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+--- Thread: lever numbers (/showthread.php?tid=1727)
lever numbers - tommy1729 - 03/20/2023
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