[PDF] The Real Numbers and the Integers PRIMITIVE TERMS

947_6axioms_for_R.pdf

The Real Numbers and the Integers

PRIMITIVE TERMS

To avoid circularity, we cannot give every term a rigorous mathematical definition; we have to accept some things as undefined terms. For this course, we will take the following fundamental notions as primitive undefined terms. You already know what these terms mean; but the only facts about them that can be used in proofs are the ones expressed inthe axioms listed below (and any theorems that can be proved from the axioms). •Real number:Intuitively, a real number represents a point on the number line, or a (signed) distance left or right from the origin, or any quantity that has a finite or infinite decimal representation. Real numbers include integers, positive and negative fractions, and irrational numbers like⎷2,π, ande. •Integer:An integer is a whole number (positive, negative, or zero).

•Zero:The number zero is denoted by0.

•One:The number one is denoted by1.

•Addition:The result of adding two real numbersaandbis denoted bya+b, and is called thesum ofaandb . •Multiplication:The result of multiplying two real numbersaandbis denoted byabora·b ora×b, and is called theproduct ofaandb. •Less than:To say thatais less thanb, denoted bya < b, means intuitively thatais to the left ofbon the number line.

DEFINITIONS

In all the definitions below,aandbrepresent arbitrary real numbers. •The numbers2through10are defined by 2 = 1+1, 3 = 2+1, etc. The decimal representations for other numbers are defined by the usual rules of decimal notation: For example, 23 is defined to be 2·10 + 3, etc. •Theadditive inverseornegativeofais the number-athat satisfiesa+ (-a) = 0, and whose existence and uniqueness are guaranteed by Axiom 9. •Thedifference betweenaandb, denoted bya-b, is the real number defined bya-b= a+ (-b), and is said to be obtained bysubtractingbfroma. •Ifa?= 0, themultiplicative inverseorreciprocalofais the numbera-1that satisfies a·a-1= 1, and whose existence and uniqueness are guaranteed by Axiom 10. •Ifb?= 0, thequotient ofaandb, denoted bya/b, is the real number defined bya/b=ab-1, and is said to be obtained bydividingabyb. •A real number is said to berationalif it is equal top/qfor some integerspandqwithq?= 0. •A real number is said to beirrationalif it is not rational. •The statementais less than or equal tob, denoted bya≤b, meansa < bora=b. 1 •The statementais greater thanb, denoted bya > b, meansb < a. •The statementais greater than or equal tob, denoted bya≥b, meansa > bora=b. •A real numberais said to bepositiveifa >0. The set of all positive real numbers is denoted byR+, and the set of all positive integers byZ+.

•A real numberais said to benegativeifa <0.

•A real numberais said to benonnegativeifa≥0. •A real numberais said to benonpositiveifa≤0. •Ifaandbare two distinct real numbers, a real numbercis said to bebetweenaandbif eithera < c < bora > c > b. •For any real numbera, theabsolute value ofa, denoted by|a|, is defined by |a|=? aifa≥0, -aifa <0. •Ifais a real number andnis a positive integer, thenth power ofa, denoted byan, is the product ofnfactors ofa. Thesquare ofais the numbera2=a·a. •Ifais a nonnegative real number, thesquare root ofa, denoted by⎷ a, is the unique nonnegative real number whose square isa(see Theorem 9 below). •Ifnandkare integers, we say thatnisdivisible bykif there is an integermsuch that n=km. •An integernis said to beevenif it is divisible by 2, andoddif not. •IfSis a set of real numbers, a real numberbis said to be amaximum ofSor alargest element ofSifbis an element ofSand, in addition,b≥xwheneverxis any element ofS. The termsminimumandsmallest elementare defined similarly. •IfSis a set of real numbers, a real numberb(not necessarily inS) is said to be anupper bound forSifb≥xfor everyxinS. It is said to be aleast upper bound forSif every other upper boundb?forSsatisfiesb?≥b. The termslower boundandgreatest lower boundare defined similarly.

PROPERTIES OF EQUALITY

In modern mathematics, the relation "equals" can be used between any two "mathematical objects" of the same type, such as numbers, matrices, ordered pairs, sets, functions, etc. To say thata=b is simply to say that the symbolsaandbrepresent the very same object. Thus the concept of "equality" really belongs to mathematical logic rather than to any particular branch of mathematics. Equality always has the following fundamental properties,no matter what kinds of objects it is applied to. In the following statements,a,b, andccan represent any mathematical objects whatsoever. (In our applications, they will usually be realnumbers.) 2

General Properties of Equality

1.(Reflexivity)a=a.

