Lattice rational arithmetic; definitions and laws
Define
addition and subtraction by the compensator rule:
a/b + c/d =
( a*(d;b) + c*(b;d) ) /
lcm(b,d)
a/b - c/d =
( a*(d;b) – c*(b;d) ) /
lcm(b,d)
Define
multiplication, reciprocal and division the usual ways:
(a/b)*(c/d) =
(a*c) / (b*d)
1/(c/d) =
(d/c)
(a/b)/(c/d) =
(a*d)/(b*d)
From
these we can define “reduction”, a.k.a.
“reciprocal addition”:
(a/b) 1/+ (c/d) = 1
/ ( (1/(a/b) )
+ (1/(c/d)) )
= lcm(a,c) / (
b*(a;d) + c*(d;a) )
Reduction
is like addition, with the roles of numerator and denominator reversed.
Therefore they follow similar rules. The following are provable:
(a/A) + (b/B) +
(c/C) = ( a (lcm(B,C);A) + b (lcm(C,A);B) + c
(lcm(A,B);C) ) / lcm(A,B,C)
(a/A) 1/+ (b/B)1/+ (c/C) =
lcm(a,b,c) / ( A (lcm(b,c);a) + B (lcm(c,a);b) + C
(lcm(a,b);c) )
Proof requires
these lemmas:
(A;B)*(C;
lcm(A,B)) = (lcm(A,C) ; B)
(
lcm(db, dc) ; da )
= ( lcm(b,c) ;
a )
Addition and reduction are
commutative.
Proof by symmetry
of definitions.
Addition and reduction are
associative.
Another proof of
symmetry of definitions.
Multiplication triple-distributes
over addition and reduction.
Proof
involves triple-distribution of multiplication over lcm.
Division distributes from the
left, and anti-distributes from the right: (trivial proof)
(A+B+C)
/ D = (A/D)
+ (B/D) + (C/D)
(A 1/+ B 1/+C ) / D
= (A/D)
1/+ (B/D) 1/+ (C/D)
A / (B+C+D) = (A/B)
1/+ (A/C) 1/+ (A/D)
A / (B 1/+ C 1/+ D) = (A/B) +
(A/C) + (A/D)
Identities (trivial proof):
(a/b) = (a/b) + (0/1) =
(a/b) 1/+
(1/0) = (a/b) * (1/1)
Near-Inverses:
(a/b)
+ (-a/b) = 0/b
(a/b) 1/+ (a/-b) =
a/0
(a/b)
* (b/a) = (ab)/(ab)
= 1 +
0/(ab)
x/x = 1
+ 0/x
0/-1 is
known as the “alternator” @; its
reciprocal is -1/0, negative infinity.
a/b + @
= -a/-b =
a/b 1/+ - 1/0
Because
of them we must weaken distribution to triple
distribution:
w(x+y+z) = wx +
wy + wz if w is finite
w(x
1/+ y 1/+ z) = wx 1/+
wy 1/+ wz if w is nonzeroid
w(x+y) = wx +
wy + 0w if
w is finite
w(x
1/+ y
) = wx 1/+ wy 1/+ w/0 if w is nonzeroid
0x
* 0y = 0xy
0x
+ 0y = 0(x+y)
= 0(lcm(x,y))
0/(nm) =
0/n + 0/(nm)
(nm)/0 =
n/0 1/+ (nm)/0
0/0 is “indefinity”, the indefinite ratio. It is
an attractor for addition, reduction, multiplication and division for all
finite nonzeroids:
a/b +
0/0 =
a/b 1/+ 0/0 = a/b * 0/0
= (a/b)/(0/0) =
(0/0)/(a/b) = 0/0
if
a and b are both not zero.
Indefinity
acts like a generic finite nonzeroid,
even though it is both an infinity and a zeroid!
0/0 is
an identity for adding infinities, and an attractor for reducing infinities:
a/0 + 0/0 =
a/b
a/0 +
-a/0 = 0/0
a/0 1/+ 0/0 =
0/0
0/0 is
an identity for reducing zeroids, and an attractor for adding zeroids:
0/a 1/+ 0/0 = a/b
0/a 1/+
0/-a = 0/0
0/a + 0/0 = 0/0
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