Is it necessary for scientific equations to be dimensionally consistent?

Bender · March 6, 2018

19 hours ago, studiot said:

But how is Science able to throw away part of the consistent maths in my example of selecting only the real part of a complete solution.

In other words what are 5 i metres, or what is the arcsin of 3 ?

What happens if I poke at my ripper trousers with a screwdriver?

Mathematics is a toolset. Not all actions executed with tools have meaningfull results. Throwing out the imaginary part is one of the mathematical tools we have to find a solution. Sometimes the imaginary part is meaningfull, but we throw it out anyway because we're not interested in it.

19 hours ago, studiot said:

Mathematics itself employs the equality sign for more than one meaning.

Could you give an example?

studiot · March 7, 2018

On ‎3‎/‎6‎/‎2018 at 6:11 PM, Bender said:

Could you give an example?

Of course.

In the expression

(x - 1) (x - 1) = x² +1 - 2x

the equals sign is used differently from its use in the expression

x2 + 4x + 4 = 0

On ‎3‎/‎6‎/‎2018 at 6:11 PM, Bender said:

What happens if I poke at my ripper trousers with a screwdriver?

Mathematics is a toolset. Not all actions executed with tools have meaningfull results. Throwing out the imaginary part is one of the mathematical tools we have to find a solution. Sometimes the imaginary part is meaningfull, but we throw it out anyway because we're not interested in it.

I have no idea about your trousers but your following exactly exemplifies my stance.

There is mathematics and there are applications (or there are other disciplines that can use Mathematics however you like to describe it).

And they are separate and distinct things.

How high did he jump?

Answer the height squared = 4 metres.

Now as an invigilator in the high jump I can say there is only one answer viz 2 metres.

But as a Mathematician I cannot pretend that -2 is not also a solution of the equation x² = 4.

Notice also this is entirely consistent usage

As an invigilator I use units of metres - ie my 4 and 2 have dimensions.

As a Mathematician no units are used - they are just numbers.

Edited March 7, 2018 by studiot

**Strange** · March 7, 2018

33 minutes ago, studiot said:

the equals sign is used differently from its use in the expression

It is?

33 minutes ago, studiot said:

There is mathematics and there are applications ...

And they are separate and distinct things.

So my cousin with a degree in Applied Maths was given a fake degree?

33 minutes ago, studiot said:

Answer the height squared = 4 metres.

In what world is height squared measured in metres?

Edited March 7, 2018 by Strange

studiot · March 7, 2018

28 minutes ago, Strange said:

n what world is height squared measured in metres?

Thank you for correcting that oversight,

Of course I should have said

Answer the height squared = 4 metres squared.

However that does not invalidate the argument.

30 minutes ago, Strange said:

So my cousin with a degree in Applied Maths was given a fake degree?

I too have a degree in applied Maths. So what?

31 minutes ago, Strange said:

It is?

What does your esteemed cousin say about this?

What happens if he substitutes x = 100 into both expressions?

**Strange** · March 7, 2018

19 minutes ago, studiot said:

I too have a degree in applied Maths. So what?

You said mathematics and applications "are separate and distinct things"

But it seems you were mistaken.

21 minutes ago, studiot said:

What happens if he substitutes x = 100 into both expressions?

So you are unable to explain how they are different?

Bender · March 7, 2018

2 hours ago, studiot said:

As a Mathematician no units are used - they are just numbers.

Since when is the mathematician allowed to casually discard the variable m?

2 hours ago, studiot said:

In the expression

(x - 1) (x - 1) = x² +1 - 2x

the equals sign is used differently from its use in the expression

x2 + 4x + 4 = 0

The context is different, but the meaning is the same.

studiot · March 7, 2018

46 minutes ago, Bender said:

Since when is the mathematician allowed to casually discard the variable m?

It must be your turn to explain what you mean, because I don't understand the question.

46 minutes ago, Bender said:

The context is different, but the meaning is the same.

Mathematics has no 'context'.

The meanings are different. Period.

I'm sorry the response was cryptic but I assume you understand enough mathematics to know that in one case the expression is true for all x, whereas in the other it is only true for two specific values of x and untrue for the rest of the infinite possibilities available.

1 hour ago, Strange said:

You said mathematics and applications "are separate and distinct things"

But it seems you were mistaken.

