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Rational Mathematics

by

James Adrian
Introduction

The terms axiom and postulate are each synonyms of the term assumption. The aim of this effort is to define a mathematics that includes calculus and complex numbers, and does not use assumptions as the foundation of that mathematics.

      Rational Mathematics is a system of scientific definitions in which rational numbers play a dominant role. The number line consists of rational numbers together with processes that produce rational numbers. It does not include transcendental numbers or other irrational numbers except as names for processes. This system contains no concept of an infinitesimal.

      Students in every branch of engineering have been forced to acquire the ability to calculate quantities by means of a mathematics that is based on axioms. Since about 1600, these axioms have been imbued with a decidedly ethereal bent - especially as it affects descriptions of very large and very small quantities.

      As an undergraduate, you probably were not told that infinity must be added as an axiom, if it is to exist in math at all. Infinity can never otherwise be established. This does not mean that derivatives and integrals and complex numbers cannot be defined without axioms.

      As of this writing, there is little recognition of the potential role of unbounded sets. You can easily verify that many websites defining mathematical terms associate unbound sets with infinite sets and fail to make a distinction between them. Many math teachers perpetuate the myth that calculus cannot be developed without infinity, and they do so without proof.

      This system of scientific definitions establishes the foundations of counting and calculating for all of engineering without employing assumptions of any kind.

      A scientific definition must name the term being defined and describe the meaning of that term without reference to the term itself, and without reference to any term not previously established, either as scientifically defined or as an experiential part of the common language.

      Terms like the and thing are undefined, but their meaning is required to be widely agreed in the culture at large. If ambiguity is possible due to multiple meanings, the meaning used for mathematical purposes must be identified. Within an intended context, the term being defined and the description of that term must be interchangeable.

      The experiential terms can be called empirical terms because these terms acquire meaning through experience in the world. The learning process is one of conditioning. Attempts to define empirical terms always results in circularity. A scientific definition is also called a formal definition.

      Formal definitions cannot be the starting point of math because they rely on preexisting terms (words or phrases previously established in the common language). Here are some well-established terms that create our ability to calculate:

      You can tell the difference between a single thing and a pair of things as surely as you can tell night from day or red from blue. Children reason and communicate with such perceptions as soon as they learn which words other people use to refer to them. We all have the ability to perceive a single thing, a pair of things, and many things. The terms amount, quantity, more than, less than, at least, no more than, at most, any, some, collection, and few refer to perceptions, not to formal definitions. These terms are too basic to be defined by more basic terms. These and related meanings are the starting point for mathematical definitions. These terms refer to empirical observations.

      There is a centuries-old reluctance among mathematicians to use the passage of time as an element in mathematical reasoning. However, this development recognizes our experience with time and includes terms such as before and after as empirical terms referring to perceptions. The order of time is thoroughly conspicuous. Events of any nameable kind come to pass in an order, with some events happening before others, and other events coming next in order relative to the event preceding it. Like the terms mentioned earlier, they have meanings that are too basic to be described by more basic terms and their attempted formal definitions, are circular. These terms can be used in formal definitions because they refer to reality in ways that are beyond dispute.

      The same perception of order occurs in our perception of space. Rocks placed in a line give us next and immediately previous or immediately neighboring rocks.

      There are many kinds of order. We often refer to an order of succession, or an order of authority, or an order of importance, and other types of order. Defining mathematical entities as having order, or having an order, or having been ordered, becomes more straight forward if these meanings (that we know so well) are acknowledged.

      The term type has a mathematical use. There are several axiomatic theories of type. A number is the name of an amount or quantity, but a quantity of cash is not the same as a quantity of debt. A quantity of minutes that have passed is not the same as a quantity of minutes spent anticipating an event. A number of miles north is not the same as a number of miles south. Although the term type cannot be assigned a formal definition, various types of things can be scientifically defined.

      When a term is defined, that definition does not imply that there is only a single possible type being defined. Terms like number and set are single-word terms. There is no need to use terms like number type A. Other types are usually named with an adjective preceding the original term, such as ordered set or whole number. Also, any defined entity may come to be known by an additional term to more effectively distinguish it from, or contrast it with, a more recently defined term.

