The Larson Research Center

In The New Math Blog, we’ve been discussing the discovery of how the discrete and continuous faces of nature are two, reciprocal, aspects of the same reciprocal number (RN). For example, the 1|2 RN has a value of -1, but the 1/2 RN has a value of 0.5, and which numerator-denominator relation we select, inverse multiplication, represented by the slash symbol, “/”, or inverse addition, represented by the pipe symbol, “|”, depends on whether we are dealing with discrete numbers, or continuous numbers.

We discovered that the essence of nature’s concept of discrete and continuous values comes down to one of interpretation, like the changing image in that famous picture that is transformed in our perception, from a young girl, to an old woman, simply by how we associate the lines and colors of the same image in our minds. It’s not that one image is more valid than the other. It’s that they are two aspects of the same thing, just as space and time are two aspects of motion. You cannot have one without the other. They are the opposite sides of the same coin.

In applying this principle to our new physics, we run into exactly the same thing. As a function, the numerical representation of the reciprocal expansion-contraction of the SUDR and TUDR combo, can be interpreted in either of two ways. It can be viewed as a discrete number, where

S|T = 1|1 = 0,

or it can be interpreted as a continuous number, where

S/T = 1/1 = 1.

As a discrete number, we are using 0 as our datum in a discrete reference system, where the numerical set of RNs is interpreted as

-n, …-1, 0, 1, ,,,n

which set is the set of positive and negative integers that is a group under addition. As a continuous number, on the other hand, we are using 1 as our datum in a continuous reference system, where the numerical set of the same RNs is interpreted as

-.n, …-.5, 1, 2, …n

which set is the set of non-zero rationals that is conventionally regarded as a field under addition and multiplication.

When we first started using RNs, as numerical representations of discrete units of motion, it was clear that we could add and subtract them, but it wasn’t clear at all how we could multiply and divide them, or even when we would want to do so. It wasn’t until we had to describe the SUDR and TUDR connections, in the fermion triplets, that we found ourselves confronted with the need to multiply and divide, because the gear ratios in a gear train are multiplied together, not added together.

However, there is more to this situation than meets the eye, because it’s not clear why the discrete numbers, called integers, should be a group, under addition, while the continuous numbers, called non-zero rationals, are a field, under both addition and multiplication. If they are indeed two, reciprocal, aspects of the same thing, why aren’t their mathematical properties reciprocally related? In other words, we seem to have lost the symmetry of reciprocity, when we consider the conventional mathematical properties of the set of RNs, as two, reciprocal, aspects of the same number.

Happily, we seem to have found the answer, and perhaps discovered along the way that the reason why this has happened goes right to the heart of the difference between the RST, as a system of physical theory, based on scalars, and the LST, as system of physical theory, based on vectors. The symmetry in the mathematical properties of discrete and continuous numbers is preserved, when one realizes that the datum, and the associated reference system, of the continuous number, is the inverse of the datum, and the associated reference system, of the discrete number.

We can easily see this when we diagram the two, reciprocal, reference systems:

Reference system of discrete RNs:     -∞ <——————-> 0 <———————> +∞
Reference system of continuous RNs: -0 <——————-> 1 <———————> +0

Clearly, one is the reciprocal of the other. The problem is that, historically, the role of reciprocity and the dual nature of RNs, has not been appropriately understood. Rational numbers, as fractions of a whole, were not recognized as having inverses, because the inverses were not interpreted as inverse fractions, but as whole numbers, in the number line. However, when we recognize that 1/2 is the inverse of 2/1 and that they have exactly the same absolute value relative to the continuous reference system, then the broken symmetry that arises in the conventional mathematical properties, is restored.

Specifically, it turns out that the discrete numbers are a group under addition, while the continuous numbers are a group under multiplication, but the group properties of continuous numbers can only be clearly recognized when we employ the RNs. Moreover, only when we join the two groups together, do we get a field of numbers under addition and multiplication! That is to say, we can add and multiply fractions and integers like we normally do, precisely because, not having the insight of the duality of RNs, we use the two groups of numbers as a field, not realizing the underlying principles involved.

To get around this, mathematicians use the idea of a ring, which is a set of integers under addition and multiplication, which is not fully a field, but almost. Essentially, this is possible in the conventional understanding of integers because they are formed in a non-reciprocal manner, such that negative numbers are defined as the mirror reflection of positive numbers: -1 is -1/1, -2 is -2/1, and so on, ad infinitum.

Nevertheless, when we employ the RNs, we see that the set of integers constitutes a group under addition, and the set of rationals constitutes a group under multiplication. Then, combining the two sets, forms a field under addition and multiplication, without the need for an ad hoc invention, such as a ring.

For example, normally we multiply .5 times 2 to get 1, but we must interpret the 2 as a discrete number and the .5 as a portion of a discrete number, or a continuous number. If we don’t do this, and instead take .5 as -.5 and 2 as +.5, which are their values, in the continuous reference system, then multiplying them would result in -.25, and confusion would be the result. In other words, we have to interpret one multiplicand as the “old woman,” and the other as the reciprocal “young woman,” in order to get consistent results, even when we don’t recognize explicitly that we are using the two, reciprocal, aspects of number, in the multiplication operation!

The same implicit use of the two groups as a field happens under the sum operation. If we add .5 and 2, we normally get 2.5, but we don’t explicitly recognize that the sum operation of the field requires one summand to be a discrete number and the other to be a continuous number. As a result, we are adding a continuous number to a discrete number, without realizing that, in doing so, we are operating in the field of the two, reciprocal, groups. However, If we stay within the group, and we add two continuous numbers, we would be adding -.5 and +.5 and getting 0, instead of 2.5, for the sum of .5 and 2!

Also, you probably noticed that 0 is not the identity element of the continuous group . The datum of the continuous reference system is 1, not 0, so the sum of (-.n) + (.n) = 0 doesn’t yield the identity element of the continuous group, which is to be expected, because the continuous group is not a group under addition, but only under multiplication, when the numerical set is the set of RNs.

Another important point is that, unlike the unit of the elements in the discrete group, which are multiples of the fixed discrete unit, |1|, the unit of the elements of the continuous group is not fixed, but it is variable, |1/n|. Consequently, the continuous group is a finite group of order n, where n is the number of elements in the group; that is, there is an infinite number of orders of this group, but each order has a finite number of elements, meaning that we can divide up a whole into as many parts as we want, ad infinitum.

Of course, there is much more to say about all this, but to discover that the principle of reciprocity leads us to the unification of the discrete and continuous aspects of nature, unified in the concept of numerical magnitude that is so analogous to the concept of physical magnitude, leaves us breathless.