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Course: MATH-H-DC03 (Real Analysis)Module - 1: The Real NumbersZafar Iqbal
Assistant Professor
Department of Mathematics
Kaliyaganj College
Kaliyaganj, Uttar Dinajpur
West Bengal - 733 129, INDIA
Email ID: zafariqbalmath@yahoo.com
WhatsApp No.: +91 9563681339
Zafar IqbalModule - 1: The Real Numbers
Development of Real Numbers
We introduce the theme in a little bit way how the real numbers are developed. To do so we start with:Zafar IqbalModule - 1: The Real Numbers
Natural Numbers
The Set of Natural Numbers
The natural numbers are the counting numbers 1;2;3;4;5;:::We denote the set of all natural numbers byN. Thus, we have N:=f1;2;3;4;5;:::g:Peano Axioms or Peano PostulatesThe setNhas the following self-evident fundamental properties:112N.2n2N=)its successorn+ 12N.31 is not the successor of any element inN.4n;m2Nhaving the same successor =)n=m.5ANsuch that 12A, andn+ 12Awhenevern2A=)A=N.Remark.The Peano axiom 5 is the basis of the mathematical induction.Zafar IqbalModule - 1: The Real Numbers
Principle of Mathematical Induction
For anyn2N, letP(n) be a mathematical statement or a mathematicalproposition which, in general, may or may not be true. If1P(n) is true forn= 1, i.e.,P(1) is true, and2P(n) is true forn=m+1 whenever it is true forn=m, i.e.,P(m+1) is true
wheneverP(m) is true,thenP(n) if true for alln2N.Algebraic Structures onNThe algebraic operations of addition `+' and multiplication `' are binary
operations on the setN, i.e., the mapsAddition (+) :NN!N
(m;n)7!m+n;Multiplication () :NN!N
(m;n)7!mn are well-dened.Zafar IqbalModule - 1: The Real Numbers
Algebraic Structures onNThe algebraic operations of addition `+' and multiplication `' on the setNhave
the following algebraic properties:1Associativity of Addition:a+ (b+c) = (a+b) +cfor alla;b;c2N.2Commutativity of Addition:a+b=b+afor alla;b2N.3Associativity of Multiplication:a(bc) = (ab)cfor alla;b;c2N.4Commutativity of Multiplication:ab=bafor alla;b2N.5Existence of Multiplicative Identity:There exists an elemen t1 2Nsuch that
a1 =a= 1afor alla2N.6Distributivity of Multiplication over Addition:a(b+c) =ab+acfor all a;b;c2N.The setNequipped with the algebraic operation addition `+' forms acommutativesemi-group, i.e., (N;+) is acommutative semi-group.The setNequipped with the algebraic operation multiplication `' forms a
commutative monoid, i.e., (N;) is acommutative monoid.Zafar IqbalModule - 1: The Real Numbers Order Structure onNLet us dene a binary relation `' the setNby abif and only if eithera=borais less thanb for alla;b2N. Then the binary relation `' onNhas the following properties:1Reexivity:F oral la2N,aa.2Anti-Symmetry:F oral la;b2N,abandba=)a=b.3Transitivity:F orall a;b;c2N,abandbc=)ac.4Connexity:F orall a;b2N,aborba.For the rst three properties, the relation `' is apartial orderonN, and (N;) is
apartially ordered set.For all the four properties, the relation `' is atotal orderor alinear orderonN,
and (N;) is atotally ordered setor alinearly ordered set.The setNis well-ordered, i.e., every non-empty subset ofNhas a least element.Zafar IqbalModule - 1: The Real Numbers
Integers
The algebraic operations of addition `+' and multiplication `' are well-dened on the setN. But if we introduce the algebraic operation of subtraction `' onN, then it becomes inadequate as the dierencemnof two natural numbersmand nmay not be a natural number. For instance, ifm= 3 andn= 5, then their dierence 35 is not a natural number. Therefore, the algebraic operations of subtraction `' is not dened onN. To make it well-dened we need to enlarge the setNby including the numbers of the formn,n2N, called the negatives of natural numbers and the number 0. The natural numbers together with their negatives and the number 0 are called integers.The Set of Integers We denote the set of all integers byZ. Thus, we have Z:=f0;1;2;3;4;5;:::g:Zafar IqbalModule - 1: The Real NumbersAlgebraic Structures onZThe algebraic operations of addition `+' and multiplication `' dened on the setZ
