Proportionality

[Deepak Kapur]

In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant.

 

i.e. y=kx

 

Doesn’t the above definition apply to

 

Y=k+x

[ajb]

I would say your second example is a linear relation. You plot the variables and you get a straight line but it does not hit (0,0). So from your example if x =0 then Y =k.

 

You see that it is not true that Y/x is constant, which is really the signal of proportionality.

 

Of course if you know that k is a constant then you can just change variables to get Y=x' and now we do have Y/x' =1 = constant.

 

 

**************

missing "not" (quite important here)

Edited June 4, 2014 by ajb

[Deepak Kapur]

 

Of course if you know that k is a constant then you can just change variables to get Y=x' and now we do have Y/x' =1 = constant.

I was just wondering......

 

Wherever any constant of proportionality (or even any co-efficient ) is used it is always multiplied and never added.

 

Is it because of our preference or is this the way nature works?

[imatfaal]

I was just wondering......

 

Wherever any constant of proportionality (or even any co-efficient ) is used it is always multiplied and never added.

 

Is it because of our preference or is this the way nature works?

 

It would not be a constant of proportionality otherwise. maths is axiomatic - you need to work within the rules of the axiomatic system you have chosen. And you need to stick to the definitions and nomenclature which that system uses.

 

Once you start re-inventing terms or affirming that a definition has a deeper non-arbitrary meaning then you are getting into very muddy waters. y= kx goes through the origin and has a slope of k; y = j+x only goes though origin if j = 0 and has a slope of 1. These are not the same lines unless k=1 and j=0 - in other cases they are easily shown to be different. Claiming that one is as good a fit to the definition of the other is meaningless.

[Deepak Kapur]

 

It would not be a constant of proportionality otherwise. maths is axiomatic - you need to work within the rules of the axiomatic system you have chosen. And you need to stick to the definitions and nomenclature which that system uses.

 

Once you start re-inventing terms or affirming that a definition has a deeper non-arbitrary meaning then you are getting into very muddy waters. y= kx goes through the origin and has a slope of k; y = j+x only goes though origin if j = 0 and has a slope of 1. These are not the same lines unless k=1 and j=0 - in other cases they are easily shown to be different. Claiming that one is as good a fit to the definition of the other is meaningless.

 

I am not affirming anything, I just want to satisfy my curiosity.

 

Another example...

 

A cube has length 2m. Its volume is 8m3.

 

Why are the units multiplied (I.e. meter cube).

 

Shouldn't they be added?

[imatfaal]

 

I am not affirming anything, I just want to satisfy my curiosity.

 

Another example...

 

A cube has length 2m. Its volume is 8m3.

 

Why are the units multiplied (I.e. meter cube).

 

Shouldn't they be added?

 

Take a large handful of cubes - place two to make a line, four to make a square and eight to make a cube. If it was additive it would be 4 to make a square and six to make a cube - BUT IT ISNT. To make it doubly clear place three to make a line, then nine to make a square and 27 to make a cube - no addition here whatsoever.

 

You need one coordinate to describe a straight line, two to describe a rectangle and three to describe the box - the units follow; metres, metres^2 and metres^3.

 

Neat ideas are just hot air if they don't comply with nature

[Deepak Kapur]

 

Take a large handful of cubes - place two to make a line, four to make a square and eight to make a cube. If it was additive it would be 4 to make a square and six to make a cube - BUT IT ISNT. To make it doubly clear place three to make a line, then nine to make a square and 27 to make a cube - no addition here whatsoever.

 

You need one coordinate to describe a straight line, two to describe a rectangle and three to describe the box - the units follow; metres, metres^2 and metres^3.

 

Neat ideas are just hot air if they don't comply with nature

 

Thanks for your answer. I also knew it to be the same, the way you described it...

 

But...

 

Somehow, I cannot wrap my head (may be an empty one) around the concepts of 'addition' and 'multiplication'.

 

 

1. When two balls collide, we multiply their masses.

 

2. When two forces act , we add forces.

 

 

My question is that when two balls collide, they are actually exerting force only.... so why to multiply?????

 

Let me try to explain what I am saying...........

 

 

Suppose there is a big ball. Two small balls collide with it from different sides.

 

a) We usually form an equation in which all the 3 masses are multiplied.

 

b) Why can't we take the collision by two small balls as forces acting on the big ball and in fact add the two small masses in our equation.

Edited June 3, 2014 by Deepak Kapur

[ajb]

Wherever any constant of proportionality (or even any co-efficient ) is used it is always multiplied and never added.

