Monday, October 21, 2013

On Reduction, 1 of 11

On Reduction

By Nathaniel S. K. Hellerstein
Adjunct Instructor, City College of San Francisco

Abstract

In this paper I discuss “reduction”, a.k.a. “reciprocal addition”; addition conjugated by reciprocal. I discuss reduction’s definition, its laws, its graphs, its geometry, its algebra, its calculus, and its practical applications. This paper contains a problem set with answer key.

1.  Definitions and Laws                          p.  2
2.  Geometry of Reduction and Protraction         p.  6
3.  Graphs of Reduction and Protraction           p.  7
4.  Applications of Reduction                     p.  8
5.  Linear and Quadratic Reciprocal Algebra       p. 10
6.  Reciprocal Complex Numbers                    p. 14
7.  When Additions Collide                      p. 16
8.  Projective Reduction                          p. 19
9.  Reciprocal Calculus                           p. 21
10. Logistic                                    p. 26
Problem Set                                     p. 32

Definitions and Laws

Define  a <+> b  =        1     .
1   +   1
a       b

Define  a <-> b  =  a <+> (-b)

<+> is called “reduction”; it is the reciprocal of the sum of the reciprocals. <-> is called “protraction”; it is the reciprocal of the difference of the reciprocals. <+> is also denoted #, or 1/+, and called “reciprocal addition”, “harmonic addition” and “parallel addition”. <-> is also denoted \$, or 1/-, and called “reciprocal subtraction”, “harmonic subtraction” or “parallel subtraction”.
a<+>b is pronounced “a with b”, and a<->b is “a without b”.

The De Morgan Identities:

1         =      1   +   1
a <+> b             a       b

1         =      1   -   1
a <-> b             a        b

1         =      1   <+>   1
a + b              a         b

1         =      1   <->   1
a - b              a         b

These follow directly from the definitions by inversion or by substituting reciprocals. I name them after the DeMorgan logic identities which distribute NOT over AND and OR, with alternation.

They imply the Reciprocal Equations Theorem:

a <+> b  =  c   if and only if     1/a + 1/b  =  1/c
a <-> b  =  c   if and only if    1/a - 1/b  =  1/c
a + b  =  c     if and only if     1/a <+> 1/b  =  1/c
a - b  =  c     if and only if    1/a <-> 1/b  =  1/c

Product Over Sum / Product Over Reverse Difference:

a <+> b  =    ab .
b + a

a <-> b  =    ab .
b - a

These follow directly from the definitions. They imply:

Complementarity:

(a+b)(a<+>b)  =       ab

(a-b)(a<->b)  =     - ab

a + b      =          ab
b <+> a

a - b      =          ab
b <-> a

Distribution:

c(a<+>b)  =  ca <+> cb

Proof:

c(a<+>b) =   cab     =    ccab     =    cacb    =  ca <+> cb
b + a        c(b + a)       cb + ca

a(a+b)  <+>  b(a+b)   =   ab,

a(b-a)   b(b-a)   =   ab,

so if   a+b = c   then   ac<+>bc = ab

and if  b-a = d   then   ad<->bd = ab

Therefore, these sums and differences:

7+3 = 10   ;   9+6 = 15  ;   7-3 = 4  ;  9-6  =  3

yield these reductions and protractions:

70<+>30 = 21; 135<+>90 = 54 ; 12<->28 = 21; 18<->27 = 54

which yield these sums and differences of reciprocals:

1/70 + 1/30  =  1/21  ;  1/135 + 1/90  =   1/54  ;

1/12 – 1/28  =  1/21  ;   1/18 – 1/27  =   1/54

Field Laws:

a <+> 0  =  0

a <+>  =  a

a <-> a  =

a <+> b  =  b <+> a

a <+> (b <+> c)  =  (a <+> b) <+> c

a (b <+> c)  =  ab <+> ac

a <-> a <+> b  =  b

(a <+> b)(c <+> d)  =  ac <+> ad <+> bc <+> bd

a  <+>  c   =    ad <+> bc
b       d            bd

Note that <+> resembles + , except that its identity element is infinity, not zero. <+> and + are mirror images of each other, but not identical. For instance, x<+>x = x/2 ,  whereas  x+x = 2x.

Consider these familiar-looking laws:

Common Denominators:

a  +  b   =    a + b
c     c          c

a  <+>  b   =    a <+> b
c       c           c

General Denominators (or, Rabbit Rule):

a  +  c   =   ad + bc                  \  /
b     d          bd                     \/
/\
a  <+>  c   =   ad <+> bc              /  \
b       d           bd

They have these less-familiar looking counterparts:

Common Numerators:

c  +  c   =       c
a     b        a <+> b

c  <+>  c   =      c
a       b        a + b

General Numerators (or, Stool Rule):

a  +  c   =       ac              ____
b     d       ad <+> bc            \  /
\/
a  <+>  c   =     ac                /\
b       d       ad + bc            /  \

The last four rules are the same as the rules for addition, with the roles of numerator and denominator reversed, and addition alternating with reduction. Fractions stand on their heads when they pass through the reciprocal looking-glass.