      Throughout this writing, the term thing is intended to be thoroughly indefinite. An idea, an object, a sentence, a mark, an action, a description, a time, a location, or anything else can be named a thing.

      Definition - K is a name if and only if K is a term that refers to and identifies a thing; and, if T is a thing and K is a term that refers to and identifies T, then K might not be the only name for T; and, K may name anything that already has a name.

      For many reasons, we are sparing with alternate names, especially for people, but this definition points to the most general meaning of the term. It does not mean an official name, or a nick name, or any particular type of name. Also, anything may be identified by more than a single name.

      Definition - A term is a word or a phrase.

      Definition - A character is a mark.

      Definition - An element is a thing contained in a collection.

      The undefined term collection is used here in the sense that it may contain a single thing or more things. Here, it is not require to contain at least a pair of things, as the term is sometimes used. The term collection is synonymous with the term collection of things, where things are of any description. Also, a collection may not contain itself. It may be an element in another collection, but it cannot be an element contained in itself.

      Definition - S is a set if and only if S is either a collection, or a named entity or space that does not contain anything; and, every element in S is unique in S; and, S does not contain itself; and, if S does not contain anything, S is said to be empty.

Notation - A set containing a pair of elements is notated this way: {a, b}. Every element in the set except the last element on the right is followed immediately by a comma and then by a space. The elements are enclosed in curly brackets.

      Definition - Set S is a non-empty set if and only if S is a collection.

      Definition - X is a variable of set S if and only if X is a name that may be assigned to any element in set S; and, any such assignment voids any previous such assignment; and, if X names element Y in S, then Y is the element referred to by X.

      A variable is a type of name. The definition of the term name allows things to have more than a single name. This enables a variable to name a thing that already has a name.

      Definition - The value of variable x of set S is the element in S that is currently named by x, and is the element in S that is referred to by x.

      Definition - For elements p and q,   p ← q or q → p   are assignments stating that p now has the value of q.

      Definition - An independent variable is a variable whose variation over time does not depend on any other variable.

      Definition - C is a subset of D if and only if C and D are sets and every element in C is also in D.

      Definition - C is a proper subset of D if and only if C and D are sets; and, every element in C is also in D; and, there is at least a single element in D that is not in C.

      Definition - The union of set T and set S is the set U of elements that are each either in set T or set S, or in both T and S.

      Definition - The intersection of set T and set U is the set S of elements that are each in both set T and set U.

      Definition - An element E is removed from set S if and only if E is in set S, and S is then redefined to exclude E.

      Definition - An element E is inserted in set S if and only if E is not in set S, and S is then redefined to include E.

      Definition - An element E is copied from set S to set T if and only if E is an element in set S; and, E is inserted in set T.

      Definition - An element E is moved from set S to set T if and only if E is copied from set S to set T, and E is then removed from set S.


Order

      There are several terms in the common language that are associated with the concept of order. The concept of order is inherent in matters of time, distance, rank, and others. One need not mention the term order when using terms such as before, after, left, right, next in line or next in time. The order of time is thoroughly evident.

      The term next can mean that something is juxtaposed to something else, or it can have the meaning associated with order in which it means next in order. This is the sense of the term that will be used here, unless explicitly stated otherwise.

      To say that an object striking the floor is an event that is next in order to dropping that object is unambiguous. The term next in order is too basic to be defined by more basic terms.

      Definition - S is an ordered set if and only if each of the following statements is true:

S is a set containing at least a pair of elements.

Every element n in S is either next in order to some element p in S; or, some element q in S is such that q is next in order to n; or, both.

A single element j in S is the only element in S that is not next in order to any other element in S.

A single element k in S is the only element in S such that there is no element in S which is next in order to k in S.

If element y in S is next in order to element x in S, then no element in S other than y is next in order to x.
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      Since an ordered set must contain at least two elements, it may also be called an ordered collection.