have the following algebraic properties:1Associativity of Addition:a+ (b+c) = (a+b) +cfor alla;b;c2Z.2Commutativity of Addition:a+b=b+afor alla;b2Z.3Existence of Additive Identity:There exists an elemen t0 2Zsuch that
a+ 0 =a= 0 +afor alla2Z.4Existence of Additive Inverse:F oreac ha2Z, there exists an elementa2Zsuch thata+ (a) = 0 = (a) +a.5Associativity of Multiplication:a(bc) = (ab)cfor alla;b;c2Z.6Commutativity of Multiplication:ab=bafor alla;b2Z.7Existence of Multiplicative Identity:There exists an elemen t1 2Zsuch that
a1 =a= 1afor alla2Z.8Distributivity of Multiplication over Addition:a(b+c) =ab+acfor all a;b;c2Z.Zafar IqbalModule - 1: The Real Numbers The setZequipped with the algebraic operation addition `+' forms acommutativegroup, i.e., (Z;+) is acommutative group.The setZequipped with the algebraic operation multiplication `' forms a
commutative monoid, i.e., (Z;) is acommutative monoid.Thus, on the setZ, the algebraic operation of subtraction `' is dened as the
inverse of the algebraic operation of addition `+'. The dierence of integers is dened by ab:=a+ (b)for alla;b2Z.Order Structure onZAs it is onN, the relation `' is atotal orderor alinear orderonZ. Therefore,
(Z;) is atotally orderedor alinearly ordered set. Here is a particular case of the property of connexity, viz., for alla2Z, exactly one ofa <0,a= 0, ora >0 is true. This property is called theTrichotomy Property.Zafar IqbalModule - 1: The Real NumbersRational Numbers
Now, all the three algebraic operations of addition `+', subtraction `' and multiplication `' are well-dened on the setZ. But if we introduce the algebraic operation of division `' onZ, then it also becomes inadequate as the quotientmn of two integersmandnmay not be an integer. For instance, ifm= 3 andn= 5, then their quotient mn is not an integer. So, we need to enlarge the setZby including the numbers of the form mn ,m;n2Z. Then the enlarged set would be well enough to dene division as an inverse operation of multiplication. But there arises a very much technical and logical error, viz., what if we takem2Zand n= 0? Well, in this case we get the quantities, namely,m0 ,m2Zwhich are conventionally undened.Denition The rational numbers are all those numbers which can be expressed in the form mn wherem2Zandn2Z rf0g. We denote the set of all rational numbers byQ.Thus, we have
Q:=nmn
:m2Z;n2Z rf0goZafar IqbalModule - 1: The Real Numbers
Algebraic Structures onQThe algebraic operations of addition `+' and multiplication `' dened on the setQ
have the following algebraic properties:1Associativity of Addition:a+ (b+c) = (a+b) +cfor alla;b;c2Q.2Commutativity of Addition:a+b=b+afor alla;b2Q.3Existence of Additive Identity:There exists an elemen t0 2Qsuch that
a+ 0 =a= 0 +afor alla2Q.4Existence of Additive Inverse:F oreac ha2Q, there exists an elementa2Qsuch thata+ (a) = 0 = (a) +a.5Associativity of Multiplication:a(bc) = (ab)cfor alla;b;c2Q.6Commutativity of Multiplication:ab=bafor alla;b2Q.7Existence of Multiplicative Identity:There exists an elemen t1 2Qsuch that
a1 =a= 1afor alla2Q.8Existence of Multiplicative Inverse of Non-Zero Rationals:F oreac h a2Q rf0g, there exists an element1a2Qsuch thata1a
= 1 =1a a.9Distributivity of Multiplication over Addition:a(b+c) =ab+acfor all a;b;c2Q.Zafar IqbalModule - 1: The Real Numbers The setQequipped with the algebraic operations of addition `+' andmultiplication `' forms aeld, i.e., (Q;+;) is aeld.Order Structure onQZafar IqbalModule - 1: The Real Numbers
Zafar IqbalModule - 1: The Real Numbers
Zafar IqbalModule - 1: The Real Numbers
Zafar IqbalModule - 1: The Real Numbers
Zafar IqbalModule - 1: The Real Numbers
Zafar IqbalModule - 1: The Real Numbers
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