Numerical Computations For Operator Axioms

The Operator axioms have produced new real numbers with new operators. New operators naturally produce new equations and thus extend the traditional mathematical models which are selected to describe various scientific rules. So new operators help to describe complex scientific rules which are difficult described by traditional equations and have an enormous application potential. As to the equations including new operators, engineering computation often need the approximate solutions reflecting an intuitive order relation and equivalence relation. However, the order relation and equivalence relation of real numbers are not as intuitive as those of base-b expansions. Thus, this paper introduces numerical computations to approximate all real numbers with base-b expansions.


Introduction
In [1], we distinguished the limit from the infinite sequence. In [2], we defined the Operator axioms to extend the traditional real number system. In [3], we promoted the research in the following areas: 1. We improved on the Operator axioms. 2. We defined the VE function and EV function. For clarity, we rename VE Function [3] to Prefix Function. For clarity, we rename EV Function [3]  and so on. In other words, real operators extend the traditional mathematical models which are selected to describe various scientific rules. Operator axioms have included infinite operators, so no other operator can be added to them. This means the Operator axioms is a complete real number system. In fact, infinite operators imply the completeness.
Thus, real operators exhibit potential value as follows: 1. Real operators can give new equations and inequalities so as to precisely describe the relation of mathematical objects.
2. Real operators can give new equations and inequalities so as to precisely describe the relation of scientific objects.
So real operators help to describe complex scientific rules which are difficult described by traditional equations and have an enormous application potential.
As to the equations including real operators, engineering computation often need the approximate solutions reflecting an intuitive order relation and equivalence relation. Although the order relation and equivalence relation of real numbers are consistent, they are not as intuitive as those of base-b expansions. In practice, it is quicker to determine the order relation and equivalence relation of base-b expansions. So we introduce numerical computations to approximate real numbers with base-b expansions.
The numerical computations we proposed are not the best methods to approximate real numbers with base-b expansions, but the simple methods to approximate real numbers with base-b expansions. The compution complexity of the numerical computations we proposed could be promoted furtherly. However, we first prove that the Operator axioms can run on any modern computer. The numerical computation we proposed blends mathematics and computer science. Modern science depends on both the mathematics and the computer science. Arithmetic is the core of both the mathematics and the science. As a senior arithmetic, the Operator axioms will promote both the mathematics and the science in the future. Theorem 1.1. Any positive number ξ may be expressed as a limit of an infinite base-b expansion sequence 0 ≤ a n < b, not all A and a are 0, and an infinity of the a n are less than (b-1). If ξ ≥ 1, then A 1 ≥ 0.
Proof. Let [ξ] be the integral part of ξ. Then we write where X is an integer and 0 ≤ x < 1, and consider X and x separately.
If X > 0 and b s ≤ x < b s+1 , and A 1 and X 1 are the quotient and remainder when X is divided by b s , then Similarly Thus X may be expressed uniquely in the form where every A is one of 0, 1, · · · , (b-1), and A 1 is not 0. We abbreviate this expression to the ordinary representation of X in base-b expansion notation.
Since a n < b, the series ∞ 1 a n b n (1.8) is convergent; and since g n+1 → 0, its sum is x. We may therefore write x = . a 1 a 2 a 3 · · · , (1.9) the right-hand side being an abbreviation for the series (1.8).
We now combine (1.2), (1.4), and (1.9) in the form . a 1 a 2 a 3 · · · ; (1.10) and the claim follows. Theorem 1.1 implies that every real number has base-b expansions arbitrary close to it. So in numerical computations for the Operator axioms, all operands and outputs are denoted by base-b expansions to intuitively show the order relation and equivalence relation.
The paper is organized as follows. In Section 2, we define the operation order for all operations in the Operator axioms. In Section 3, we construct the numerical computations for binary operations. In Section 4, we define some concepts in the Operator axioms. In general, an operation includes many binary operators. For example, the operation "[[1 + [1 + 1]] − − − −[1 + 1]]" includes three "+" and one "− − −−". Since each operator produces a binary operation, n operator in an operation will produce n binary operations. It is better to compute all binary operations in an operation in order. The order is denoted as Operation Order.
[4, §5.3.1] stores tradition operations as an expression tree and then applies traversal algorithm to evaluate the expression tree. Likewise, each operation of the Operator axioms can be stored as an expression tree in which each number '1' become a leaf node and each operator become an internal node. Then Operation Order is just the traversal order of the expression tree. In this paper, we choose inorder traversal as Operation Order. Figure  1 illustrates an expression tree for the number " It is supposed that the numerical computation applies base-10 expansions. Then the numerical computation for "[[1 + [1 + 1]] − − − −[1 + 1]]" will proceed with the following Operation Order: In summary, every operation in the Operator axioms can divide into many binary operations by Operation Order. So numerical computations focus on the binary operations.

Numerical Computations For Binary Operations
In [3], Operator axioms have expressed the real number system. In this section, "real number" refer to the real number deduced from Operator axioms [3].

Division Of Binary
Operations. According to the complexity of numerical computations, we divide binary operations into low operations, middle operations and high operations. Table 1 lists their elements in detail. Let e be Euler's number. It is supposed that a ∈ −∞, +∞ is a base-b expansion, n ∈ Z and k ∈ N . The numerical computation for [e + + + a] can be constructed with the Taylor-series expansion as follows.
It is supposed that a

Numerical Computations For High Operations.
It is supposed that the constants n, a 1 , It is supposed that the constant e ∈ R with 1 < e. For clarity, we rename VE Function [3] to Prefix Function. For clarity, we rename EV Function [3] to Suffix Function.     It is supposed that the constant t, u, v ∈ R. Brent's method converges faster than the bisection method and thus acts as the main root-finding method for Root Equations.
In the following, we construct all numerical computations for high operations by induction. It is supposed that the constant p 1 ∈ R with 1 ≤ p 1 . It is supposed that the constant p 2 ∈ R. It is supposed that the constant q 1 ∈ R with 1 ≤ q 1 . It is supposed that the constant q 2 ∈ R with [1 − 1] ≤ q 2 . It is supposed that the constant r 1 ∈ R with [1 − 1] < r 1 . It is supposed that the constant r 2 ∈ R with 1 < r 2 . In the following, we approximate these constants with those fractions such as [a 1 − −a 2 ].

Some Concepts In The Operator Axioms
We define some replacements for the notations of the Operator axioms, as is shown in Table 2.
We define the pronunciations for some expressions in the Operator axioms, as is shown in Table 3.
We divide the real operators into an ordered level with the natural numbers. Table 4 lists the levels of the real operators in detail.
The order of real operators is listed as follows: level-1 < level-2 < level-3 < · · · < level-n We define the operations of real operators as follows. According to the Definition 4.1, all operations such as a + n b, a − n b, a/ n b compose the complete operations. · · · · · · n + n , − n , / n