2.(Symmetry)Ifa=b, thenb=a.

3.(Transitivity)Ifa=bandb=c, thena=c.

4.(Substitution)Ifa=b, thenbmay be substituted forain any mathematical statement

without affecting that statement"s truth value. In addition, for real numbers, we have the following properties. The first five statements say roughly that if you start with a true equation between two real numbers, you can "do the same thing to both sides" and still have a true equation. The last two say that if you start with two true equations, you will still have a true equation after adding them together, multiplying them together, subtracting one from the other, or dividing one bythe other (provided you are not dividing by zero). All of these statements can be proved using only reflexivity of equality and substitution. In these statements,a,b,c,drepresent arbitrary real numbers.

Properties of Equality of Real Numbers

1. Ifa=b, thena+c=b+c,ac=bc, anda-c=b-c.

2. Ifa=bandcis nonzero, thena/c=b/c.

3. Ifa=b, then-a=-b.

4. Ifa=b, andaandbare both nonzero, thena-1=b-1.

5. Ifa=b, thena2=b2.

6. Ifa=bandc=d, thena+c=b+d,ac=bd, anda-c=b-d.

7. Ifa=bandc=d, andcanddare both nonzero, thena/c=b/d.

AXIOMS FOR THE REAL NUMBERS AND INTEGERS

We assume that the following statements are true.

1.(Existence)There exists a setRconsisting of all real numbers. It contains a subsetZ?R

consisting of all integers.

2.(Closure ofZ)Ifaandbare integers, then so area+bandab.

3.(Closure ofR)Ifaandbare real numbers, then so area+bandab.

4.(Commutativity)a+b=b+aandab=bafor all real numbersaandb.

5.(Associativity)(a+b) +c=a+ (b+c) and (ab)c=a(bc) for all real numbersa,b, andc.

6.(Distributivity)a(b+c) =ab+acand (a+b)c=ac+bcfor all real numbersa,b, andc.

7.(Zero)0 is an integer that satisfiesa+ 0 =a= 0 +afor every real numbera.

8.(One)1 is an integer that is not equal to zero and satisfiesa·1 =a= 1·afor every real

numbera.

9.(Additive inverses)Ifais any real number, there is a unique real number-asuch that

a+ (-a) = 0. Ifais an integer, then so is-a.

10.(Multiplicative inverses)Ifais any nonzero real number, there is a unique real number

a -1such thata·a-1= 1.

11.(Trichotomy law)Ifaandbare real numbers, then one and only one of the following

three statements is true:a < b,a=b, ora > b.

12.(Closure ofR+)Ifaandbare positive real numbers, then so area+bandab.

13.(Addition law for inequalities)Ifa,b, andcare real numbers anda < b, thena+c <

b+c.

14.(The well ordering axiom)Every nonempty set of positive integers contains a smallest

integer. 3

15.(The least upper bound axiom)Every nonempty set of real numbers that has an upper

bound has a least upper bound.

SELECTED THEOREMS

These theorems can be proved from the axioms in the order listed below.

1.Properties of zero

(a)a-a= 0. (b) 0-a=-a. (c) 0·a= 0. (d) Ifab= 0, thena= 0 orb= 0.

2.Properties of signs

(a)-0 = 0. (b)-(-a) =a. (c) (-a)b=-(ab) =a(-b). (d) (-a)(-b) =ab. (e)-a= (-1)a.

3.More distributive properties

(a)-(a+b) = (-a) + (-b) =-a-b. (b)-(a-b) =b-a. (c)-(-a-b) =a+b. (d)a+a= 2a. (e)a(b-c) =ab-ac= (b-c)a. (f) (a+b)(c+d) =ac+ad+bc+bd. (g) (a+b)(c-d) =ac-ad+bc-bd= (c-d)(a+b). (h) (a-b)(c-d) =ac-ad-bc+bd.

4.Properties of inverses

(a) Ifais nonzero, then so isa-1. (b) 1 -1= 1. (c) (a-1)-1=aifais nonzero. (d) (-a)-1=-(a-1) ifais nonzero. (e) (ab)-1=a-1b-1ifaandbare nonzero. (f) (a/b)-1=b/aifaandbare nonzero.

