The Extension of Number Set Driven by Inverse Operations

Abstract:

The reviewing of number set extension has developed a new, to be proven statement: The （none zero）multicomponent vectors, m-vectors, are numbers. The m-vectors one by one corresponds to the dots in multidimensional variable space, m-space. The m-vectors have the four operations: addition, subtraction, multiplication, and division. These operations satisfy the three laws: communitive law, distributive law, and associative law. The m-vectors are the extension of scalars, and the scalars are a special case of m-vectors: scalar (i=1) ó m-Vectors (i=m). The m-vectors formed a multiplication Group, and can be used for m-systems analysis.

Key Words: number set, inverse, m-vectors, m-systems, R^n, R^m

By reviewing the history of number set development, we see that initially people recognized the natural number, N:1,2,3, .... The first operation they discovered was addition: 1+1=2. With the result of the addition, in order to find the origin (1+X=2), the ancient people found the inverse of subtraction: 2-1=1. Subtraction leads to a great leap in the understanding of the numbers that was recognition of the negative numbers (and 0): 1-2=-1. People's knowledge of number set also extended from the natural number N to the integer Z. Integers included 0, positive integers (N), and negative integers (-N). While they used a number axis to represent the set, the axis was like bamboo, and the integers and the sections of the bamboo one by one corresponded.

On the basis of repeated additions, people defined the operation of multiplication. That was from 2+2=4, they defined 2*2=4. With the result of multiplication (product), in order to find the origin (2*X=4), they defined the inverse of division: 4/2=2.

Division leads to another great leap in the understanding of the numbers that was the recognition of fractions: 1/2=0.5. The number set was therefore extended from integers to fractions, and the number set was extended to rational numbers and was noted as Q. The rational numbers consist of 0, positive and negative rational numbers. The rational numbers were finite decimals or cyclic decimals[ ]. Each rational number had a corresponding dot on the axis, but there were still infinite many dots on the axis, no corresponding rational numbers.

On the basis of repeated multiplications, people found the operation of power, or exponentiation: 2*2=2^2=4. With the result of the power operation (exponentiation), in order to find the origin, X^2=4, they defined the inverse of radical: 4^ (1/2) = 2. The radical caused another great leap on understanding of irrational numbers. The number set also extended to real numbers. The real number set R included zero, rational numbers, and irrational numbers. Irrational numbers were infinite decimal numbers, and there were positive irrational numbers and negative irrational numbers. Thus, all the dots representing the real numbers spread on the axis, and the dots and the real numbers one by one corresponded through the whole axis. They considered the “numbers” were the coordinates of the “dots” on the axis, and the “dots” were the projections of “numbers” on the axis. The set of numbers developed to this stage, the dots on the axis were very dense, almost filling the entire axis. Each irrational number had a corresponding dot on the axis, but there were still infinite many dots on the axis no corresponding irrational numbers.

The dots on the line were one dimensional, so in order to correspond to the following discussion on vectors, we also called the real number of a single component a “scalar”. The dots (projections) and scalars (numbers) on the axis one by one corresponded. All real numbers (scalars) formed a number axis.

From here, the extension of the number set came to a bifurcation: complex plane and (real) plane. In order to make algebraic equations, for example X^2 +1=0, soluble,  people introduced "imaginary numbers." The set of numbers extended from one axis to the complex plane. Moreover, in this direction, they have developed the Quaternion,  and defined i^2=j^2=k^2=ijk=-1. But the Quaternion gave up the community law, jk=-kj≠kj. Therefore, in this direction (imaginary numbers), the development of the number set ended at Quaternion.

Another direction of the development of the number set was that not to engage imaginary numbers, nor the complex plane, but to extend to a real 2-dimensional plane, and further 3-dimensional, ... to numerical-dimensional space, n-space. By solving numeric variate equations, the projection of numbers extended from one-dimensional line to two-dimensional plane, three-dimensional space, ..., and extended to an abstract n-space. The combination of ordered n real numbers was called "array", or n-array vectors, which was a dot in n-dimensional space. Thus, the dot on the number axis corresponded to a real number one by one, the dot on the 2-plane corresponded to a 2-tuple array (2-vector) one by one, the dot in 3-dimensional space corresponded to a 3-vectors one by one, …, and the dot in n-space corresponded to a n-vectors one by one.

