03/20/2023, 12:16 AM

This idea might have occured here before and it also relates to many known ideas.

In particular :

mediant and weighted mediant.

center of mass.

mass point geometry.

" extended reals " and "hyperoperator numbers " and such terms , which are not universally agreed upon hence the " ".

I might have mentioned it before.

It also related to numerical methods and averaging.

and some number theory.

So forgive me if I am repeating myself or others.

Often we only see its sum defined but it can be extended.

And it has nice properties !

commutative , distributive , associative , solvable in a sense etc

although it may not agree with the generalized binomium theorem.

( it works for the binomium case of squares for sure see later )

I call them lever numbers because they compute the balancing point of levers for weights and distances.

That is imo the best definition.

With special thanks to Archimedes.

( I also considered them as norms for other numbers but lets ignore that complication for now )

( do not confuse with my spiral numbers or hypercomplex ideas )

Ok time to define and rationalize :

For cases where we do not devide by zero :

(a,p) + (b,q) = ( a+b , (ap + bq)/(a+b) )

a,b are called weights and p and q positions.

examples :

(1,0) + (2,1) = (3, (1*0 + 2*1)/3 ) = (3,2/3)

(1,1) + (2,2) = (3, (1*1 + 2*2)/3 ) = (3,5/3) = (3, 1 + 2/3)

(notice the distance shift, this is why I picked this example. Basically the lever and weights are all placed a unit further )

conjecture

|p,r|/|q,r| = b/a

[p - (ap + bq)/(a+b)]/[q - (ap + bq)/(a+b)] =

[pa + pb - ap - bq]/[qa + qb - ap - bq] =

[pb - bq]/[-pa + aq] = b(p-q)/[a(q-p)] = b/(-a)

so

|p,r|/|r,q| = b/a

Qed

( this shows the mediant and mass point geometry interpretations agree )

***

(a,x) * (b,q) = (ab,q)

(a,x) = a

***

c [(a,p) + (b,q)] = ( c(a+b) , (ap + bq)/(a+b) )

c [(a,p) + (b,q)] = ( c(a+b) , (c/c)(ap + bq)/(a+b) )

***

(c,z) * [(a,p) + (b,q)] = ( c(a+b) , z<(ap + bq)/(a+b)> )

z<(ap + bq)/(a+b)> = (a z<p> + b z<q>)/(a+b)

***

(c,z)*(a,p) = (ca,zp)

(1,1)*(a,p) = (a,p)

(c,1)*(a,p) = (ca,p)

(c,p) + (a,p) = (a+c,p)

***

This is how the pairs are defined.

you can take the variables real or complex or ... or increase the variables or the dimensions etc.

But I wanted to start simple with a couple of 2 variables.

Notice the product is designed to be distributive over addition.

And also it follows (a,p)^0 = (1,1) what is the multiplicative unit.

( the idea of adding mod to the party occured )

ok now how about squares and the (a+b)^2 = a^2 + 2ab + b^2 formula ? ( binomium square case )

binomium :

[(a,p) + (b,q)]^2 = ( (a+b)^2 , (ap + bq)^2/(a+b)^2 )

= (a^2,p^2) + (2ab,pq) + (b^2,q^2)

= (a^2 + 2ab, (a^2 p^2 + 2abpq)/(a^2 + 2ab) ) + (b^2,q^2)

= ( (a+b)^2 , [(a^2 + 2ab)*(a^2 p^2 + 2abpq)/(a^2 + 2ab) + b^2 q^2]/(a+b)^2 )

so

(a^2 p^2 + 2abpq) + b^2 q^2

must =

(ap + bq)^2 = a^2 p^2 + 2 apbq + b^2 q^2.

and it does !

And probably this ( binomium ) works for higher powers too ( by induction ).

How about taylor or fourier ? analytic analogues ? Iterations ? hyperoperators ?

Defining exp , log , sine etc ?

What do you think ?