 

Is it because of our preference or is this the way nature works?

As explained it is an axiom or a convention that proportionality means Y/x = constant.

 

Please note that often in physics we do have y= mx + c.

[imatfaal]

 

...

Somehow, I cannot wrap my head (may be an empty one) around the concepts of 'addition' and 'multiplication'.

 

 

1. When two balls collide, we multiply their masses.

 

2. When two forces act , we add forces....

 

Forces - yep. Not sure what you are getting at with multiplying masses - can you give me simple concrete example. FYG When two balls collide I would look to the conservation of linear momentum, and depending on the form of the collision perhaps energy conservation; I cannot see a obvious reason to be multiplying masses.

...

Let me try to explain what I am saying...........

 

 

Suppose there is a big ball. Two small balls collide with it from different sides.

 

a) We usually form an equation in which all the 3 masses are multiplied.

 

b) Why can't we take the collision by two small balls as forces acting on the big ball and in fact add the two small masses in our equation.

 

I would set up an equation with the total momentum (vector remember) before collision and afterwards. If elastic I would do similar with kinetic energy

 

We do not use force as momentum is a conserved quantity and much easier to use

[Deepak Kapur]

 

Forces - yep. Not sure what you are getting at with multiplying masses - can you give me simple concrete example.

 

Actually, I had culled a couple of equations involving collision and product of masses, but the links have been lost. I will fix it soon.

 

 

The following is the example where masses are multiplied. This is not collision but interaction between masses ( something like indirect collision).

 

F₁₂ = G

Edited June 3, 2014 by Deepak Kapur

[imatfaal]

This is the force between particle 1 and 2 due to Gravity. And in this case you do multiply the masses. The gravitational attraction due to object 1 on object 2 is dependent on the mass of object 1, the mass of object 2 and the distance between them. For this to work we multiply.

 

Take some numerical examples and play with them - ignoring G and units; let's use two simple guesses at the equation

 

F=(M1m2)/r^2 or F= (M1+m2)/r^2

 

 

- we want the force to double if M1 doubles OR if m2 doubles; this is from observation and cannot be changed.

 

So put in M1=40 m2=1 and r=2 into both attempts at the equation. Then double M1 , then double m2. In the first equation the force will double each time - in the second erroneous equation it will not double.

[Deepak Kapur]

This is the force between particle 1 and 2 due to Gravity. And in this case you do multiply the masses. The gravitational attraction due to object 1 on object 2 is dependent on the mass of object 1, the mass of object 2 and the distance between them. For this to work we multiply.

 

Take some numerical examples and play with them - ignoring G and units; let's use two simple guesses at the equation

 

F=(M1m2)/r^2 or F= (M1+m2)/r^2

 

 

- we want the force to double if M1 doubles OR if m2 doubles; this is from observation and cannot be changed.

 

So put in M1=40 m2=1 and r=2 into both attempts at the equation. Then double M1 , then double m2. In the first equation the force will double each time - in the second erroneous equation it will not double.

 

I am just a curious person, nothing more...

 

 

I think when two balls collide, almost the same scenario as discussed above happens...

 

1. If we double the mass of 1st ball, its impact on the 2nd ball increases (or doubles, I don't know) .

 

2. If we double the mass of 2nd ball, its impact on the 1st ball increases. ( of course, distance is zero here).

 

 

And this fact was used in the equation that I lost (may be the equation was by some crackpot, but I just got curious.. and so drew parallel between masses and forces).

 

To my mind, the point still remains, that in the gravitation formula, if instead of masses we had forces, our treatment would have been different. We would not have multiplied the forces in question. (This is only a guess as per my limited intelligence, other intelligent minds may quantify such a hypothetical interaction where instead of two masses, we use two forces).

[imatfaal]

 

I am just a curious person, nothing more...

 

 

I think when two balls collide, almost the same scenario as discussed above happens...

 

1. If we double the mass of 1st ball, its impact on the 2nd ball increases (or doubles, I don't know) .

 

2. If we double the mass of 2nd ball, its impact on the 1st ball increases. ( of course, distance is zero here).

 

...

 

Because the momentum doubles if the mass doubles but the velocity remains constant.

 

 

 

And this fact was used in the equation that I lost (may be the equation was by some crackpot, but I just got curious.. and so drew parallel between masses and forces).

 

To my mind, the point still remains, that in the gravitation formula, if instead of masses we had forces, our treatment would have been different. We would not have multiplied the forces in question. (This is only a guess as per my limited intelligence, other intelligent minds may quantify such a hypothetical interaction where instead of two masses, we use two forces).

