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Is it necessary for scientific equations to be dimensionally consistent?


studiot
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4 hours ago, studiot said:

I am not sure where to post this so have used Philosophy (of Science) to allow latitude in exploring this subject.

I hope it will make a welcome change from the current Philosophy subject we have surely now done to surely death.

Dimensional analysis is a very powerful technique in Science and is one of the things that distinguishes Science from Pure Mathematics, but to repeat the title,

Is it necessary for all equations in Science to be dimensionally consistent?

7 apples = 3 bananas

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Thank you John Cuthber for the typographical correction.

 

Now we have all calmed down and reached the stage of more reasoned discussion how about this equation

 


[math]\frac{{M{L^2}{T^{ - 2}}}}{{M{L^2}{T^{ - 2}}}} = \frac{{Energy}}{{Moment}} = 1a[/math]


Where a is a constant depending upon units.

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I'm going to take a little bit of a contrarian position on the OP.  Scientific equations do not have to be dimensionaly consistent, per se.  They relate terms that the user has to make dimensionally consistent.  For example, F = ma.  If the user is calculating in metric units they have use newtons, kg and acceleration in m/sec^2.  The user has to make the correct choices in order to assure dimensional consistency, and they have to know the definition of the Newton in order to know this usage is consistent. The same equation could be used in the English system using pounds of force, mass in pounds mass and acceleration in ft/sec^2-- but the user will have to introduce conversion factors to attain dimensional consistency.

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2 hours ago, OldChemE said:

I'm going to take a little bit of a contrarian position on the OP.  Scientific equations do not have to be dimensionaly consistent, per se.  They relate terms that the user has to make dimensionally consistent.  For example, F = ma.  If the user is calculating in metric units they have use newtons, kg and acceleration in m/sec^2.  The user has to make the correct choices in order to assure dimensional consistency, and they have to know the definition of the Newton in order to know this usage is consistent. The same equation could be used in the English system using pounds of force, mass in pounds mass and acceleration in ft/sec^2-- but the user will have to introduce conversion factors to attain dimensional consistency.

If they are using the metric system, they will be using s, not sec.

Anyway the equation is dimensionally consistent, because the conversion factor has dimensions.  

 

To return to the previous example, it occured to me that all chemical reactions are usually written with arrows. Eg Wikipedia 

6 hours ago, studiot said:

Thank you John Cuthber for the typographical correction.

 

Now we have all calmed down and reached the stage of more reasoned discussion how about this equation

 


ML2T2ML2T2=EnergyMoment=1a


Where a is a constant depending upon units.

 a has dimensions. Its value changes according to its units.

Also: dimensional analysis does not guarantee a result which makes sense.

Edited by Bender
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7 hours ago, OldChemE said:

I'm going to take a little bit of a contrarian position on the OP.  Scientific equations do not have to be dimensionaly consistent, per se.  They relate terms that the user has to make dimensionally consistent.  For example, F = ma.  If the user is calculating in metric units they have use newtons, kg and acceleration in m/sec^2.  The user has to make the correct choices in order to assure dimensional consistency, and they have to know the definition of the Newton in order to know this usage is consistent. The same equation could be used in the English system using pounds of force, mass in pounds mass and acceleration in ft/sec^2-- but the user will have to introduce conversion factors to attain dimensional consistency.

I think it's a given that you have to do the analysis correctly, and as John Cuthber has pointed out, you can reduce this to the base dimension: mass, length, time, etc. where you don't have to worry about which system you are using.

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8 hours ago, OldChemE said:

I'm going to take a little bit of a contrarian position on the OP.  Scientific equations do not have to be dimensionaly consistent, per se.  They relate terms that the user has to make dimensionally consistent.  For example, F = ma.  If the user is calculating in metric units they have use newtons, kg and acceleration in m/sec^2.  The user has to make the correct choices in order to assure dimensional consistency, and they have to know the definition of the Newton in order to know this usage is consistent. The same equation could be used in the English system using pounds of force, mass in pounds mass and acceleration in ft/sec^2-- but the user will have to introduce conversion factors to attain dimensional consistency.

Good morning and thank you for your thoughts.

By dimensions I don't mean units I mean the MLT etc referred to by swansont in the previous post.


 

14 hours ago, Bender said:

It would work better with an arrow where the = sign is

 

The question was Is it necessary.....?

This was inspired because we write equations differently in different scientific disciplines.

Many still write a 'chemical equation' with an equals sign. Sometimes they include other information in the chemical equation such as energy production or state of reactants or products etc.

My second example just shows that even in mechanics odd things happen, although that equation is dimensionally consistent.

Here is a third example from computer programming.

 

i = i + 1

 

Please all remember this is a discussion subject, not a battlefront.

So the question is really about should we (try to) impose rigid definitions from one branch of Science on another?

