## Tuesday, October 29, 2013

### On Reduction, 7 of 11

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

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)

(xN+1, yN+1) =    ( (xN+yN)/2 , 2(xN<+>yN) )

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