 

But you cannot just switch forces and masses without manipulation. If we are considering two forces that arise from gravitation we would use the vector sum of those forces. But when we calculate a single force we need to consider both masses and multiply.

 

Remember we don't just make this up - I has to, HAS TO, fit observations from nature.

[studiot]

 

In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant.

 

i.e. y=kx

 

Doesn’t the above definition apply to

 

Y=k+x

 

In short , no the definition in green does not.

 

 

But the statement in black is not correct either.

 

A word of friendly advice here.

I have noticed in your threads that you bring in far too many ideas far too quickly.

The result is that you confuse yourself and possibly others as well.

Further you have scattered your subject over several questions, where it is apparent that a difficulty in one also comes out in each thread.

 

So let us stick to proportion in this thread. It will help in the others as well.

 

When two quantities, say B and A are in proportion we say that B is proportional to A.

 

This means that B depends upon A in a particularly specified manner.

You should realise that there are many other ways for B to depend upon A where they are not in proportion.

 

What is meant by in proportion is that if we double A we double B, if we triple A we triple B, if we halve A we halve B and so on.

 

We use this as follows:

 

If A is 10 and B is 14, then

For A = 20 B = 28

 

and if A=10 and B = 5

For A = 20 then B = 10

 

Note I have not needed to employ a constant of any sort in this definition.

The above is always true regardless of the constant I shall introduce below.

But I can go on pairing values of A and B in the manner above without any constant.

 

We can convert the above to a general equation connecting A and B by introducing a multiplicative constant.

We call this the constant of proportionality.

 

So B = pA, where p is the constant of proportionality.

 

In the examples above the first p = 1.4 and the second p = 0.5.

 

Note I have said a multiplicative constant.

This is very important.

If I do anything else then B is no longer proportional to A

In particular if I add anything at all, even a constant, B is not proportional to A.

 

So if I have a second additive constant such that B = pA + D then I loose proportionality.

 

ajb has mentioned 'linear', but I recommend you avoid the term since linear and proportional are not always the same.

 

So we move on to what happens if B is proportional to two quantities, A and C.

 

So B is proportional A and B is proportional to C

 

How can I write an equation to combine A and C so that my definition will hold?

 

Well if I add C to A what happens?

That is if I double A but do not alter C for the moment.

 

so if B = p(A+C)

I want B2 to be double B1, when I double A

 

Let us work out B1 = p(A+C) and B2 = p(2A+C)

 

B1 = pA+pC and B2 = 2pA+pC

 

This shows that B2 is not double B1 so an equation adding C to A will not work in proportion.

 

OK so what about multiplication?

 

If B = pAC

 

B1 = pAC and B2 = p2AC

 

So B2 = 2B1 as required.

 

So far, I have kept C constant but if the constant of proportionality for C is q then

 

B = pqAC, maintains proportionality.

 

We can then combine p and q into one constant of proportionality r = pq.

 

So B = rAC

 

Finally I have councilled avoiding the term 'linear'. This is because we can extend the concept of proportionality to more complicated (non linear) expressions so that we can say B is proportional to the reciprocal of A, or B is proportional to the square of A.

 

Our equations now become

 

B = p(1/A) and B = p(A²)

 

To cope with this extension we add qualifying words to the original statement.

 

We now call our original simple proportionality direct proportionality,

 

Then we say that being proportional to the reciprocal is inverse proportionality and

 

Being proportional to any other expression is proportional to any other expression so our example with the square of A says that B is proportional to the square of A.

 

We often omit the 'direct' when we mean simple proportionality.

 

Does this help?

Edited June 4, 2014 by studiot

[Deepak Kapur]

Does this help?

 

 

A word of friendly advice here.

I have noticed in your threads that you bring in far too many ideas far too quickly.

The result is that you confuse yourself and possibly others as well.

Further you have scattered your subject over several questions, where it is apparent that a difficulty in one also comes out in each thread.

 

Does this help?

 

A lot.... In India it is called.......... 'Saagar in a Gaagar'. Saagar=Sea , Gaagar= An earthen pot used for storing water.

 

 

 

 

 

 

A word of friendly advice here.

I have noticed in your threads that you bring in far too many ideas far too quickly.

 

The result is that you confuse yourself and possibly others as well.

Further you have scattered your subject over several questions, where it is apparent that a difficulty in one also comes out in each thread.

