Dale G. Jungst

THE SYSTEM OF WHOLE NUMBERS

This chapter discusses the system of whole numbers. The concept of number develops from the repeated associations with collections of real objects, that is, collections of blocks, stones, and fingers. Every set that is an element of a given class is called a representative of that class or is said to represent the class. If equals are multiplied to equals, the results are equal. A whole number, x , is divisible by two only if the number named by the ones digit in the Hindu–Arabic numeral for x is divisible by two. A whole number, p , is prime only if it has exactly two factors. There is not a largest prime whole number; that is, the set of prime whole numbers is infinite. A pair of whole numbers can have many common multiples. A method for computing the greatest common factor of a pair of whole numbers that involves repeated application of the division algorithm is called the Euclidean Algorithm.

THE SYSTEM OF RATIONAL NUMBERS

This chapter discusses the system of rational numbers. Every equation of the form a + x = b , where a and b are elements of the system and x is a variable whose domain is that system, has a nonempty solution set. Every element of the system has an inverse in regard to addition. Construction of the rational-number system employs the system of integers exactly as the construction of the fractional numbers employed the system of whole numbers. Every rational number has an inverse in regard to addition and every nonzero rational number has an inverse in regard to multiplication; this is the property that distinguishes the system of rational numbers from the other number systems. The system of integers is isomorphic to a subsystem of the rationals, that is, the integers are embedded in the rational numbers as the whole numbers are embedded in the fractional numbers.

THE SYSTEM OF INTEGERS

This chapter discusses the system of integers and describes the need for a number system that is more extensive than the whole-number system and different from the fractional numbers. The system of whole numbers can be used in the construction of the system of integers. The construction of the integers parallels to some extent the construction of the whole numbers, and more so, the construction of the fractional numbers. A similar construction can be used to construct a more extensive number system than the integers. Every element of an equivalence class is called a representative of that class. If an integer and its inverse in regard to addition are added, the result is the identity in regard to addition. An important function on the set of integers, called the absolute value function, maps every integer onto a nonnegative integer.

THE SYSTEM OF FRACTIONAL NUMBERS

This chapter discusses the system of fractional numbers. A distinguished-partition is identified by, or associated with, an ordered pair of whole numbers. To determine the multiplicative inverse of a given nonzero fractional number, the fractional number is determined, such that the product of it and the given fractional number is the identity for multiplication. The sum or product of two fractional numbers can be determined by choosing any element or representative of each of the fractional numbers, and then determining the sum or product of these representatives. Fractions are used to determine sums and products of fractional numbers in a way that is analogous to the way finite sets are used to determine sums and products of whole numbers. The chapter describes an isomorphism between the system of whole numbers and a subsystem of the fractional numbers. The system of whole numbers is isomorphic to a subsystem of the system of fractional numbers.

SETS, RELATIONS, FUNCTIONS, AND OPERATIONS

This chapter discusses sets, relations, functions, and operations. The modern approach to the study of mathematics is to concentrate on mathematics as a system or structure, or as a collection of substructures. Four essential components are identified in a mathematical or logical system or structure; they are undefined terms, definitions, postulates, and theorems. The role of postulates is fundamental in a mathematical system. Postulates are assumptions that provide the foundation for the structure. The role and the use of postulates are not unique to mathematics. A one-to-one correspondence between two sets A and B is a pairing of elements of A and B, such that with every element of A exactly one element of B is paired, and with every element of B, exactly one element of A is paired. If two sets are equal, they are equivalent. Equivalent sets need not be equal. A relation is a set of ordered pairs. The set of first components is called the domain of the relation. The set of second components is called the range.

DECIMAL NUMERALS, INTRODUCTION: REAL NUMBERS

This chapter discusses decimal numerals and real numbers, and describes the construction of a number system that is more extensive than the system of rational numbers. The set of fractional numbers that can be named with terminating decimals is a subset of the set of fractional numbers. The number of decimal places in the decimal numeral of the product of two numbers is the sum of the number of decimal places in the decimal numerals for the two factors. The sum, difference, and product of two numbers, named by terminating decimals, is also a number that has a terminating decimal name. To every fractional number, there corresponds a unique repeating decimal. There is a one-to-one correspondence between the set of fractional numbers and a set of repeating decimal numerals. The set of irrational numbers can be characterized as the set of numbers that can be named by nonrepeating decimals.

ALGORITHMS FOR COMPUTATION WITH WHOLE NUMBERS

This chapter discusses algorithms for computation with whole numbers. In the Hindu–Arabic numeration system, each digit in a numeral indicates a particular number of ones, tens, and hundreds; therefore, the fundamental idea of computational algorithms, that use the properties of this numeration system, is to determine in an efficient manner the total number of ones, tens, and hundreds. The modern Roman numeration system employs the subtractive principle. The Hindu–Arabic system employs the positional and additive principles. The chapter describes how the computation algorithms depend on the properties of the Hindu–Arabic numeration system and on the properties of the whole-number system.

AN INTRODUCTION TO LOGIC AND MATHEMATICAL REASONING

Publisher Summary This chapter discusses logic and mathematical reasoning. A variable is a symbol that holds a place for any member of a specified set. The specified set of permissible replacements for a variable is called its domain or replacement set. An open sentence is a sentence that contains one or more variables and becomes a statement when each of the variables is replaced by a member of its domain. Logical connectives can be used to form new statements from given statements. A statement variable is a variable whose domain is a set of statements. A statement pattern is a symbolic expression containing one or more statement variables and one or more of the symbols such that whenever the variables are replaced by statements and a replacement is made for the symbols, the result is a statement. The resulting statement is called an instance of the pattern.

10 – ABSTRACT SYSTEMS

Publisher Summary This chapter focuses on abstract systems. For all sets A, every function f on A is a transformation on A, only if f is one to one from A onto A. The set of transformations on a set has the following properties in regard to the operation composition of transformations; there is an identity, every element has an inverse, and the operation is not commutative. For every set S, the transformation i on S, that maps every element to itself is called the identity transformation on S. The set of all transformations on a set S forms a group in regard to the operation of composition of transformations. A ring is a mathematical system that consists of a nonempty set Q on which two binary operations are defined.

4 – NUMERATION SYSTEMS

Publisher Summary This chapter discusses numeration systems. A numeration system is a scheme in which basic symbols are combined, in a systematic way to name numbers. To have a numeration system, at least two symbols are needed. A numeration system uses the additive principle if the number named by a combination of basic symbols is determined by finding the sum of the numbers named by each of the basic symbols in the combination. A numeration system uses the positional principle if the number named by a basic symbol depends on its position in a combination of basic symbols. A numeration system employs the multiplicative principle if some method exists whereby the number named by a basic symbol is multiplied by some number. A numeration system employs the subtractive principle if it contains a method, whereby the number named by a combination of symbols is determined by subtracting the number named by one symbol from that named by another.