Notation - In the case of an ordered set, the curly brackets that are normally used to enclose elements in a set are replaced by parentheses. Unless otherwise specifically required, if y is next in order to x, then y is written to the right of x. Here is an example ordered set: S = (t, u, v, w, x, y, z).

      Definition - R is a reverse-ordered set if and only if R is an ordered set; and, the element in set R that is not next in order to any other element in set R is notated on the right end of the horizontal line of elements written; and, the order of elements written is such that if element h in set R is next in order to element j in set R, then element h is written immediately to the left of element j.

      In 1921, Kazimierz Kuratowski offered the now-conventional definition of the ordered pair (a, b):

(a, b) = {{a}, {a, b}}

      The above equation cannot be used in this system of definitions because the axioms of traditional set theory that give this equation meaning are missing.

      Definition - An ordered pair is an ordered set containing exactly a pair of elements.

      Definition - An unordered set is a set that is not an ordered set.

      Definition - A is the first element of ordered set S if and only if A is the single element in S that is not next in order to any other element in S.

      Definition - Z is the last element of ordered set S if and only if Z is the single element in S such that there is no element in S that is next in order to Z.

      Definition - C is a circular ordered set if and only if C is a set containing at least a pair of elements; and, every element in C is next in order to some other element in C; and, if element y in C is next in order to element x in C, then no element in C other than y is next in order to x.

Notation - A circular ordered set is not enclosed in parentheses alone but rather these: (_a, b, c_). The underline follows the first parenthesis and precedes the second.

      Definition - L is a labeled set if and only if L is an ordered pair containing a circular ordered set C of single-mark numerals, and a variable of C; and; the value of the variable of C in L is an element in C; and C is next in order to the variable of C in L; and, L may not contain any other elements.

Notation - A labeled set is enclosed in parentheses, and the variable of C in L is separated from the defined notation of C by a comma and a space in these ways:

(1, (_0, 1_))

(c, (_a, b, c, d_))

(t, (_s, t, u, v, w_))

(0, (_0, 1, 2_))

(r, (_p, q, r, s, t, u, v_))

      Definition - J is appended to set S if and only if each of the following statements is true:

J is an ordered set of variables; or J is a set containing a single variable.

If set S is empty, then J is inserted into set S.

If set S contains exactly a single element k, then J is inserted into set S as an element that is next in order to k in S, or the elements of ordered set J are inserted in their order where the first element of J is next in order to k in S.

If set S is an ordered set, and k is the element in S such that no element in S is next in order to k, then J is inserted into set S as an element that is next in order to k in S, or the elements of ordered set J are inserted in their order where the first element of J is next in order to k in S.

If set S is a reverse-ordered set, and k is the element in S such that no element in S is next in order to k, then J is inserted into set S as an element that is next in order to k in S, or the elements of ordered set J are inserted in their order where the first element of J is next in order to k in S.
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      Notice that the element appended is not required to come from any particular source; and, the appended element must not be a duplicate of any element already in S; and, an element cannot be appended to an ordered set that has no last element (such as a circular ordered set); and, an element cannot be appended to an element that may not contain any elements other than specified by its definition.

      Definition - S is an ordered succession of marks if and only if each of the following statements is true:

S is a succession of marks consisting of at least a pair of marks written on a horizontal line without spaces or punctuation between the marks of S.

Every mark n in S is either next in order to some mark p in S; or, some mark q in S is such that q is next in order to n; or, both.

A single mark j in S is the only mark in S that is not next in order to any other mark in S.

A single mark k in S is the only mark in S such that there is no mark in S that is next in order to k in S.

If mark y in S is next in order to mark x in S, then no mark in S other than y is next in order to x.
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      Notice that the definition of an ordered succession of marks does not require that a mark in an ordered succession of marks be unique in such a succession. Repeated instances of a given mark are distinguished by the marks that may be next in order to it or those marks to which it is next in order. A succession of marks of marks can be an element in a set, but only if it is unique in that set. Here is an example of a succession of marks: 2011021. Unlike a set of marks, it is not enclosed in other marks and the marks in the succession of marks are not separated by commas or spaces. If two successions of marks are in a set that include identical marks, the combination of marks or their order must be different. 2011021 is different from 2101021.

