Friday, April 18, 2014

Lattice Rationals, 5 of 10

LCM and compensator for negative numbers and ratios

For simplicity’s sake, and to resemble the 2-denominator rule as much as possible, I propose that  lcm is an odd function, and gcf is an even function:

lcm(-a,b) = lcm(a,-b)   =  - lcm(a,b)
gcf(-a,b) = gcf(a,-b)   = gcf(a,b)
so
lcm(-a,-b) =  lcm(a,b)      ;     gcf(-a,-b)  =  gcf(a,b)
and  as before:
lcm(a,b)*gcf(a,b)  =   a*b

We no longer have double-distribution of multiplication over lcm:
-1 * lcm(2,3)  =  -6      but     lcm(-2,-3)  =  6
But we still have triple-distribution:
and lcm and gcf still double-distribute over each other.
We still have:
lcm(ab,ac)    =   |a|*lcm(b,c)
gcf(ab,ac)    =   |a|*gcf(b,c)
a * lcm(b,c)    =   sign(a)*lcm(ab,ac)

We also have “Alternation”:
lcm(a,a)        =    gcf(a,a)        =     |a|
lcm(a,|a|)    =    gcf(a,|a|)    =       a
The compensator is odd in the first variable and even in the second:
(-a;b)    =    - (a;b)      for a not equal to zero.
(a;-b)    =      (a;b)
All this, plus the Compensated Addition Rule, implies:
(a/-b)  +  (c/-d)      =     (-a/b)  +  (-c/d)
In particular, consider 0/-1; call it the “alternator” @. Then:
(a/-b)   + @       =      (-a/b)
(a/b)   +  @       =      (-a/-b)
Note that 0 and @ form a modulo-2 group under both addition and multiplication:
0+0      =    @+@    =    0*0      =    @*@    =    0  ;
0+@    =    @+0      =    0*@    =    @*0     =     @
Multiplication by @ does not double-distribute over addition;
@ *(0+0)    =   @*0   =   @  ;     but   @*0 + @*0    =   @+@   =   0
But it does triple-distribute:
@*(a+b+c)    =    @*a  +  @*b  +  @*c

Define lcm, gcf and compensator of ratios as ratios:
Lcm(a/A,  b/B)     =   lcm(a,b) / lcm(A,B)
gcf(a/A,  b/B)     =   gcf(a,b) / gcf(A,B)
(a/A ;  b/B)     =   (a;b) / (A;B)
We get many of the same rules as above; for instance lcm*gcf = product.