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:
                              a * lcm(b,c,d)      =    lcm(ab,ac,ad)
               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.

Thursday, April 17, 2014

Lattice Rationals, 4 of 10

LCM and compensator for zero


               To define sums for ratios with denominator zero, we need to define gcf, lcm and compensator for zero. Since every number divides into zero, and zero divides into none, it is at the top of the divisibility lattice; therefore zero is an attractor for lcm and an identity for gcf:
                              lcm(a, 0)  =  0                     ;              gcf(a,0)  =    a
               Since (a;b) = lcm(a,b)/b  =  a/gcf(a,b),  it follows that   
(a;0)   =   0/0   =   a/a
The first equation is useless; 0/0 is indefinite; but a/a equals one; so let us take as a rule:
               (a;0)   =   1
Now (0;a) = 0/a   ; this is 0 if a is not zero, indefinite if a=0. So what is (0;0)? If we take the rule that (0;0)=0, then we get the equation:
               (a/0)  +  (c/0)   =   (0/0)
This is the two-denominators result. But if we assume that (0;0) = 1, then:
(a/0)  +  (c/0)   =   (a+c)/0
               This is the one-denominator result, as requested. Therefore in this paper I shall take the rule:
                              (0;a)       =             0             if a does not equal zero; and
                              (0;0)       =             1
               Then in general:
                              (a/0) + (b/c)      =       (a/0)    if c does not equal zero;
                              (a/0) + (b/0)      =       (a+b)/0
               Infinities absorb finite quantities, but add like integers, by adding numerators.




Wednesday, April 16, 2014

Lattice Rationals, 3 of 10



GCF, LCM, compensators and the Euclidean Algorithm           

               Here are some rules uniting addition with gcf, lcm, and the compensator:
                              gcf(a,b)  =  gcf(a-b, b)  =  gcf(a+b, b)  =  gcf(a+nb, b)  for any integer n.
               Define (a mod b) to be the remainder of a when divided by b. (A mod B)  equals A+nB for some n; therefore:
                              gcf(a, b) = gcf( a mod b, b)  =  gcf(a, b mod a)
               This, along with the rule:
                              gcf(a,0)  =  gcf(0,a)  =  a
               implies the Euclidean algorithm. For instance:
                              gcf(52,20)  =  gcf(12,20)  =  gcf(12,8)  =  gcf(4,8)  =  gcf(4,0)  =  4
               Modulation ping-pongs across gcf until resolution. From these rules:
                              Lcm(a,b)   =    a*b/gcf(a,b)
                              (a;b)          =    a / gcf(a,b) 
               we can derive these Euclidean-algorithm-like rules:
                              lcm(a,b)    =     (a/(a mod b)) * lcm(a mod b, b)       =     (b/(b mod a)) * lcm(a, b mod a) 
lcm(ab,a)    =    lcm(a,ab)    =    ab
                              (a;b)            =     (a ; b mod a)   =   (a/(a mod b)) * (a mod b ; b)
                              (a;ab)          =     1
                              (ab;b)          =    a
               Therefore, for instance:
               Lcm(52,20) = (52/12)*lcm(12,20) = (52/12)*(20/8)*lcm(12,8) = (52/12)*(20/8)*(12/4)*lcm(4,8)  =   (52/12)*(20/8)*(12/4)*8  =   260
               (52;20)  =  (52/12)*(12;20) =  (52/12)*(12;8) =  (52/12)*(12/4)*(4;8) =  (52/12)*(12/4)*1  =  13 
               (20;52)  =  (20;12)  =  (20/8)*(8;12) =  (20/8)*(8;4) = (20/8)*2   =  5

Tuesday, April 15, 2014

The Federal Receipt: a Modest Proposal



The Federal Receipt: a Modest Proposal

I interrupt this blogging of “Lattice Rationals” to make a modest proposal.

I propose that there be a Federal Receipt. This Receipt is to be mailed to each taxpayer soon after April 15; detailing, for each taxpayer, both taxes received, and how much of those funds went to which federal program. It would go something like this:

***
Dear Joe Blow:

We got from you:

Income tax:       $ X
Social Secuity:  $ X
other taxes:       $ X

total:                   $ X

We will spend that on:

Interest payments:  $ X
Social Security:          $ X
Medicare:                   $ X
Medicaid:                   $ X
DOD:                           $ X
Veterans benefits:   $ X
... (many more) ...

****

The spending receipt entries will equal total taxes taken, times respective fractions of the federal budget. How far to break it down is, I suppose, a matter of policy. Let's say, just enough detail to cover a side of a page. No doubt the political parties will quarrel over details. The point is to clarify and demystify. Let the general public be more accurately aware of our nation's true budgetary priorities. Such a receipt will, for instance, dispel the popular illusion that NASA or foreign aid are major programs in federal terms.