      Marks are also known as characters.

      Definition - V is a variable of ordered succession of marks J if and only if V is a name that may be assigned to any mark in ordered succession of marks J; and, any such assignment voids any previous such assignment; and, if V names mark y in S, then y is the mark referred to by V.

      Definition - H is a numeral if and only if each of the following statements is true:

S is a set of marks containing only the lower-case alphabet in any language, and the upper-case alphabet in any language, and the marks 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

H is not a set

H is a single mark identical to a single mark in set S, or, H is an ordered succession of marks, each identical to a single mark in set S; and, such marks are written on a horizontal line without spaces or punctuation between the marks of H.
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      The possible multiplicity of marks in a numeral is only to facilitate the use of numerals as names for many possible things and not to establish multi-mark numerals as sets of marks. Elements of a set are notated with each element separated from other elements by a punctuation mark and a space.

Notation - These are examples of sets of numerals:

{101, A0FT, 0, 7G}

(101, A0FT, 0, 7G)

(_101, A0FT, 0, 7G_)

      Definition - L is a digit if and only if each of the following statements is true:

S is a set of marks containing only the lower-case alphabet in any language, and the upper-case alphabet in any language, and the marks 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

L is either a single-mark identical to a single mark in set S; or, L is a single mark of a ordered succession of marks, each identical to a single mark in set S.

If L is a single mark, it is a numeral.

If L is a single mark of an ordered succession of marks, L is not a numeral.

      Definition - Y is a variable of numeral h if and only if Y is a name that may be assigned to any digit in numeral h; and, any such assignment voids any previous such assignment; and, if p names a digit q in numeral h, then q is the digit referred to by p.

      A variable is a type of name. The definition of the term name allows things to have more than one name. This enables a variable to name a thing that already has a name.

      Definition - The value of variable x of numeral h is the digit in numeral h that is currently named by x, and is the digit in numeral h that is referred to by x.

      Definition - For digits p and q,   p ← q or q → p   are assignments stating that p now has the value of q.

      Definition - H is a single-mark numeral if and only if H is a numeral; and, H is a single mark.

      Definition - H is a multi-mark numeral if and only if H is a numeral; and, H is an ordered succession of marks written on a horizontal line without spaces between the marks of H.

      Definition - A numeral name is a name assigned to a numeral in a set containing at least a single numeral.

      Definition - A digit name is a name assigned to a mark of a numeral.

      Definition - Set S a variable set if and only if set S is a set of variables, where each variable in set S has a value which is a digit; and, S contains no other type of element.

      Definition - Numeral j is the numeral of variable set S if and only if each of the following statements is true:

S is a variable set.

If set S contains exactly a single variable, then j is the value of that variable.

If set S contains more than a single variable, then S is an ordered set; and, j is the multi-mark numeral that is the ordered succession of marks formed by the values of the variables in set S taken in the order of those variables in sets S.
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Notation - If Q = (Va, Vb); and, Va = 1, and Vb = 0, then the numeral of set Q is 10.

      In the case of a reverse-order set such as K = ((Vb, (_0,1_)), (Va, (_0,1_))), and Va = 1, and Vb = 0, then the numeral of set Q = 01.

      Definition - E replaces G if and only if each of the following statements is true:

If G is an element in an unordered set U, then G is removed from set U and E is inserted in set U.

If G is an element in ordered set S, and y in S is next in order to G, then y becomes next in order to E, and G is removed from set S.

If G is an element in ordered set S, and G is next in order to y in S, then E becomes next in order to y in S, and G is removed from set S.

If G is a particular instance of a mark in an ordered succession of marks J, and an instance of y in J is next in order to that instance of G, then that instance of y in J becomes next in order to E, and J is redefined to exclude that instance of G.

If G is a particular instance of a mark in an ordered succession of marks J, and that instance of G is next in order to an instance of y in J, then E becomes next in order to that instance of y in J, and J is redefined to exclude that instance of G.
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      Notice that the single mark E is not required to come from any particular source.