Regards

tommy1729

Tom Marcel Raes

In particular :

mediant and weighted mediant.

center of mass.

mass point geometry.

" extended reals " and "hyperoperator numbers " and such terms , which are not universally agreed upon hence the " ".

I might have mentioned it before.

It also related to numerical methods and averaging.

and some number theory.

So forgive me if I am repeating myself or others.

Often we only see its sum defined but it can be extended.

And it has nice properties !

commutative , distributive , associative , solvable in a sense etc

although it may not agree with the generalized binomium theorem.

( it works for the binomium case of squares for sure see later )

I call them lever numbers because they compute the balancing point of levers for weights and distances.

That is imo the best definition.

With special thanks to Archimedes.

( I also considered them as norms for other numbers but lets ignore that complication for now )

( do not confuse with my spiral numbers or hypercomplex ideas )

Ok time to define and rationalize :

For cases where we do not devide by zero :

(a,p) + (b,q) = ( a+b , (ap + bq)/(a+b) )

a,b are called weights and p and q positions.

examples :

(1,0) + (2,1) = (3, (1*0 + 2*1)/3 ) = (3,2/3)

(1,1) + (2,2) = (3, (1*1 + 2*2)/3 ) = (3,5/3) = (3, 1 + 2/3)

(notice the distance shift, this is why I picked this example. Basically the lever and weights are all placed a unit further )

conjecture

|p,r|/|q,r| = b/a

[p - (ap + bq)/(a+b)]/[q - (ap + bq)/(a+b)] =

[pa + pb - ap - bq]/[qa + qb - ap - bq] =

[pb - bq]/[-pa + aq] = b(p-q)/[a(q-p)] = b/(-a)

so

|p,r|/|r,q| = b/a

Qed

( this shows the mediant and mass point geometry interpretations agree )

***

(a,x) * (b,q) = (ab,q)

(a,x) = a

***

c [(a,p) + (b,q)] = ( c(a+b) , (ap + bq)/(a+b) )

c [(a,p) + (b,q)] = ( c(a+b) , (c/c)(ap + bq)/(a+b) )

***

(c,z) * [(a,p) + (b,q)] = ( c(a+b) , z<(ap + bq)/(a+b)> )

z<(ap + bq)/(a+b)> = (a z<p> + b z<q>)/(a+b)

***

(c,z)*(a,p) = (ca,zp)

(1,1)*(a,p) = (a,p)

(c,1)*(a,p) = (ca,p)

(c,p) + (a,p) = (a+c,p)

***

This is how the pairs are defined.

you can take the variables real or complex or ... or increase the variables or the dimensions etc.

But I wanted to start simple with a couple of 2 variables.

Notice the product is designed to be distributive over addition.

And also it follows (a,p)^0 = (1,1) what is the multiplicative unit.

( the idea of adding mod to the party occured )

ok now how about squares and the (a+b)^2 = a^2 + 2ab + b^2 formula ? ( binomium square case )

binomium :

[(a,p) + (b,q)]^2 = ( (a+b)^2 , (ap + bq)^2/(a+b)^2 )

= (a^2,p^2) + (2ab,pq) + (b^2,q^2)

= (a^2 + 2ab, (a^2 p^2 + 2abpq)/(a^2 + 2ab) ) + (b^2,q^2)

= ( (a+b)^2 , [(a^2 + 2ab)*(a^2 p^2 + 2abpq)/(a^2 + 2ab) + b^2 q^2]/(a+b)^2 )

so

(a^2 p^2 + 2abpq) + b^2 q^2

must =

(ap + bq)^2 = a^2 p^2 + 2 apbq + b^2 q^2.

and it does !

And probably this ( binomium ) works for higher powers too ( by induction ).

How about taylor or fourier ? analytic analogues ? Iterations ? hyperoperators ?

Defining exp , log , sine etc ?

What do you think ?

Regards

tommy1729

Tom Marcel Raes