The Larson Research Center

We’ve been discussing the two groups of operationally interpreted (OI) reciprocal numbers. The group under addition is isomorphic to the group of integers, under addition, and the group under multiplication is isomorphic to the group of nonzero rational numbers, under multiplication.

The binary operations of these two groups is actually subtraction, symbolized by the pipe symbol, “|”, and division, symbolized by the usual slash symbol, “/”. As the properties of these numbers differs substantially from those of the integers and rational numbers, we need to designate them some way that will help us refer to them uniquely.

The easiest way to to do this, although probably not the best way, is to resort to acronyms. Accordingly, we will refer to the difference interpretation of reciprocal numbers as DRNs, and the quotient interpretation as QRNs.

The two, different, OI magnitudes of the hypotenuse of the right triangle, with sides a = b = 1, are shown in figure 1 below.

Figure 1. The Geometry of QRN and DRN. Two Operational Interpretations of a Unit Reciprocal Number

While the zero evaluation of the hypotenuse might seem strange at first, when it clearly has a definite length, this difficulty can be overcome, when it is noted that, here, the number c is not necessarily a measure of the length, but of the relative angles, of the hypotenuse. The difference value obtained is tantamount to measuring the difference between the two acute angles of the right triangle. When these angles are the same, we may say that their difference is 0, just as legitimately as we may say that they are identical.

Taking the unit triangle, and its two OI values, as the genesis of a discrete number system, there are two, and only two, unit increments possible. These are:

1. 1/2 = .5, or 1|2 = -1
2. 2/1 = 2, or 2|1 = 1

In the DRN value, the unit increments are linear (-1, 1), while in the QRN value, they are not (.5, 2), even though the change in geometry is identical in both cases, as shown in figure 2 below:

Figure 2. The Two Possible Unit Increments of the Unit Triangle

It’s easy to see that a set of DRNs, consisting of all combinations of 1|2 = -1 and 2|1 = 1, is a representation of a group under addition, isomorphic to the set of integers, which is a group under addition: Let D be the set of DRNs, n|n, 1|n, n|1, where n is a positive whole number. The subset of 1|n = {…, -4, -3, -2, -1, 0}, and the subset of n|1 = (0, 1, 2, 3, 4, …}. Let the symbol “+” indicate the operation of addition. Then (D,+) is a group. The proof is:

Closure : If a and b are DRNs then a + b is a DRN.

Associativity : If a, b, and c are DRNs, then (a + b) + c = a + (b + c).

Identity element : 0 is a DRN and for any DRN a, 0 + a = a + 0 = a.

Inverse elements : If a is a DRN, then the inverse DRN, −a, satisfies the inverse rules: a + (−a) = (−a) + a = 0.

In the case of QRNs, consisting of all combinations of 1/2 = .5 and 2/1 = 2, these form a set that is a representation of a group under multiplication, isomorphic to the set of nonzero rationals, which is a group under multiplication: Let Q be the set of QRNs, n/n, 1/n, n/1, where n is a positive whole number. The subset of 1/n = {…, .2, .25, .33…, .5, 1), and the subset of n/1 = {1, 2, 3, 4, …}. Then (Q,*) is a group. The proof is:

Closure : If a and b are QRNs then ab is a QRN.

Associativity : If a, b, and c are QRNs, then (ab)c = a(b c).

Identity element : 1 is a QRNI and for any QRN a, 1 * a = a * 1 = a.

Inverse elements : If a is a QRN, then the inverse QRN, a, satisfies the inverse rules: aa = aa = 1.

One of the many interesting aspects of this development is that the group of QRNs contains an infinite number of finite subgroups of order (2n)-1, where n/n is the identity element of the group. As figure 2 above shows, this is akin to dividing the unit circle into parts, from 0 to 1 and from -0 to 1.  As pointed out in the previous post, if we regard 1 as π radians of rotation, then the finite QRN groups are tantamount to a subset of rotations of size n. In essence, we are dividing the unit circle into clockwise and counterclockwise rotations.

For example, when n = 2, the identity element is 2/2 = 1, and the other elements of the subgroup are 1/2 = .5 and 2/1 = 2, which are actually two, equipollent, intervals between -0 and 1, and 0 and 1, respectively. This is what enables the rotation representation, as clockwise and counterclockwise rotations from 0 to π. However, to recognize this point, it is necessary to understand how .5 and 2 can constitute -0 and 0, respectively, when the denominator, or the numerator, of the identity QRN, 2/2, is reduced from 2 to 1; that is, we don’t increment one side of the initial unit triangle from 1 to 2, as in figure 2, but we start with a two-unit triangle and reduce one side to 1 unit.

In the RST, this is a consequence of “direction” reversals, because there are two half-cycles in each cycle of oscillation.  However, when we just deal with numbers, we are accustomed to adding one discrete number to another, since there is no physical constraint to prevent this.  Yet, in the RST, we can’t just add a unit of time, or a unit of space, to the unit progression, to obtain ds/dt = 1/2, or 2/1, because units of space and units of time are not independent entities, but they are nothing more than the reciprocal aspects of the same entity.

Thus, the only way ds/dt can change from 1/1 to 1/2 or 2/1, physically, is for the progression “direction” reversals to take place over two cycles, effectively reducing one aspect, or the other, from 2 increases every two units, to one increase every two units. Of course, from the number perspective of physical magnitudes, it makes no difference how we get to 1/2, or 2/1, these values are simply possibilities, but from the progression perspective, we arrive at the conclusion that only 2/2 has these possibilities, intuitively. 

Indeed, from the progression perspective, we see that a decrease offsets an increase, so the only way to get to the 1/2, or the 2/1, space|time progression ratio, in the RST, also effectively halts the increase in the reversing aspect.  Like a soldier marching “in place,” the “direction” reversals effectively reduce the space, or time, progression from unit progression, to zero progression, in the reversing aspect.  This is clearly demonstrated in the progression algorithms and the world line charts, here and here.

However, within additional units of progression, it’s possible to divide the pie into additional pieces, as it were.  Thus, when ds/dt = 4/4, there are twice as many units of space and time available to form units of displacement, or elements of the subgroup:

1/4, 2/4, 3/4, 4/4, 4/3, 4/2, 4/1

Therefore, with n/n, we obtain n-1 negative and positive elements, and the identity element is still 1, because n/n is always equal to 1. However, to meet the closure requirement of the group, we have to include x dimensions of elements.  I’m sure the mathematicians will recognize this and have a name for the notion, but with my limited knowledge of groups, the only way that I can describe it is with the notion of x dimensions.

On this basis, for example, (1/4)*(1/4) = 1/16, or 1/4², which is an element of the group, so that the group elements are n⁰/n⁰, n¹/m^x, and m^x/n¹, where n and m are positive whole numbers, m > n, and x = 1 -> ∞. This follows from the fact that the denominator in the negative elements of the subgroup, and the numerator in the positive elements of the subgroup, will always be factors of m. I’m not sure what this does to the order of the group, or whether this means that the subgroups are infinite after all.

I’ve yet to contemplate the consequences of all this, or even to validate it completely.  I may have to stand corrected, at some point, but right now it seems that this approach gives us the intuitive route to the ad hoc formalism used in quantum mechanics.