# Equal Exchange

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The equal sign is a third dimension to the equation. It is the exchange of the reaction to become the product.

example

1+1                                =                       2

exchange

Gravitational energy example

Potential Energy = Kinetic Energy

PE                         =      KE

mgh                    =       (1/2)m(v^2)

The ability to change or the equal sign

(2)/(v^2)(g)(h)

This is 2 divided by velocity of of object squared multiplied by gravitational acceleration of earth multiplied by height of the object

This third part of the equation is what I call the Gravitational process of the earth. It is the thing reacting upon the conversion between Kinetic and Potential Energy of a falling object.

Other examples: The equal sign is another dimension yet to be explored. The conversion or process can be used to calculate the process of a chemical reaction and process of a mechanical spring.

These values will be highly useful in finding new values associated with the study of physics and chemistry.

I think that the calculation of these values is highly important. It is another dimension of mathematics and science that has yet to be explored.

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Posted (edited)

The equal sign itself has no value it is not an independent variable. A dimension is defined as an independent variable or other math object that can vary in value without affecting any other value.

Edited by Mordred

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Posted (edited)

It helps to show the difference between chemical processes. As well as this it can show the difference between two celestial objects. Example the gravitational process on mars and then on earth.

Examples

You can calculate the speed of a falling object with the value of the gravitational process of the earth.

(Gravitational process of earth)(mass of object)(ability of object)=speed of object

The ability of an object has three units (m^4)/(s^4)(g) meters to the fourth power divided by seconds to the fourth power multiplied by grams

You can calculate the ability of a reaction to become a product in chemistry.

(Chemical process)(mass of chemical)(ability of chemical)= rate of chemical reaction

ability of chemical has units of (M^4)/(s^4)(g) moles to the fourth power divided by seconds to the fourth power multiplied by grams

With the ability of a falling object you can calculate the ability of an object to cover a distance.

(ability of falling object)(mass of object)= Ability of object to cover a distance. The units of this is meters per second to the fourth power. (m/s)^4

Edited by kenjimckinstry

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Posted (edited)
1 hour ago, Mordred said:

The equal sign itself has no value it is not an independent variable. A dimension is defined as an independent variable or other math object that can vary in value without affecting any other value.

Again it doesn't count as a dimension.

It is a relation not a dimension.

Edited by Mordred

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!

Moderator Note

You haven't demonstrated any new physics here. If your "insight" is to have value, it has to do something that we don't already have. Otherwise you're just re-hashing standard physics.

What do you have that's new? And give specific demonstrations of it.

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Posted (edited)
6 hours ago, Mordred said:

Again it doesn't count as a dimension.

It is a relation not a dimension.

To amplify Mordred's statement ( +1 )

Mathematically and logically a relation is a particular conncetion between pairs of (mathematical) objects.

Equality as represented by the equals sign = is characterised by three properties.

Where A B and C are three mathematical objects

1) Reflexivity

A = A

2) Symmmetry

If A = B then B = A

3) Transitivity

If A = B and B = C then A = C

These may seem obvious but they are fundamnetal and very important.

Another stronger reelation is identity. This is different from equality and should be carefully distinguished.
All identities are also equalities, but not all equalities are identities.

An easy way to see this is to compare the following

${x^2} - 1 \equiv \left( {x + 1} \right)\left( {x - 1} \right)$

This is an identity. Note the different symbol.

It is true for all x or each and every possible value of x.

But

${x^2} - 1 = 0$

is only an equality. It is only true for certain values of x and not true for many more.

To pick up on the remark about chemical equations.

7 hours ago, kenjimckinstry said:

It helps to show the difference between chemical processes.

You noted that chemical reactions represent a process as well as an equality (mass balance charge balance etc)

These are more properly shown with various arrows for this reason

$\to$

etc

Edited by studiot

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It only works for certain science equations. It doesn't work on all equations.

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1 minute ago, kenjimckinstry said:

It only works for certain science equations. It doesn't work on all equations.

How do you determine which it works for and which it doesn't?

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It works for equations with a multiplicative ability. The values have to be quantifiable by a variable. It has to be very well proportioned.

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Posted (edited)

Can you argue that the equality sign is a relation ? Not a dimension under mathematical definition ?

Mathematics doesn't require words.

The proof must be based upon the math

Edited by Mordred

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I think the math proves that it is something that exists. You're correct, it isn't a dimension but a relation.

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On 3/25/2020 at 6:09 AM, studiot said:

To amplify Mordred's statement ( +1 )

Mathematically and logically a relation is a particular conncetion between pairs of (mathematical) objects.

Equality as represented by the equals sign = is characterised by three properties.

Where A B and C are three mathematical objects

1) Reflexivity

A = A

2) Symmmetry

If A = B then B = A

3) Transitivity

If A = B and B = C then A = C

These may seem obvious but they are fundamnetal and very important.

Another stronger reelation is identity. This is different from equality and should be carefully distinguished.
All identities are also equalities, but not all equalities are identities.

An easy way to see this is to compare the following

x21(x+1)(x1)

This is an identity. Note the different symbol.

It is true for all x or each and every possible value of x.

But

x21=0

is only an equality. It is only true for certain values of x and not true for many more.

