**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.

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