So you are unable to explain how they are different?

Why do you think I will answer your questions, when you do not answer mine?

**Strange** · March 7, 2018

2 minutes ago, studiot said:

I'm sorry the response was cryptic but I assume you understand enough mathematics to know that in one case the expression is true for all x, whereas in the other it is only true for two specific values of x and untrue for the rest of the infinite possibilities available.

That is not a function of the equals sign but of the expressions that it is used to show the equality of. The equals sign means “equals” in both cases.

koti · March 8, 2018

I stuck to reading like I said I would 2 weeks ago and I’m confused like I predicted I would be. I don’t have a degree in applied maths like the cousin or studiot, would you @studiot explain this like you would to a 12 year old? After all Einstein said „If you can’t explain it simply, you don’t know it well enough”

StringJunky · March 8, 2018

6 minutes ago, koti said:

I stuck to reading like I said I would 2 weeks ago and I’m confused like I predicted I would be. I don’t have a degree in applied maths like the cousin or studiot, would you @studiot explain this like you would to a 12 year old? After all Einstein said „If you can’t explain it simply, you don’t know it well enough”

He also said: “Everything should be as simple as it can be, but not simpler”

Eise · March 8, 2018

12 hours ago, studiot said:

In the expression

(x - 1) (x - 1) = x² +1 - 2x

the equals sign is used differently from its use in the expression

x2 + 4x + 4 = 0

As this seems to be important... The meaning depends on the context. One way to see this is that you can change the '=' into '=>', in the other that is definitely wrong.

But you mustn't. The problem could be given as:

Solve:

(x - 1) (x - 1) = x² +1 - 2x

So the 'derivation' would be:

x² +1 - 2x = x² +1 - 2x

So it is true for every x.

(Why do I have faint memories of 'see1' and 'see2'?)

Some already said it: mathematics is (amongst others) with dimensionless objects, e.g. numbers and vectors. So you can add x² and x, because there is no dimension there. But still you cannot add 2 and the vector (1,2), because they also in mathematics have different dimensions.

But in physics all quantities measured have dimensions: length, surface, mass etc. And here the same is true as in math: you cannot add numbers with different dimensions.

The reaction equation:

HCL + 10H₂O = HCL(aq) + 16.61 kcal

is just sloppy use of the '=' sign. Literally, i.e. in the mathematical sense, you cannot add molecules and warmth. But we know what is meant. The word 'equation' is not in the mathematical, or even in the physical sense.

'Allowed symbols in reaction equations are:

Quote

Symbols are used to differentiate between different types of reactions. To denote the type of reaction:^[1]

" $=$ " symbol is used to denote a stoichiometric relation.

" $\rightarrow$ " symbol is used to denote a net forward reaction.

" $\rightleftarrows$ " symbol is used to denote a reaction in both directions.^[3]

" $\rightleftharpoons$ " symbol is used to denote an equilibrium

Edit: OK, that citation does not quite work. Follow the link to see the symbols in their full glory...

Edited March 8, 2018 by Eise

John Cuthber · March 8, 2018

12 hours ago, studiot said:

Answer the height squared = 4 metres squared.

Since that answer is ambiguous, it is not correct.

studiot · March 8, 2018

1 hour ago, John Cuthber said:

Since that answer is ambiguous, it is not correct.

Once again thank you for the typographical correction.

Yes 4 square metres would be better.

**Strange** · March 8, 2018

Just now, studiot said:

Yes 4 square metres would be better.

Why would anyone measure height in square metres. (Rhetorical question; no answer required.)

John Cuthber · March 8, 2018

8 minutes ago, studiot said:

Once again thank you for the typographical correction.

Yes 4 square metres would be better.

It's not the square meters bit that's a problem

Saying the square of something is two is an ambiguous definition of that thing.

8 minutes ago, Strange said:

Why would anyone measure height in square metres. (Rhetorical question; no answer required.)

I might measure the length of a pendulum in some strange unit like square root of metres.

studiot · March 8, 2018

Thank you Eise for you input, which as alwys respects the material and views posted by others whether you disagree or not.

I note you have picked up on several points raised by various others.

2 hours ago, Eise said:

As this seems to be important... The meaning depends on the context. One way to see this is that you can change the '=' into '=>', in the other that is definitely wrong.

