If you have been following these posts, you should understand how we are merging the two groups of operationally interpreted numbers, one infinite under addition, the other finite, under multiplication, to form a union of the two that constitutes a field under multiplication and addition.
The theoretical representations of these two groups are:
- Representation of infinite group: The set of SUDRs, their inverses, the set of TUDRs, and the S|T identity element.
- Representation of finite group: The set of positive S|Ts, their inverses, the set of negative S|Ts, and the S:T identity element
Recall that the S:T unit takes the form
S|T/S|T,
where the vertical pipe symbol “|” denotes the subtraction operation, and the diagonal slash symbol “/” denotes the division operation. We decided to use the colon symbol “:” to represent the division operation, and to abbreviate negative S|Ts as S and positive S|Ts as T. A negative S|T unit is simply one that has more discrete SUDR units than discrete TUDR units, while in a positive S|T unit, the inequality is in the other direction; It has more discrete TUDR units than SUDR units.
The unusual aspect of this is that, while the identity element of the representation of the infinite group is 0, formed by the addition of any element in the group with its inverse, each such pair represents a unique form of the identity element that is a basis for a unique finite group of order |n|-1, where |n| is the quantity of discrete units denoted by the numerator and denominator of the identity element. For example, 1|1, 2|2, 3|3, …, n|n = 1|1 = 0. Thus, we see that there is an infinite set of finite groups of order |n|-1, where n/n = 1/1 = 1 is the identity element of that group.
This is very different from the way integers are normally interpreted, where the addition of -n with its inverse +n results in a single value of the identity element, 0. Yet, with these new reciprocal number (RN) groups, we can see this connection between the two groups, in the operation of their identity elements; that is, by selecting an identity element from the infinite group, we define a finite group of a given order, which is actually contained within the infinite group, as indexed by the unique value of the selected identity element.
To be more precise: Let a be an element of the infinite group paired with b, its inverse element; then, if c = a|b, is selected as the identity element of the finite group of order n, where n = (a -1), then there are (2n + 1) elements in the group. For instance, if a = b = 2, then there are (2(|a|-1) + 1) = 2((2-1)+1) = 2+1 = 3 elements in the finite group, negative .5 and it’s inverse 2 (positive .5), and the identity element itself:
1/2, 2/2, 2/1
If a = b = 9, then there are (2(9-1) + 1) = 17 elements in the group:
1/9, 2/9, 3/9, 4/9, 5/9, 6/9, 7/9, 8/9, 9/9, 9/8, 9/7, 9/6, 9/5, 9/4, 9/3, 9/2, 9/1
However, while the value of c = a|b, and the value of d = b|a, as identity elements of both the infinite group and the corresponding finite group of order n = a-1, are equal under both addition and multiplication, the values of their constituent magnitudes do not have the same sign, one is positive and the other is negative. Hence, another, reciprocal, finite group is formed, when the identity element, selected from the infinite group, is d = b|a. The order n of this group is n = b-1, but to distinguish this group from its inverse group of order n, we denote its order as -n.
On this basis, pairing these inverse finite groups of order n and -n, we form the elements of a field under multiplication and addition, where
c/d * d/c = (c*d)/(d*c) = 0/0 * 0/0 = 0,
which is the identity element of the field (i.e. an element in the union of the infinite and finite groups) under multiplication. On the other hand, the addition operation gives the same result.
c/d + d/c = (c+d)/(d+c) = 0/0 + 0/0 = 0,
which is the identity element of the field under addition.
Hence, as long as a|b = b|a = a-b = 0, then the field identity element is zero, under both multiplication and addition, but if a|b is less than zero, then b|a is necessarily greater than zero, and vice versa. Therefore, in the non-zero case, c and d will always have opposite polarities. Then, in either case where a > b, or a < b,
c/d + d/c = (c+(-d))/((-d)+c) = 0/0 = 0..
In other words, under addition, the field element doesn’t change from 0, regardless of the size (n*1) of a and b, but, on the other hand, under multiplication, the field element does change:
c/d * d/c = (c*(-d))/(-d*(c)) = -(n*1)/-(n*1) = (n*1),
when a > b, and
c/d * d/c = (-c*d)/(d*(-c)) = -(n*1)/-(n*1) = (n*1),
when a < b.
This, then, is the reason that we can use the “meshed gears” analogy to quantify the numerical relationship of the S|T units, at the apexes of the triplets, although it’s now clear that increasing the diameter of the gears, as the ratios change, is not the correct representation of the ratio. There is another way that corresponds to the field values we’ve just been describing. To see this, consider the graphic below, which illustrates how the finite groups of order n and -n are embedded in the infinite group.
Figure 1. Finite Group of Order 1 and Order -1
In the figure above, we can see that selecting -2 and 2, from the infinite group of RNs gives us two finite groups, one generated by the identity element c, where a|b = 2|2, and one generated by the inverse identity element d, where b|a = 2|2. However, the order of the two groups should be 3; that is, the order should be 2(|a|-1)+1 = 2(2-1)+1 = 3, but it isn’t. Instead, we get a finite group of order 11.
Of course, the order has increased because we are using the RNs, not their integer equivalents. In the case of RNs, two out of four units are required to reach a unit value of -2, from 0, but four of these same units are required to reach 2, from -2. In other words, the distance between -2 and 2 is four units, or 4(.5) = 2.
Thus, the total distance from 0 to -2, and from -2 to 2, or vice versa, is 2+4 = 6. Plugging this number into our equation for order gives us
2(6-1) + 1 = 10 + 1 = 11,
the number of elements in the two finite groups shown above. However, this non-intuitive result stems from the mathematical properties of the group, because we have combined the inverse elements of the infinite group to get the identity element of both the infinite group and the corresponding finite group. Moreover, we see that the identity element of the infinite group is 0, which is now understood as the reciprocal of the identity element of finite groups; that is, 0 is the reciprocal of infinity, and discrete is the reciprocal of continuous. This stuff is mind-blowing, but it all proceeds from the symmetry of reciprocity, the most fundamental property of the universe.