At this stage, the n-space, called “linear space” or “sample space”, was noted as R^n. In this n-dimensional ‘linear space ', only the addition (subtraction) was defined: “The components of the sum are the sum of the corresponding components” (and, the components of the difference are the differences of corresponding components). But there was no strict definition of vector multiplication, nor vector division. Following the matrix, a dot product and cross product (inner product and outer product) were defined, and with great success on applications. However, dot product and cross product were not closed, that is, the product did not belong to the same set, not in the same n-space anymore. Thus, no inverse existed. Like the complex plane, the extension of the number set seemed to go into a dead end, again. Although, someone developed a so called “Hadamard product”, component-wise multiplication which was restricted for matrix use only. It did not have one to one projection in n-space, so it was not considered as a number extension.

However, the world is made of multi-dimensional systems, instead of one-dimensional variables. In order to understand systems and describe the dynamics of systems, people paralleled the addition to define the multiplication of vectors as ' the components of the product are the products of the corresponding components. [ ] '. So defined vectors multiplication does satisfy the commutative law, the associative law, and the distributive law. Furthermore, such defined multiplication has an Identity element: An m-vectors is the Identity element of multiplication that of all m components are 1. Also, there exist Inverse elements: If the components of two none-zero vectors were reciprocal to each other, then the two vectors were mutually reversed. Thus, the multiplication defined in this way has an inverse: division. And the division was also closed to the set. Thus, the vectors product and vectors quotient were still in the same set, same m-space. People noted this multicomponent vectors, with multiplication and division, as m-vectors. This multidimensional variable space on which they defined multiplication and division was noted as R^m, and the R^m was promoted to Group, called “m-Vectors Multiplication Group” or “m-Vectors Exchange Group”.

In this m-Vectors Multiplication Group, in R^m, m-vectors had addition and subtraction, multiplication and division, as well as power operations. Because the operations were closed, the results of these operations were uniquely determined and corresponded to a dot in the m-space. The “numbers” and the “dots” have reached a new harmony in m-space. They called a “m-vectors” a set of coordinates of a dot in “m-space”, and a dot in “m-space” a projection of an “m-vectors” in the m-variable space.

In this way, driven by inverse operations, the knowledge of the number set has been extended to achieve several major leaps: from positive to negative, from integer to fractional, from rational to irrational, from scalars to m-vectors; the number set extended from dots to line, from line to plane, from plane to 3-dimensional space, …, to n-sample space or m-variable space. This knowledge revealed that the numbers and their operation results corresponded to the dots one by one in the m-space.  At the same time, people's understanding and analysis of the objective world has also risen from "variable analysis" to "systematic analysis".

One of the most direct applications of m-Vectors Multiplication Groups is the solution of the ' State Transition ' of the m-systems. That is, a state of m-systems can be expressed by an m-vectors, and the succession of the m-systems can be described by the trajectory of the dot in the m-space and expressed by the operation of the m-vectors:

If Y_(k)*T_(k)=Y_(k+1)，

Then,

T_(k)=Y_(k+1)/Y_(k).

Where both Y_(k)，T_(k) are m-vectors, and the subscript k represents the time; Y_(k)  indicates the state of the system at time k; T_(k) represents the state transfer of the m-systems at time k.

In other words, the state Transition, T_(k), is the quotient of the present over previous state vectors, Y_(k+1)/Y_(k), of the m-systems.

An m-systems described by m-vectors is the most natural and sustainable systems. The components in these m-systems are independent of each other. So, it would be the basic systems, and the most worthy of our research. We are calling this basic systems "simple giant systems" for the time being. The reason we call it "simple" is because of their independency among the components; and the reason we call it "giant" is because of its huge component number. Most (if not all) of the natural systems, such as grassland, stock market, human society, …, can be considered as "simple giant systems", because of their resource competing features. These natural m-systems can run with a minimum energy input.

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