On Reduction
By Nathaniel S. K. Hellerstein
Adjunct Instructor, City
College of San Francisco
paradoctor@aol.com
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.
Table of Contents
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
Answer Key p.
34
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
Complementarity
and Distribution imply Triads:
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.
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