To pick up on the remark about chemical equations.

You noted that chemical reactions represent a process as well as an equality (mass balance charge balance etc)

These are more properly shown with various arrows for this reason

etc

That's a great post.

Yeah, equivalence relations are much looser than identity. Any bivalent relation will give you an equivalence relation. You can actually even define an equivalence relation out of any function.

Take a function f, and you can define the equivalence relation ~ such that a~b iff f(a)~f(b). This is part of a process of decomposing a function. If the domain (what the function is taking as an argument) is A, and the the codomain (what the function points to) is B, then you can decompose a function via ~ in the following way.

Make a function f* that takes A and maps each element to the set of other elements of A that it is equivalent to under ~. So, an a in A maps under f* to all of the other a's that map to the same thing under f. For example, take the rule for f to be f(x) = x2. f*, then, will map 1 to {-1, 1}, 2 to {-2, 2}, 3 to {-3, 3} etc, because -1 and 1 both go to the same place under f. We typically write the codomain of f* as A/~.

You can then go from A/~ to what is called the "image" of f. And that's just the collection of things in B that actually get pointed to from things in A by f. Often you'll see this just imf.
So we have A -> A/~ -> imf. But there's something neat here. Since we have in essence collapsed A into things that don't map to the same thing in imf and all of the things in imf get mapped to, this function is reversible. So we have A -> A/~ <-> imf. Since it goes back and forth like that (assuming the function is structure preserving), A/~ and imf are isomorphic. That's a big word for "for all current intents and purposes, they're the same structurally". A nice fact about isomorphisms is that they are all equivalence relations.

We can go further into canonical decomposition and go from imf to B, but, for our purposes, we've gotten what we need for this discussion.

Any function gives us two different equivalence relations. So in our example of f(x) = x2 from the Real numbers to the Real numbers, we get the Real numbers being equivalent to sets of Real numbers that have the same absolute value. Those clearly aren't the same thing, in terms of identity. We also get this set of sets of numbers with the same absolute value being equivalent to the set of the squares of the absolute values of the previously mentioned set. Clearly sets containing things like {-2, 2} and sets containing things like 4 aren't identical, but they are equivalent.

Any nice back and forth will satisfy the axioms for the equivalence relation. Identity, on the other hand, has extra cool stuff like substitution. Identity is much stronger than equivalence.

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Thank you ydoaPs

But we should be careful using the word 'equivalence' as it does not mean 'equal', and of course certainly not identity.

Mathematical Equivalence is a very specialised way of using partitions of sets, as I'm sure you know.

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17 minutes ago, studiot said:

Thank you ydoaPs

But we should be careful using the word 'equivalence' as it does not mean 'equal', and of course certainly not identity.

Mathematical Equivalence is a very specialised way of using partitions of sets, as I'm sure you know.

If I'm understanding you correctly, I think you're using different terminology. It might be a geographic thing.

As far as I learned, an Equivalence Relation has the properties you detailed in your post, and, by the process I discussed, partitions a set into Equivalence Classes resulting in the quotient set (or monoid or group or category etc). I'm not sure what there is to gain by using "equivalence" only in terms of equivalence classes rather than for either equivalence relations and equivalence classes. If we had to pick, I'd probably make the relation "Equivalence", but that's probably just the categorial bias I have.

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3 hours ago, ydoaPs said:

If I'm understanding you correctly, I think you're using different terminology. It might be a geographic thing.

As far as I learned, an Equivalence Relation has the properties you detailed in your post, and, by the process I discussed, partitions a set into Equivalence Classes resulting in the quotient set (or monoid or group or category etc). I'm not sure what there is to gain by using "equivalence" only in terms of equivalence classes rather than for either equivalence relations and equivalence classes. If we had to pick, I'd probably make the relation "Equivalence", but that's probably just the categorial bias I have.

No I don't think our terminology is different but my point was that 'equivalence' hides more levels of meaning.

Consider the set of positive integers, n.

This may be partitioned into two subsets, which form equivalence classes.

n is odd

n is even

No two members of either subset are numerically equal.

No two members of either subset are identical.

Yet any even (or odd) number is equivalent to any other even or odd number.

You can also demonstrate this with functions and mappings if you please.

So in that case there are three levels of meaning to the word "equivalence"

And before I was only describing two.

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5 hours ago, studiot said:

No I don't think our terminology is different but my point was that 'equivalence' hides more levels of meaning.

Consider the set of positive integers, n.

This may be partitioned into two subsets, which form equivalence classes.

n is odd

n is even

No two members of either subset are numerically equal.

No two members of either subset are identical.

Yet any even (or odd) number is equivalent to any other even or odd number.

You can also demonstrate this with functions and mappings if you please.

So in that case there are three levels of meaning to the word "equivalence"

And before I was only describing two.

And that's not even counting all of the other forms of sameness like adjoint situations. There's so many different forms of "eh, that's sort of the same as this other thing in certain ways".

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14 hours ago, ydoaPs said:

And that's not even counting all of the other forms of sameness like adjoint situations. There's so many different forms of "eh, that's sort of the same as this other thing in certain ways".

We are agreed now  ?  +1

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