# Relative Mathematics

#### Conway

Archie

Thank you.

I however have already done as you asked. I posed 1 simple field axiom to be taken as an addition to all current field axiom. The nature of which will allow for division by zero (among other things). It does so with out changing ANY OTHER axiom. Therefore I have altered the definition of a "number" with-in the current system without changing the current system. It can only be said that there is a "failure" in the current system with division by zero. It can not be said that it is "ITERGRAL to the proper functioning of the system". It's that the system can not do it. Except the Riemann Sphere, where division by zero finds meaning.

Also I would disagree with your very first statement. Numbers are already defined exactly as I am posing. It is only that I have made a clear distinction between a "numbers" pieces, where mathematics has currently not.

I look forward do more of your thoughts.

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#### chiro

MHF Helper
It's not specific enough for me Conway.

You should specify some set theoretic definitions and/or definitions with numeric structures [like linear objects in some field].

You can't rely on vague notions and vague means that you can't put it in specific terms.

Remember - the algebra involves intersections and unions for sets and arithmetic for everything else.

If you can explain this [and do so specifically without using convoluted language] then you will get the point across.

Specifics Conway - more specifics.

1 person

#### Conway

Chiro

I understand you, I appreciate you. You feel that I was/am not specific enough. If I may say one last thing as we part ways....

Leave all "theoretic definitions and/or definitions with numeric structures [like linear objects in some field", exactly as they are.

I have stated things that only affect multiplication and division by zero. All other multiplication and division yields their current sums......so unless you are using liner objects that are related to zero (which your not), then there is not a problem with what you are posing is a problem......

Also one can not get more specific then a field axiom...maybe you should apply my field axiom to some "linear object in some field".

Sincerely thank you for your time.

Conway

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#### chiro

MHF Helper
I'll make this as easy to understand as possible.

There are two main structures in mathematics - numbers and sets.

Sets have two main operations [and they can be related to one another] which are intersections and unions.

Example - A and B or A or B.

Numbers have five main operations - addition, subtraction, multiplication, division and the modulus.

Can you tell me what structure you are using and what operations in terms of the AND, OR, +, -, *, /, and MOD?

If you can't do that then nobody will understand you.

Also - if you are using numbers then you will run into problems with zero's which means it's highly likely that you are using sets and if so you will need to describe what operations you are using in terms of the unions and intersections of sets.

It's very simple to do the above.

1 person

#### Conway

Chiro

I think I see where our problem has been.

I am NOT using sets
I am NOT using numbers

I am using composites of numbers...for example

(_) = one space = z2
(1) = one value = z1

when a value (z1) is "placed" into a space (z2) a number is created.

in a binary operation of multiplication or division one symbol represents z1, and the other symbol represents z2.

2 x 3

2(value,z1) x 3(space,z2)

(_,_,_) = three spaces
(1,1) = 2 values, or alternatively (2)

I place the values into the spaces.....then add

(1,1+1,1+1,1)...or alternatively (2 + 2 + 2)

After and only After placing value into spaces do I arrive at a number.

I hope here you can see...I am not using numbers....Nor am I using sets.....I am using pieces of numbers.

Thanks...

#### topsquark

Forum Staff
Chiro

I think I see where our problem has been.

I am NOT using sets
I am NOT using numbers

I am using composites of numbers...for example

(_) = one space = z2
(1) = one value = z1

when a value (z1) is "placed" into a space (z2) a number is created.

in a binary operation of multiplication or division one symbol represents z1, and the other symbol represents z2.

2 x 3

2(value,z1) x 3(space,z2)

(_,_,_) = three spaces
(1,1) = 2 values, or alternatively (2)

I place the values into the spaces.....then add

(1,1+1,1+1,1)...or alternatively (2 + 2 + 2)

After and only After placing value into spaces do I arrive at a number.

I hope here you can see...I am not using numbers....Nor am I using sets.....I am using pieces of numbers.

Thanks...
I'm totally lost. You need to get to the very basics here. What are "pieces of numbers" and how do you combine them to become numbers. Be specific and give examples where you can.

-Dan

#### Conway

Dan

Of course, I understand your request here.

The pieces of a number are

Value
Space

A value is placed into a space to create a number

(_) = one space
(1) = one value
(1) = 1 (the number)

Did you read post #26....I felt we may have been close to an understanding at this point.....I am more than willing to return to notation.....albeit it presents difficulty

Better? If not.... I understand if you no longer wish to continue. I certainly hope this is not the case.

#### chiro

MHF Helper
Everything should be isomorphic to a set or number in some way.

Numbers are sets with specific organization [i.e. they organize information so that arithmetic and geometry make sense].

If you can't represent what you are doing with some sort of set structure and algebra then you aren't doing mathematics.

#### Conway

Chiro

I can and have, though much work has to be done in this regard. Please refer to post #27.....ill copy and paste the relevant parts for you actually.....

(V + S) → A: ∀A, S ∈ V, S→(V,S) ∈ A, V ∈ S, V→(V,S) ∈ A
∀A ≠ 0: (V,S) = (A,A)
∀A = 0: (V,S) = (0,1)

Or alternatively

(V,S) ∈ ∀A
∀A ≠ 0 : (V,S) = (A,A)
∀A = 0 : (V,S) = (0,1)

V= value, or z1
S= space, or z2
A= any number in a given set

#### chiro

MHF Helper
How do you construct the mappings in terms of set unions and intersections?

1 person