      Notice that a single-mark numeral can become a non-numeral digit in a multi-mark numeral. A digit is a numeral only when it is a single-mark numeral and not when it is a digit in a multi-mark numeral.

      Definition - Single mark j prepends single mark k if and only if S is a set; and, k is in S; and, ordered succession of marks jk, replaces k in S.
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      Definition - Single mark j prepends ordered succession of marks K if and only if S is a set; and, K is in S; and, K is an ordered succession of marks in S; and, ordered succession of marks jK replaces K in S.
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Subscripts

      Definition - Kq is a subscripted name if and only if each of the following statements is true:

K is a mark and j is a mark.

K together with q form a name distinct from K and distinct from j; and, this name is pronounce K sub j.

The name formed by K and j is written Kj.

If Kj is a name, K may be said to be subscripted; and, K may be said to be subscripted by j; and, Kj may be said to be a name formed by subscripting; and, Kj may be said to be a subscripted name; and, j is a subscript; and, j is K's subscript.
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      Consider these ordered sets:

(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z)

(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z)

(Α, Β, Γ, Δ, Ε, Ζ, Η, Θ, Ι, Κ, Λ, Μ, Ν, Ξ, Ο, Π, Ρ, Σ, Τ, Υ, Φ, Χ, Ψ, Ω)

(α, β, γ, δ, ε, ζ, η, θ, ι, κ, λ, μ, ν, ξ, ο, π, ρ, σ/ς, τ, υ, φ, χ, ψ, ω)

      Consider this ordered set of subscripted names:

a, ψb, ψc, ψd, ψe, ψf, ψg, ψh, ψi, ψj, ψk, ψl, ψm, ψn, ψo, ψp, ψq, ψr, ψs, ψt, ψu, ψv, ψw, ψx, ψy, ψz)

      Because the alphabet is an ordered set, the set of names subscripted by the alphabet is an ordered set of names, but the alphabet only serves to establish the order of the names. This ordered set of names could be a property list where ψc is the family car and ψw is the washing machine. Further, even if these names are all quantities, the quantities might not be in any particular order. This set could be a record of speeds detected by a roadside machine. The timestamps might be used as subscripts, but the speed quantities would likely not be an ordered set.


Number

      Definition - A number is the name of an amount or a quantity of any type of thing; or, it is the name of the amount or quantity of things represented by a set that is empty.

      Definition - One is the name of the amount or quantity of elements in a set containing exactly a single thing.

      If set S contains a single thing, then set S contains one thing. One is a number because it is the name of an amount or a quantity of some type. The numeral 1 is a name for the number one. If A is a name for B and B is a name for C then A is a name for C.

      Definition - Two is the name of the amount or quantity of elements in a set containing exactly a pair of things.

      If set S contains a pair of things, then set S contains two things. Two is a number. The numeral 2 is a name for the number two.

      Definition - Zero is the name of the amount or quantity of elements in a set that is empty.

      If set S is empty, then set S contains zero things. Zero is a number. The numeral 0 is a name for the number zero.

      Definition - P is a procedure if and only if P is a set of written statements which specify initial conditions, and specify subsequent actions which are to take place. [Explanatory notes or comments are enclosed in brackets. They do not contribute to the functioning of a procedure.]

      Definition - X is equal to Y, or X equals Y if and only if X describes or names an element, collection, set, numeral, number, digit, quantity, quantitative description, ordered succession, procedure, or defined entity, and Y describes or names the same element, collection, set, numeral, number, digit, quantity, quantitative description, procedure, or defined entity.
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      Notice that if A equal B, then B equals A. This is true because they describe or name the same thing. A is equal to B or A equals B may be written A = B. By repeatedly applying the definition, it follows that if A = B and B = C then A = C. If A → B, then thereafter B = A. If x names an element w, then x = w.

      Cuneiform writing in Mesopotamia was primarily devoted to recording amounts of things being traded. Comparisons needed to be made. There is an ancient procedure for comparing the amounts of individually separate items. Originally, containers were used where the otherwise identical procedure below is defined in terms of sets and elements.