 

:)

 

Edited by studiot
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1 hour ago, studiot said:

My second example just shows that even in mechanics odd things happen, although that equation is dimensionally consistent.

Can you explain what odd thing happened? I missed it.

1 hour ago, studiot said:

Here is a third example from computer programming.

i = i + 1

In most languages, that is not an equation. In languages where it is an equation, the behaviour would depend on the semantics of the language. It might be an error, for example.

1 hour ago, studiot said:

So the question is really about should we (try to) impose rigid definitions from one branch of Science on another?

Obviously not.

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1 minute ago, studiot said:

Rearrange that equation to show that energy = 5 times moment (in suitable units)

 

Is that so?

Sorry. I guess I am being slow but I have no idea what you are saying.

Are you saying that the equation is dimensionally correct but meaningless?

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52 minutes ago, Strange said:

But is that relevant?

Science doesn't usually deal with equations that are meaningless :)

Forgive me.

I thought I posted in the Philosophy section.

 

20 hours ago, studiot said:

am not sure where to post this so have used Philosophy (of Science) to allow latitude in exploring this subject.

And haven't we got a live thread discussing the fact that Philosophy and Science can each address matters unavailable to the other?

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1 minute ago, studiot said:

I thought I posted in the Philosophy section.

Yes, but the question was about scientific equations. And, if they are scientific, they shouldn't be meaningless. (Depending on the definition of "meaningless" perhaps.)

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4 minutes ago, Strange said:

Yes, but the question was about scientific equations. And, if they are scientific, they shouldn't be meaningless. (Depending on the definition of "meaningless" perhaps.)

Perhaps there was a message rather than a meaning in that equation

Mindlessly following rules can lead to difficulties.

 

:)

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So it is more a discussion about different uses of the = sign?

I guess in chemistry, it can be used for a transition, and in programming it can be used as assignment.

In my experience, chemists are less strict/pedantic about their notation than physicists or mathematicians. But I don't know whether that is general. I see no reason to lecture others for how they use symbols.

4 hours ago, studiot said:

Yes exactly. +1

Not necessarily. It can be useful as a dimensionless number in a specific case. Engineers regularly use similar numbers to describe the general behaviour of a system, especially in fluid dynamics.

Also: changing the units in such a way that it stays dimensionless, does not change the value.

List of examples

I've also made my own to describe the stability behaviour of a generalised solenoid actuator. 

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12 minutes ago, Bender said:

So it is more a discussion about different uses of the = sign?

I guess in chemistry, it can be used for a transition, and in programming it can be used as assignment.

In my experience, chemists are less strict/pedantic about their notation than physicists or mathematicians. But I don't know whether that is general. I see no reason to lecture others for how they use symbols.

Not necessarily. It can be useful as a dimensionless number in a specific case. Engineers regularly use similar numbers to describe the general behaviour of a system, especially in fluid dynamics.

Also: changing the units in such a way that it stays dimensionless, does not change the value.

It is a discussion, not a battleground.

All views are welcome.

I hope to learn something from the discussion; if others do vas well, that's great.

 

Exactly.

The ratio of energy to moment in say a static lever is meaningless since they are different things even though they have the same dimensions.

But the ratio of say inertial forces to viscous ones has meaning and is dimensionless since both numerator and denominator are of the same nature.

The ratio of distance to time is also not meaningless.
Since they are not the same and their dimensions are different the ratio produces a new quantity.

But even that may not be enough to have real meaning in some cases.

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6 hours ago, studiot said:

Here is a third example from computer programming.

i = i + 1

It's not equation. It's assignment operator.

http://en.cppreference.com/w/cpp/language/operator_assignment

http://en.cppreference.com/w/cpp/language/copy_assignment

Programmer can overload assignment and/or copy-assignment operators to implement custom operation executed after assigning something to object. Things that have basically nothing to do with mathematics.

Edited by Sensei
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9 minutes ago, studiot said:

The ratio of energy to moment in say a static lever is meaningless since they are different things even though they have the same dimensions.

But the ratio of say inertial forces to viscous ones has meaning and is dimensionless since both numerator and denominator are of the same nature.

I still don't see the relevance of this. Any equations using these relationships (meaningful or not) will still need to be dimensionally consistent.

10 minutes ago, studiot said:

The ratio of distance to time is also not meaningless.
Since they are not the same and their dimensions are different the ratio produces a new quantity.

And the equation describing the relationship of that new quantity to distance and time will be dimensionally consistent.

I have lost track of the point of the thread. The answer to the original question is, obviously, "yes". Everyone, including you seems to agree with that. You haven't produced any counter-examples. But you have taken it off in various random tangents for no apparent purpose. (Apart, I suppose, from the fun of a free ranging discussion.)

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I'll be more blunt than that.
Any useful physical relationship, where one variable is a function of another ( or more variables ), needs to be dimensionally consistent.
That is what R. C. Shukla taught me in 4th year at Brock University.
I've never doubted him.

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