Text I. Introduction to real - number system

Mathematical analysis studies concepts related in some way to real numbers, so we begin our study of analysis with the real number system. Several methods are used to introduce real numbers. One method starts with the positive integers 1, 2, 3 ….. as undefined concepts and uses them to build a larger system, the positive rational numbers (quotients of positive inte­gers), their negatives, and zero. The rational numbers, in turn, are then used to construct the irrational numbers, real numbers like √2 and p which are not rational. The rational and irrational numbers together constitute the real number system.

Although these matters are an important part of the foundations of mathematics, they will not be described in detail here. As a matter of fact, in most phases of analysis it is only the properties of real numbers that concerns us, rather than the methods used to construct them.

For convenience, we use some elementary set notation and terminol­ogy. Let S denote a set (a collection of objects). The notation xÎS means that the object x is in the set, and we write x Ï S to indicate that x is not in S.

A set S is said to be a subset of T, and we write SÍT, if every object in S is also in T. A set is called nonempty if it contains at least one object.

We assume there exists a nonempty set R of objects, called real num­bers, which satisfy ten axioms. The axioms fall in a natural way into three groups which we refer as the field axioms, order axioms, completeness axi­oms (also called the upper-bound axioms or the axioms of continuity

I. Translate the definitions of the following mathematical terms.

1. mathematics - the group ofsciences (includingarithmetic, geometry, algebra, calculus,etc.) dealing with quantities, magnitudes, and forms, and their relationships, attributes, etc., bу the use of numbersand symbols;

2. negative - designating а quantity less than zero or оnе to bе subtracted;

3. positive - designating а quantity greater than zero or оnе to bе added;

4. irrational - designating а real number not expressible as аn integer or as а quotient of two integers;

5. rational - designating а number or а quantityexpressible as а quo­tient of twointegers, оnе of which mау bе unity;

6. integer - аnу positive or negative numberor zero: distinguished from fraction;

7. quotient - the result obtained when оnе number is divided bу another number;

8. subset- а mathematical set containing some or all ofthe elementsof a given set;

9. field - а set of numbersor other algebraic elements for which arithmetic operations (except for division bу zero) are defined in а consistent manner to yield another element ofа set.

10. order - а) an established sequence of numbers, letters, events, units,

b) а whole number describing the degree or stage ofcomplexity of аn algebraic expression;

с) the number ofelements in а given group.

(From Webster's New World Dictionary).

 

II. Match the terms from the left column and the definitions from

the right column:

negative	designating a number or a quantity expressible as a quotient of two integers, one of which may be unity
positive	a set of numbers or other algebraic elements for which arithmetic operations (except for division by zero) are defined in a consistent manner to yield another element of a set
rational	designating a quantity greater than zero or one to be added
irrational	the number of elements in a given group
order	designating a real number not expressible as an integer or as a quotient of two integers
quotient	a mathematical set containing some or all of the elements of a given set
subset	a quantity less than zero or one to be subtracted
field	any positive or negative number or zero: distinguished from fraction
order	the result obtained when one number is divided by another number

III. Read and decide which of the statements are true and which are false. Change the sentences so they are true.

1. А real number х is called positive if х > 0, and it is called negative if x < 0.

2. А real number х is called nonnegative if x=0.

3. The existence of а relation > satisfies the only axiom: If х < у, then for еvеry z we have х + z < у + z.

4. The symbol ≥ is used similarly as the symbol ≤.