for mathematics.

Furthermore, those are all physics texts. We should look at mathematical logic texts for the actual answer.

 

[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."

-----------------------
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.