By Nathaniel Hellerstein
This paper redefines the addition of rational numbers, in a way that allows division by zero. This requires defining a “compensator” on the integers, plus extending least-common-multiple (LCM) to zero and negative numbers. “Compensated addition” defines ordinary addition on all ratios, including the ‘infinities’ n/0, and also ‘zeroids’ 0/n. The infinities and the zeroids form two ‘double ringlets’. The lattice rationals modulo the zeroids yields the infinities plus the ‘wheel numbers’. Due to the presence of the ‘alternator’ @ = 0/-1, double-distribution does not apply, but triple-distribution still does.
Table of contents:
1. Adding infinities and Compensated Addition p. 2
2. LCM and compensator for positive numbers p. 3
3. GCF, LCM, compensators and the Euclidean Algorithm p. 5
4. LCM and compensator for zero p. 6
5. LCM and compensator for negative numbers and ratios p. 7
6. LCM and compensator: review of definitions and laws p. 9
7. Lattice rational arithmetic; definitions and laws p. 11
8. Double ringlets of zeroids and infinities p. 13
9. The Wheel Numbers p. 14
10. Lattice Rationals modulo zeroids and infinities p. 15
Adding infinities and Compensated Addition
This paper began with my desire to add infinities. In particular I wanted the following to be valid:
2/0 + 3/0 = 5/0
This is consistent with the one-denominator rule for adding fractions:
a/b + c/b = (a+c)/b
but not with the two-denominators rule:
a/b + c/d = (ad+bc)/(bd)
for then 2/0 + 3/0 = 0/0, the indefinite ratio.
How to harmonize the two rules? Well, how about finding a rule that covers both cases?
Consider the following addition:
5/12 + 7/18 = (5*3 + 7*2)/36 = 29/36
Where did that 36 come from? It is the lowest common multiple of 12 and 18: lcm(12,18)=36. But where did the 3 and the 2 come from? I call these ‘compensating factors’ or ‘compensators’. They compensate for the new denominator:
5/12 = (5*3)/36 : 7/18 = (7*2)/36
Therefore let us define the “compensator of b to d”, a.k.a. “b;d”, thus:
b;d = lcm(b,d) / d = b / gcf(b,d)
where gcf is greatest common factor. Then we have the rule:
a/b + c/d = ( a*(d;b) + c*(b;d) ) / lcm(b,d)
This is the Compensated Addition Rule.