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.