      Definition - This is the one-to-one correspondence procedure which is specified by the following statements taken in their written order unless otherwise specified:

A pair of sets, j and k, each contain elements, while another pair of sets, p and q, are empty.

S - Move a single element from set j to set p.

Move a single element from set k to set q.

If either set j or set k is empty, conclude that set p and set q contain the same number of elements, and that set p and set q have a one-to-one correspondence; otherwise, continue by performing statement S.
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      Definition - The number of elements in set S is the name of the quantity or amount of elements in set S.

      Definition - The number of elements in set C is less than the number of elements in set D if and only if each of the following statements is true:

Set A and set C have a one-to-one correspondence.

Set B and set D have a one-to-one correspondence

A is a proper subset of B.
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      Definition - The number of elements in set D is greater than the number of elements in set C if and only if each of the following statements is true:

Set A and set C have a one-to-one correspondence.

Set B and set D have a one-to-one correspondence

A is a proper subset of B.
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      Definition - Number p is less than number q if and only if each of the following statements are true:

S is a set and T is a set.

The number of elements in set S is p; and, the number of elements in set T is q.

S is a proper subset of T
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      Definition - Number q is greater that number p if and only if each of the following statements are true:

S is a set and T is a set.

The number of elements in set S is p; and, the number of elements in set T is q.

S is a proper subset of T
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      Definition - S is an ordered set of numbers if and only if the following statements are true:

S is a set containing at least two numbers.

A single number j in S is the only number in S that is not next in order to any other number in S.

A single number k in S is the only number in S such that there is no number in S that is next in order to k in S.

If number y in S is next in order to number x in S, then no number other than y is next in order to x in S.

If number y in S is next in order to number x in S, then y is greater than x.
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      Definition - Number q is one more than or one greater than number p if and only if each of the following statements is true:

O is a set containing exactly a single element.

S is a set.

The number of elements in S is p.

U is the union of S and O.

The number of elements in U is q.
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Opposite Type

      A pair of sets may contain numbers having opposite meaning. A few examples include meters north versus meters south, seconds that have passed versus seconds that have yet to pass, units of weight loss versus units of weight gain, units of magnetic repulsion versus units of magnetic attraction, and financial debts to others versus financial debts owed to oneself. There are a great many pairs of opposites in the world.

      Definition - Element p and element q are of opposite types if and only if they are of opposite meaning.

      Rather than name the types of individual numbers with terms such a debt owed, weight gained, meters left, etc., one set of the pair is arbitrarily called positive and the other is called negative. Whether one calls seconds of the past as positive or seconds of the future as positive is arbitrary. Either choice can be made. If p is the positive type, then q is the negative type.

      Definition - Set S and set T are opposite sets if and only if each of the following statements is true:

S is an ordered set of numbers; and, T is an ordered set of numbers.

S and T each contain the number zero.

Zero in S is the only number in S such that there is no number in S that is next in order to zero in S.

Zero in T is the only number in T such that it is not next in order to any other number in T.

Each of the elements in S other than zero are of an opposite type from such elements in T.

S is called the negative set; and, T is called the positive set.

The intersection of S and T contains only the single number zero.
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      Definition - D is a dimension if and only if each of the following statements is true:

D is an ordered set of numbers.

Sets S and T are opposite sets; and, S is the negative set.

S is a proper subset of D.

T is a proper subset of D

D is the union of S and T.

Zero is the element in D that is next in order to the last element in S.
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Notation - D = (-Sz, -Sy, -Sx . . . -Sa, zero, TA, TB, TC, . . . . TZ)

      The notation of the negative subset includes a dash on the left side of the name of each element. Also, the use of the alphabet as the ordered set of names of the subscripts is notated in the opposite direction from that of the positive set.

      The notation of set D need not be along a horizontal line. It could be vertical or in any other orientation. It is meant to satisfy the notational needs of a Cartesian Coordinate System without limiting the number of dimensions represented.

      The definition of the term dimension does not restrict the types of opposite elements in S and T; nor does it require than the amounts named by adjacent numbers in these sets differ by a constant amount.