But you mustn't. The problem could be given as:

Solve:

(x - 1) (x - 1) = x² +1 - 2x

So the 'derivation' would be:

x² +1 - 2x = x² +1 - 2x

So it is true for every x.

10 hours ago, koti said:

I stuck to reading like I said I would 2 weeks ago and I’m confused like I predicted I would be. I don’t have a degree in applied maths like the cousin or studiot, would you @studiot explain this like you would to a 12 year old? After all Einstein said „If you can’t explain it simply, you don’t know it well enough”

Between say 1900 and 1920 the distinction would have been taught to a 12 year old in primary school. Until about 1960 this was pushed up the ladder to early secondary school at about 13 or 14. I am not sure what happened after that.

The point is that the first of those expressions is an identity and the second an equality.

The distinction as taught to 12 -14 year olds is that an identity is true for every x, an equality is only true for some x.

At university deeper understanding is studied in the set theory of equivalence realtions.

But it is far from unusual in Mathematics for words or symbols to perform multiple duties.
Look in any Dictionary of Mathematics and you will see that many of the terms have multiple meanings and use.
This is the place where 'context' is of use in Mathematics.

3 hours ago, Eise said:

(Why do I have faint memories of 'see1' and 'see2'?)

Yes indeed multiple meanings.

3 hours ago, Eise said:

Some already said it: mathematics is (amongst others) with dimensionless objects, e.g. numbers and vectors. So you can add x² and x, because there is no dimension there. But still you cannot add 2 and the vector (1,2), because they also in mathematics have different dimensions.

Actually you can add 2 to the vector (1,2). All that happens is that you move from the mathematics of a vector space to the mathematics of an affine space.
One application of this has importantance in modern relativity.

But the mathematical definition of a vector is very different from the one normally employed in Physics, and you have used here.

So you can, for instance, add a constant (which is not a vector) to a vector in a Fourier series.

3 hours ago, Eise said:

But in physics all quantities measured have dimensions: length, surface, mass etc. And here the same is true as in math: you cannot add numbers with different dimensions.

Maybe so, but Physics is not the only Science and the question here was about Science in general.

This is not an exercise in "Physik uber alles"

3 hours ago, Eise said:

The reaction equation:

HCL + 10H₂O = HCL(aq) + 16.61 kcal

is just sloppy use of the '=' sign. Literally, i.e. in the mathematical sense, you cannot add molecules and warmth. But we know what is meant. The word 'equation' is not in the mathematical, or even in the physical sense.

That is a matter of opinion.

The person who originally wrote that equation was a very famous Professor of Chemical Thermodynamics at the University of California, Santa Barbara.

I used it because it is now standard and I did not measure the themochemistry part.

So in short, some do some don't.

The liberating part of Science is that, like the English Language, no one has domain over expression.

Long may that continue.

3 hours ago, Eise said:

Quote

Symbols are used to differentiate between different types of reactions. To denote the type of reaction:^[1]

" $=$ " symbol is used to denote a stoichiometric relation.

" $\rightarrow$ " symbol is used to denote a net forward reaction.

" $\rightleftarrows$ " symbol is used to denote a reaction in both directions.^[3]

" $\rightleftharpoons$ " symbol is used to denote an equilibrium

Well that is so as far as it goes, but it is woefully incomplete.

What symbol is used for the electrochemical neutrality of a chemical equation?

What symbol is used for a nonstoichiometric equation? for instance in the presence of excess of one reagent?

If a reagent appears also as a product, can you legitimately 'cancel' part or all of that quantity, as you most assuredly can in a mathematical expression for example

2x + 3 = 6 + 3 yields x = 3

But if 3 is the concentration of a necessary reagent, what happens if you cancel it?

Did you attempt the nonstochiometric question I asked Strange?

You can extract 6 dimensionally correct equations from the nonstoichiometrical statement of the reaction and solve them for the 6 unknowns in a proper mathematical way.

This is an example of Applied Mathematics in action, whereby the Mathematics is separated from the Science of the situation and solved, providing useful information about the Science of the situation.

If you answered the chemical question I posted earlier

**Strange** · March 8, 2018

11 minutes ago, studiot said:

The point is that the first of those expressions is an identity and the second an equality.

The distinction as taught to 12 -14 year olds is that an identity is true for every x, an equality is only true for some x.

