The Larson Research Center

In the previous post, I discussed the combination of 1/2 and 2/1 as combining two, equal, numbers with opposite “directions.” Since, in ordinary arithmetic, the sum of these two numbers is taken as the sum of a fraction and a whole number that can’t be equal to one another, by definition, this seems strange.

However, as soon as it’s understood that the interpretation of the reciprocal numbers is not a quantitative interpretation, but an operational interpretation, and that there are two operations that can be found, things begin to clear up. The first operational interpretation is the ordinary interpretation of division of whole numbers. Thus, under this interpretation, 1/2 = .5 and 2/1 = 2, but, under the second interpretation, 1/2 = -1 and 2/1 = +1, which shows us that, under the first interpretation of division, 2/1 = 2 is actually +.5, the inverse of 1/2 = -.5, when we take reciprocal “directions” into account. 

Of course, it’s true that we can’t ignore the difference that the “direction” of a reciprocal number makes in the relative value, because 2 (+.5 in disguise, we might say) is four times greater than -.5. For example, if we divide +.5 by -.5 on a calculator, we get

.5/-.5 = -1,

not 4. Hence, we have to recognize that +.5 is the operational interpretation of 2/1, but that

1/2 * 2/1 = 2/2 = 1/1 = 1,

just as

.5 * 2 = 1.

In other words, In using reciprocal numbers, it’s best not to interpret the value of the reciprocal relation, until the arithmetic is completed, in order to avoid error and confusion. Indeed, to help eliminate confusion, as much as possible, we use a different symbol, the pipe symbol, to indicate the reciprocity of the number, when it is to be interpreted under addition,

1|2 + 2|1 = 3|3 = 1|1 = 0 (i.e. -1 + 1 = 0).

When the reciprocal relation is to be interpreted under multiplication, we use the customary symbol, the slash symbol, to indicate the reciprocity of the number,

1/2 * 2/1 = 2/2 = 1/1 = 1 (i.e. .5 * 2 = 1).

It has been suggested that we need a different symbol for the sum operation to avoid the confusion with ordinary arithmetic, where

1/2 + 2/1 = .5 + 2 = 2.5

and

-1 * 1 = -1.

However, it’s clear that this is not necessary, if we understand that the addition operation is always used with the reciprocity indicated by the pipe symbol, and multiplication operation is always used with the reciprocity indicated by the slash symbol.  In both cases, as long as numerators are combined with numerators, and denominators with denominators, under the appropriate binary operation for the indicated reciprocity, no confusion results. 

What about combined operations? For example, what is the meaning of

(1|2)/(2|1) = -1/1 = -1, or (1/2)|(2/1) = (-.5)|(2) = -1.5?

This is problematic, because “direction” is only defined in the reciprocal relation of whole numbers, not in the numbers themselves. Since the numerators and denominators are inverses of each other, the operations should yield the appropriate identities of the respective groups (i.e. 1 and 0).  However, if we do what we have always done in the ordinary arithmetic of fractions, invert and multiply the denominator, we get, for the slash reciprocity,

(1|2)/(2|1) = (1|2) * (1|2) = (-1) * (-1) = 1.

If we do the same thing for the piped reciprocity; that is, if we invert the denominator and the operation, which means we invert and add, instead of subtract, we have to recognize that the inversion doesn’t change the “direction” of the denominator,

(1/2)|(2/1) = (1/2) + (1/2) = (-.5) + (+.5) = 0,

which yields the respective identities in each case, as required. 

I won’t be able to go into this level of detail in the presentation, certainly, but if the question comes up, I’ll be prepared with the answer: The “directions” of numbers are conserved in the sum and multiplication (subtraction and division) operations of reciprocal numbers. Once that is established, I will proceed to show how we find the third property of numbers, multiple dimensions, clarifying the difference between the powers of a number, as multiple factors, and the dimensions of a number, as independent sets of reciprocal, or dual, “directions.”

That will be in the next post for sure this time.