On Reduction, 7 of 11

When Additions Collide

     Addition and reduction are both linear relative to themselves, but curved relative to the other. They’re nonlinear when combined. For instance, any quadratic equation is an alternating-addition equation:

     ax² + bx + c      =   0     if and only if

     (x + b/a) <-> c/b   =   x  if and only if

     (x <+> c/b) - b/a   =   x

     Conversely,

     (x + A) <-> B   =   x     if and only if

     (x <+> B) - A   =   x     if and only if

     x² + Ax + AB  =   0

     Alternating addition and reduction yields linear fractional functions:

                        

      (Ax + B) <+> C   =     ACx  +   BC

                              Ax  + (B+C)

      (Ax <+> B) + C   =     A(B+C)x +  BC  

                              Ax   +  B

       ax + b       =  ( (   a²   ) x  +   ab    )  <+>   a    

       cx + d              ad-bc         ad-bc            c

       ax + b      =  ( ( ad-bc ) x  <+>  ad-bc )  +   b  

       cx + d               d²               cd         d

     Continued fractions arise from an infinite alternation of addition and reduction:

             

           1

      a +                    =  a + ( 1/b <+> ( c + ( 1/d <+> ... )))

           b +  1

                           

                c +  1

                          

                     d ... 

     We can simplify these series via the Flipper Identities:

     a + ( b <+> c ) =  a + b + ( -b <+> -b²/c )

     a <+> ( b + c ) =  a <+> b <+> ( -b + -b²/c )

     I call these “flipper identities” because they flip c upside down.

     We can define multiplication, squaring and inverse by alternating addition and reduction.

     ((u-v) <-> (u+v))     =    (  u² – v²   ) / (2v)

     ((u<->v) - (u<+>v))        =    ( 2u²<->2v² ) /  v

Therefore:

     ((a+b-c) <-> (a+b+c))      =    ( a² + 2ab + b² – c² ) / (2c)

     ((a-b-c) <-> (a-b+c))      =    ( a² - 2ab + b² – c² ) / (2c)

     ((a<+>b<->c)-(a<+>b<+>c)) =    ( 2a² <+> ab <+> 2b² <-> 2c² ) / c

     ((a<->b<->c) - (a<->b<+>c))     =    ( 2a² <-> ab <+> 2b² <-> 2c² ) / c

Therefore:

     2ab/c      =    ((a+b-c) <->(a+b+c)) + ((a-b+c) <->(a-b-c))

     ab/2c      =

((a<+>b<->c)-(a<+>b<+>c)) <+> ((a<->b<+>c)-(a<->b<->c))

     ab         =    ((a+b-2)<->(a+b+2)) + ((a-b+2) <->(a-b-2))  

     ab         =   

((a<+>b<->½)-(a<+>b<+>½)) <+> ((a<->b<+>½)-(a<->b<->½))  

     a/c        =    ((a+½-c) <->(a+½+c)) + ((a-½+c) <->(a-½-c))

     a/c        =  

((a<+>2<->c)-(a<+>2<+>c)) <+> ((a<->2<+>c)-(a<->2<->c))

     a²         =    2((a-1)<->(a+1)) + 1  = ((a<->2)-(a<+>2)) <+> 4

     a²         =    ½((a<->1)-(a<+>1)) <+> 1 = ((a-½) <-> (a+½)) + ¼

     There are other formulas. For instance:

     - a²/x     =    ((x-a)<+>a)- a

                =    ((x+a)<->a)+ a

                =    ((x<->a)+a) <->a

                =    ((x<+>a)-a)<+> a

    

       1/x      =    (((-x)-1)<+>1)-1     

=    (((-x)+1) <->1)+1 

=    (((-x)<->1)+1)<->1 

=    (((-x)<+>1)-1)<+> 1

     I speculate that any formula defining x*y, x/y or x² from the additions must have at least one minus, at least one constant, and be at least three levels deep. Also, a formula defining x*y/z can lack constants, but it must have a minus, and be at least four levels deep.

     Consider this iteration:

     (x₀, y₀)    =    (a, b)  

     (x_N+1, y_N+1) =    ( (x_N+y_N)/2 , 2(x_N<+>y_N) ) 

     The pair takes arithmetic and harmonic means of itself. If a and b are both positive, then the sequence rapidly converges to sqrt(ab); if a and b are both negative, then the sequence converges to -sqrt(ab); and if a and b have opposite signs, then the sequence is chaotic.

     Arithmetic mean equals (x_N+y_N)<+>(x_N+y_N); harmonic mean equals (x_N<+>y_N)+(x_N<+>y_N); so addition and reduction, without constants, but with recursion, suffice to generate square roots and chaos.