 

It seems to be a result of some flaw in the synaptic firing in my brain. Or it is the result of some flaw in the 'thing/phenomenon' that leads to synaptic firing. Or it is the result of some flaw in the thing/phenomenon that leads to the 'thing/phenomenon' that leads to synaptic firing. and so on................

 

Anyway, I will try to win over this...though neurologists say that we are not the originator of thoughts in our brain.

 

 

 

Now, if you don't consider it 'arguing' I have something to submit below....

 

Let y=kx, Let k=5

 

when x=1 , y=5 , y/x=5

x=2 , y=10 , y/x=5

x=3 , y=15, y/x=5

 

 

Let y=k+x Let k=5

 

when x=1 , y=6 , y-x=5

x=2 , y=7 , y-x=5

x=3 , y=8, y-x=5

 

In 1st example we have y/x = a constant.

 

In 2nd example we have y-x = a constant.

 

Why should we prefer y/x over y-x?

 

There might be many phenomenon that fall in the category of y-x=constant. So, why the second example cannot be a proportionality?

 

Does it mean that there can be no other definition of proportionality other than in which y/x is a constant?

 

 

I hope I am not annoying anyone...........

 

 

 

 

But you cannot just switch forces and masses without manipulation. If we are considering two forces that arise from gravitation we would use the vector sum of those forces. But when we calculate a single force we need to consider both masses and multiply.

 

 

 

Please consider a hypothetical phenomenon where we have two force fields A & B. (May be such examples are already there in nature)

 

Force field A exerts some kind of force on Force field B and Force field B exerts the same kind of force on Force field A.

 

 

The force reduces when we take the force fields apart either one at a time or both together. The force increases when we bring the force fields together either one at a time or both together.

 

In this case, I doubt we would multiply the two force fields as opposed to the two masses, if we were to reach at some kind of equation.

 

 

Thanking you in anticipation of your patience and effort...........

[studiot]

 

I hope I am not annoying anyone...........

 

 

Not at all so long as you have a genuine interest.

 

 

Why should we prefer y/x over y-x?

 

There might be many phenomenon that fall in the category of y-x=constant. So, why the second example cannot be a proportionality?

 

Does it mean that there can be no other definition of proportionality other than in which y/x is a constant?

 

 

You are nearly there, but remember that x is the independent variable and y the dependent variable.

 

The idea is that changes in y depend upon changes in x ie they only happen because we change x.

This is because we normally know x but not y. We obtain y by calculation from the equation or formula.

 

So an expression of the type y/x or y-x contain an unknown.

 

However you are skirting around the fact that we can sometimes change things (ie find a different x) so that we can recover proportionality.

 

for example

 

The length of a stretched spring is not proportional to the force required to stretch it.

 

but

 

Another variable, the extension, e, is proportional.

If we double F we double e and so on

e=kF.

 

e, of course is the difference between the current length at force F and the original that is e = (L-L₀)

 

 

Does it mean that there can be no other definition of proportionality other than in which y/x is a constant?

 

This is not a satisfactory definition of proportionality. What happens when x=0? We do not allow division by zero.

I have already given you one that works in all cases.

Use it!

 

Please consider a hypothetical phenomenon where we have two force fields A & B. (May be such examples are already there in nature)

 

Force field A exerts some kind of force on Force field B and Force field B exerts the same kind of force on Force field A.

 

 

Force fields do not exert forces on each other.

 

They exert forces on material objects placed in the field(s).

 

So what you are asking is how do the effects of the fields combine when they both act on the same object.

 

Do you understand this, this is essential before proceeding.?

Edited June 4, 2014 by studiot

[imatfaal]

 

Force fields do not exert forces on each other.

 

They exert forces on material objects placed in the field(s).

 

So what you are asking is how do the effects of the fields combine when they both act on the same object.

 

Do you understand this, this is essential before proceeding.?

 

 

Deepak - I couldn't agree more with the quoted section from Studiot's post above. You need to take it on board.

[Deepak Kapur]

 

Not at all so long as you have a genuine interest.

 

 

 

You are nearly there, but remember that x is the independent variable and y the dependent variable.

 

The idea is that changes in y depend upon changes in x ie they only happen because we change x.

This is because we normally know x but not y. We obtain y by calculation from the equation or formula.

 

So an expression of the type y/x or y-x contain an unknown.

 

However you are skirting around the fact that we can sometimes change things (ie find a different x) so that we can recover proportionality.

 

for example

 

The length of a stretched spring is not proportional to the force required to stretch it.

 

but

 

Another variable, the extension, e, is proportional.

If we double F we double e and so on

e=kF.

