Tuesday, April 15, 2014

Lattice Rationals, 2 of 10

LCM and compensator for positive numbers

We can define compensator, lcm and gcf from prime factorizations:

lcm( product(pi^ai),  product(pi^bi)  )   =    product(pi^ max(ai, bi)  )
gcf( product(pi^ai),  product(pi^bi)  )   =    product(pi^ min(ai, bi)  )
( product(pi^ai) ;  product(pi^bi)  )   =    product(pi^ max(ai - bi , 0)  )

Compensator, lcm and gcf follow these rules for positive numbers:
lcm(b,d)      =      (d;b)*gcf(b,d)*(b;d)      =      (d;b)*b      =     (b;d)*d
d   =   (d;b)*gcf(b,d)                          :              b   =   (b;d)*gcf(b,d)
gcf(b,b)   =   b          ;          lcm(b,b)   =  b                     ;              (b;b)  =  1
gcf(b,bc)   =   b        ;          lcm(b,bc)   =  bc                 ;              (b;bc)  =  1           ;              (bc;b)  =  c
gcf(1,c)   =   1          ;          lcm(1,c)   =  c                      ;              (1;c)  =  1              ;              (c;1)  =  c

Distribution laws:
gcf(a, lcm(b,c))   =  lcm(gcf(a,b),  gcf(ac))
lcm(a, gcf(b,c))   =  gcf(lcm(a,b),  lcm(ac))
a*lcm(b,c)   =  lcm(a*b,  a*c)
a*gcf(b,c)   =  gcf(a*b,  a*c)

These last two laws are because addition distributes over minimum and maximum, and lcm and gcf are defined by minima and maxima of exponents. For instance:
(2^3 * 3^5) * lcm(2^2 * 3^1,  2^1 * 3^4)
=   (2^3 * 3^5) * 2^max(2,1)  * 3^max(1,4))
=   2 ^ (  3 + max(2,1) )  *  3 ^ ( 5 + max(1,4) )
=   2 ^  max(3+2, 3+1) )  *  3 ^ max(5+1, 5+4)
=  lcm ( 2^(3+2) * 3^(5+1)  ,   2^(3+1) * 3^(5+4)  )
=  lcm (( 2^3 * 3^5)*(2^2  * 3^1)  ,   ( 2^3 * 3^5)*(2^1  * 3^4)   )

Also, the distribution of multiplication over lcm implies Cancellation:
(a*b ; a*d)    =   (b ; d)

Commutativity and associativity:
lcm(a,b) = lcm(b,a)            ;              lcm(a,lcm(b,c))    =    lcm(lcm(a,b),c))
gcf(a,b) = gcf(b,a)              ;              gcf(a,gcf(b,c))    =    gcf(gcf(a,b),c))

These laws, plus the compensator definition of addition, yield “Semicancellation”:
(a*c)/(b*c)                         =                            a/b    +    0/(bc)
I also call this law “casting out zeroids”, where a ‘zeroid’ is a ratio with numerator zero.
This in turn implies “Compensated Reduction”:
a / b                                     =                            (a;b) / (b;a)   +  0/b
Since lcm(b,b) = b   and  (b;b)=1, we recover the one-denominator rule:
(a/b) + (c/b)   =   (a*(b;b) + c*(b;b)) / lcm(b,b)   =   (a+c)/b
But if b and d are relatively prime, then lcm(b,d)=b*d;   (b;d)=b;  and (d;b)=d; so we recover the two-denominators rule:
(a/b) + (c/d)   =   (a*d + c*b) / (b*d)