5.Properties of quotients

(a)a/1 =a. (b) 1/a=a-1ifais nonzero. (c)a/a= 1 ifais nonzero. (d) (a/b)(c/d) = (ac)/(bd) ifbanddare nonzero. (e) (a/b)/(c/d) = (ad)/(bc) ifb,c, anddare nonzero. (f) (ac)/(bc) =a/bifbandcare nonzero. (g)a(b/c) = (ab)/cifcis nonzero. (h) (ab)/b=aifbis nonzero. (i) (-a)/b=-(a/b) =a/(-b) ifbis nonzero. (j) (-a)/(-b) =a/bifbis nonzero. (k)a/b+c/d= (ad+bc)/(bd) ifbanddare nonzero. (l)a/b-c/d= (ad-bc)/(bd) ifbanddare nonzero. 4

6.Transitivity of inequalities

(a) Ifa < bandb < c, thena < c. (b) Ifa≤bandb < c, thena < c. (c) Ifa < bandb≤c, thena < c. (d) Ifa≤bandb≤c, thena≤c.

7.Other Properties of inequalities

(a) Ifa≤bandb≤a, thena=b. (b) Ifa < b, then-a >-b. (c) 0<1. (d) Ifa >0, thena-1>0. (e) Ifa <0, thena-1<0. (f) Ifa < bandaandbare both positive, thena-1> b-1. (g) Ifa < bandc < d, thena+c < b+d. (h) Ifa≤bandc < d, thena+c < b+d. (i) Ifa≤bandc≤d, thena+c≤b+d. (j) Ifa < bandc >0, thenac < bc. (k) Ifa < bandc <0, thenac > bc. (l) Ifa≤bandc >0, thenac≤bc. (m) Ifa≤bandc <0, thenac≥bc. (n) Ifa < bandc < d, anda,b,c,dare nonnegative, thenac < bd. (o) Ifa≤bandc≤d, anda,b,c,dare nonnegative, thenac≤bd. (p)ab >0 if and only ifaandbare both positive or both negative. (q)ab <0 if and only if one is positive and the other is negative. (r) There is no smallest positive real number. (s)(Density)Ifaandbare two distinct real numbers, then there are infinitely many rational numbers and infinitely many irrational numbers betweenaandb.

8.Properties of squares

(a) For everya,a2≥0. (b)a2= 0 if and only ifa= 0. (c)a2>0 if and only ifa >0. (d) (-a)2=a2. (e) (a-1)2= 1/a2. (f) Ifa2=b2, thena=±b. (g) Ifa < bandaandbare both nonnegative, thena2< b2. (h) Ifa < bandaandbare both negative, thena2> b2.

9.Properties of Square Roots

(a) Ifais any nonnegative real number, there is a unique nonnegative real number⎷ asuch that?⎷ a?2=a. (b) Ifa=bandaandbare both nonnegative, then⎷ a=⎷b. (c) Ifa < bandaandbare both nonnegative, then⎷ a <⎷b. (d) Ifa2=bandbis nonnegative, thena=±⎷ b.

10.Properties of Absolute Values

(a) Ifais any real number, then|a| ≥0. (b)|a|= 0 if and only ifa= 0. (c)|a|>0 if and only ifa?= 0. (d)| -a|=|a|. (e)|a|=⎷ a2. 5 (f)|a|= max{a,-a}. (g)|a-1|= 1/|a|ifa?= 0. (h)|ab|=|a||b|. (i)|a/b|=|a|/|b|ifb?= 0. (j)|a|=|b|if and only ifa=±b. (k) Ifaandbare both nonnegative, then|a| ≥ |b|if and only ifa≥b. (l) Ifaandbare both negative, then|a| ≥ |b|if and only ifa≤b. (m)(The triangle inequality)|a+b| ≤ |a|+|b|. (n)(The reverse triangle inequality)??|a| - |b|??≤ |a-b|.

11.Order properties of integers

(a) 1 is the smallest positive integer. (b) Ifmandnare integers such thatm > n, thenm≥n+ 1. (c) There is no largest or smallest integer.

12.Properties of Even and Odd IntegersIn each of the following statements,mandnare assumed to be integers.

(a)nis even if and only ifn= 2kfor some integerk, and odd if and only ifn= 2k+ 1 for some integerk. (b)m+nis even if and only ifmandnare both odd or both even. (c)m+nis odd if and only if one of the summands is even and the other isodd. (d)mnis even if and only ifmornis even. (e)mnis odd if and only ifmandnare both odd. (f)n2is even if and only ifnis even, and odd if and only ifnis odd.

13.Properties of Exponents

In these statements,mandnare positive integers.

(a)anbn= (ab)n. (b)am+n=ambn. (c) (am)n=amn. (d)an/bn= (a/b)nifbis nonzero. 6