      Each number in a dimension is the name of an amount and, in addition, it has a type.

      Definition - M is the absolute value of N if and only if set D is a dimension; and, N is in D; and, M is of the positive type regardless of the type of N.

      Definition - N is an integer if and only if each of the following statements is true:

D is a dimension containing opposite sets S and T as proper subsets.

S is named the negative set, and T is named the positive set.

If y in S is next in order to x in S, then y is one greater than x.

If w in T is next in order to v in T, then w is one greater than v.

U is the union of S and T.

N is in U.
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      Notice that zero is one greater that the last element of the negative set. Also notice that positive 1 and positive 2 have the same absolute value as the last two elements of the negative set. These may be called negative 1 and negative 2. Every element k in the negative set that is next in order to j in the negative set is one greater than k. The naming of integers of greater absolute value must be addressed after defining ordered sets of named numerals.

      Definition - Element j is incremented in set S if and only if each of the following statements is true:

S is an ordered set of numbers or S is an ordered succession of marks.

If j is a variable of S, and t is in S, and, j = t, and k in S is next in order to t, then variable j is reassigned to k in S.
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      Definition - Element j is decremented if and only if each of the following statements is true:

S is an ordered set of numbers or S is an ordered succession of marks.

If j is a variable of S, and t is in S, and, j = t, and t is next in order to k in S, then variable j is reassigned to k in S.
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Base

      The term integer has been defined. This enables the identification of a number as being an integer, and a set as being a set of integers. To use integers in calculations, individual names for integers must be chosen. There are several naming systems in use. The most popular systems, use the binary numerals {0, 1}, and Hindu-Arabic numerals {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}.

      Definition - A base is an ordered set of numerals.

      The numerals of a base are each unique in that set. Recall that the elements of a set are required to be unique in the set.

      Definition - A binary digit is a digit that is in the set {0, 1}.

      Definition - A binary base is a base such that the digits of its numerals are binary digits.

      This type of base is commonly called base two. Here is an example of a binary base:

(0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001)

      This example has a pattern to the numerals that is not required by the definition of a binary base. This ordered set is also a binary base:

(1001, 100, 1, 10, 111, 0, 101, 110, 11, 1000)

      Computer scientists have shown that, in such sets, there exists more than one useful order of numerals.

      Definition - A decimal base is a base such that the digits of its numerals are Hindu-Arabic numerals.

      This type of base is commonly called base ten. Here is an example of a decimal base:

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21)

      Notice that in the examples of bases that the order of the elements could be randomly scrambled to form a different ordered set, and yet that different ordered set would satisfy the definition of a base of the type named. Nonetheless, each of these bases has a standard order that needs to be defined.



      Definition - Set S is a standard set of binary numerals if and only if set S is defined by the following procedure, which is the described by the following collection of statements performed in their written order unless otherwise indicated. [Set S is also called a binary base.]

Stop = 0 [The value of Stop is determined by a hardware switch.]

Set S = (0, 1) [This is where the numerals accumulate.]

C = (_0,1_)

The mark x is a variable of set S; and, initially, x = 0 in S [The value of x must be a numeral in set S at each time that the variable x is used in this procedure. The variable x os used as a subscript of the variables in set T.]

Y is a variable of set S; and, Y = 0 in S

T is the variable set where each variable in set T is a variable of set C. Initially, T = (V1, V0). T is a reverse-ordered set.

R is a variable set; and, initially R = (W1, W0), each equal to 0; and, R is a reverse-ordered set. [The value of the variables in set R indicate whether a variable in set T having the same subscript has been prepended in set T. 0 indicates not and 1 indicates that it has.]

Start - Append the numeral of variable set T to set S.

If Stop = 1, go to END

Increment V0, a variable of set C.

Star - If Vx = 0, prepend subscript x of Vx in set T with 1 only if Rx is 0; increment x, increment Vx, continue to Star; otherwise x = 0 and continue at Start.

End

      The foregoing procedure produces numerals until physically stopped by the hardware switch. Here is an example output of numerals assigned to set S:

S = (0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010).