**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.