 

e, of course is the difference between the current length at force F and the original that is e = (L-L₀)

 

 

This is not a satisfactory definition of proportionality. What happens when x=0? We do not allow division by zero.

I have already given you one that works in all cases.

Use it!

 

 

Force fields do not exert forces on each other.

 

They exert forces on material objects placed in the field(s).

 

So what you are asking is how do the effects of the fields combine when they both act on the same object.

 

Do you understand this, this is essential before proceeding.?

Well........ if you say so.........it's okay.

But........

 

 

I think......the hypothetical situation should have been taken into account.

 

 

especially when.......

 

Mass can be defined as the total energy content of a body.

and

Force fields also contain energy.

 

 

Anyway, very many thanks.......for spending so much time on a curious lay man.

[studiot]

 

I think......the hypothetical situation should have been taken into account.

 

 

especially when.......

 

Mass can be defined as the total energy content of a body.

and

Force fields also contain energy.

 

None of this is necessarily true. It is all, as you said, hypothetical.

 

 

Well........ if you say so.........it's okay.

But........

 

Further it suggests you have not bothered to work through the material I carefully spent time writing out for you.

 

You started this thread about proportionality.

 

So let us keep on topic and discuss that.

[Deepak Kapur]

 

 

 

 

Further it suggests you have not bothered to work through the material I carefully spent time writing out for you.

 

You started this thread about proportionality.

 

So let us keep on topic and discuss that.

 

 

Further it suggests you have not bothered to work through the material I carefully spent time writing out for you.

 

This is a totally baseless allegation. In fact, I have gone through graphs of various kinds of equations to understand your point. ( and have understood it to quite a large extent)

 

 

 

You started this thread about proportionality.

 

I brought the other example because somehow that dealt with multiplication and not addition. (the same is the case with proportionality, it deals with multiplication and not addition)

[studiot]

 

 

Quote

 

Does it mean that there can be no other definition of proportionality other than in which y/x is a constant?

 

This is not a satisfactory definition of proportionality. What happens when x=0? We do not allow division by zero.

I have already given you one that works in all cases.

Use it!

 

 

Quoted is your last statement on proportionality, followed by my reply and further explanation.

 

You have not responded to this.

 

Further your last statement was a contradiction of my long post explaining proportionality.

 

I have suggested we discuss forces, masses and energy when we have finished proportionality.

 

Proportionality is a very very important concept in Physics, that we use whenever we can, so it is vital to fully understand it and to be able to get it right.

[overtone]

 

 

In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant.

You may be confusing the changes with the variables themselves. They are not the same thing.

 

It is the variables that are defined to be proportional, and "related by the use of a constant", not the changes. Only certain kinds of changes reliabiy maintain proportionality, and adding stuff is not one of them.

 

In a like manner, you seem to be overlooking the distinction between a mathematical form and a physical reality it happens to be useful in describing. The usefulness of any particular mathematics in describing physical situations is a valuable but essentially coincidental or auxiliary feature of the math itself. If something works, great, if it doesn't you don't change the math or screw around with the definitions etc - you go find some other math that works.

 

 

[Deepak Kapur]

 

Quoted is your last statement on proportionality, followed by my reply and further explanation.

 

You have not responded to this.

 

 

I have thought long on this topic.......it's still at the back of my mind.

 

I am afraid you won't like my response...........

 

 

My response is as follows.............

 

 

I think that proportionality is the result of a choice rule......and.......choices can be many.

[overtone]

 

 

I think that proportionality is the result of a choice rule......and.......choices can be many.

It is the name of a "choice rule". You can of course choose other rules and setups - they will have their own names.

 

Why would you want to give different mathematical "rules" the same name?

[studiot]

 

studiot, on 04 Jun 2014 - 4:22 PM, said:

 

Quoted is your last statement on proportionality, followed by my reply and further explanation.

 

You have not responded to this.

 

 

I have thought long on this topic.......it's still at the back of my mind.

 

I am afraid you won't like my response...........

 

 

My response is as follows.............

 

 

I think that proportionality is the result of a choice rule......and.......choices can be many.

 

 

It is not a question of whether I like or dislike your response - actually neither.

 

I don't understand your response.

 

I have no idea what you mean by a choice rule.

 

The definition of proportionality was set at least four hundred years ago.

If you are proposing an alternative, I have pointed out it's arithmetical flaw since we cannot divide by zero.

 

A further arithmetical flaw is that is does not define what happens when the constant is zero.

 

Do you think an alternative definition for a relationship between three quantities that is undefined for two of them in certain important cases is worthwhile?