But, in both cases, the "=" means that the LHS is equal to the RHS. It is the nature / relationship of the LHS and RHS that defines what the equals sign means.

If the convention was to use a different symbol for identity and equality (as is done in some programming languages) then changing the symbol would not (necessarily) turn an equality into an identity; it would just be wrong.

Edited March 8, 2018 by Strange

studiot · March 8, 2018

Just now, Strange said:

But, in both cases, the "=" means that the LHS is equal to the RHS. It is the nature / relationship of the LHS and RHS that defines what the equals sign means.

No it doesn't.

Did you do what I asked and substitute x = 100 into both equations.

Were both sides the same in both cases?

Here is a Physics question for you to chew on, since you eschew pure maths and chemistry ones.

Why is inductance measured in Henries in the MKS system and centimetres in the cgs system

Why is EMF measured in voltage in the MKS system and ergs in the cgs system since they are different physical quantities?

What happened to the dimensions?

**Strange** · March 8, 2018

33 minutes ago, studiot said:

No it doesn't.

So you are saying the two sides are not equal in both cases?

koti · March 8, 2018

12 hours ago, StringJunky said:

He also said: “Everything should be as simple as it can be, but not simpler”

I acknowledge that hence I like to learn. That aside, I'm sure Einstein would smack studiot with a ruler in this thread.

Edited March 8, 2018 by koti

Carrock · March 8, 2018

7 hours ago, studiot said:

No it doesn't.

Did you do what I asked and substitute x = 100 into both equations.

Were both sides the same in both cases?

Here is a Physics question for you to chew on, since you eschew pure maths and chemistry ones.

Why is inductance measured in Henries in the MKS system and centimetres in the cgs system

Why is EMF measured in voltage in the MKS system and ergs in the cgs system since they are different physical quantities?

What happened to the dimensions?

Why do you claim inductance is a length in CGS.

Why do you claim voltage is energy in CGS?

23 hours ago, studiot said:

How high did he jump?

Answer the height squared = 4 [square] metres.

Now as an invigilator in the high jump I can say there is only one answer viz 2 metres.

But as a Mathematician I cannot pretend that -2 is not also a solution of the equation x² = 4.

Is jumping from 3 metres below ground level to 2 metres below ground level possible only in maths, not in the physical world?

Edited March 8, 2018 by Carrock

studiot · March 8, 2018

18 minutes ago, Carrock said:

Is jumping from 3 metres below ground level to 2 metres below ground level possible only in maths, not in the physical world?

Is jumping over a hole a high jump?

However you are right to point out that I need to be more careful with my choice of examples, thank you +1

19 minutes ago, Carrock said:

Why do you claim inductance is a length in CGS.

Why do you claim voltage is energy in CGS?

Because it's true?

Maybe swansont remembers and used these units, but anyway here are some tables.

Note very carefully that certain quantities (as I mentions) have different units and dimensions in different systems.

(The equations are also different )

**Strange** · March 8, 2018

7 minutes ago, studiot said:

Because it's true?

Even if true, I fail to see the relevance. An equation would normally use one type of units, and would need to be consistent in those units. An equation converting between units would need conversion factors that maintained dimensional consistency.

Carrock · March 8, 2018

42 minutes ago, studiot said:

Maybe swansont remembers and used these units, but anyway here are some tables.

Note very carefully that certain quantities (as I mentions) have different units and dimensions in different systems.

(The equations are also different )

The tables are pretty useless, but a voltage erg(e.s. system) is pretty clearly different from an energy erg.

I don't know much about c.g.s., but I'm certain it was dimensionally valid if you use its peculiar conventions.

[edit] There seems to be a mixture of three different conventions on p.330 col3.

Edited March 8, 2018 by Carrock

Bender · March 8, 2018

22 hours ago, studiot said:

It must be your turn to explain what you mean, because I don't understand the question.

Mathematically, you can work with dimensions as if they are variables that don't have a value (or can have any value, if you want). Works great.

22 hours ago, studiot said:

Mathematics has no 'context'.

The meanings are different. Period.

I'm sorry the response was cryptic but I assume you understand enough mathematics to know that in one case the expression is true for all x, whereas in the other it is only true for two specific values of x and untrue for the rest of the infinite possibilities available.

Both expressions are true for all values of y.

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