Construction Of Real Numbers

Construction Of Real NumbersAll mathematicians know (or r tout ensembley they know) all(prenominal) about the corporeal rime. However usually we just accept the real verse as being there rather than considering precisely what they atomic number 18. In this project I will attempts to answer that question. We shall begin with optimistic integers and whence successively construct the rational and finally the real falls. Also showing how real verse satisfy the axiom of the f number jump, whilst rational verse do non. This shows that all real song converge towards the Cauchys range.1 IntroductionWhat is real outline real analysis is a field in mathematics which is applied in m both areas including number theory, probability theory. All mathematicians know (or think they know) all about the real be. However usually we just accept the real numbers racket as being there rather than considering precisely what they are. The aim of this get is to analyse number theory to sho w the difference between real numbers and rational numbers.Developments in calculus were mainly made in the ordinal and eighteenth century. Examples from the literature put forward be given such as the proof that derrierenot be rational by Lambert, 1971. During the development of calculus in the 17th century the entire organize of real numbers were used without having them defined clearly. The first person to release a definition on real numbers was Georg Cantor in 1871. In 1874 Georg Cantor revealed that the set of all real numbers are uncountable innumerable but the set of all algebraic numbers are countable infinite.As you can see, real analysis is a somewhat theoretical field that is closely related to mathematical concepts used in most branches of economics such as calculus and probability theory. The concept that I have talked about in my project are the real number system.2 DefinitionsNatural numbersNatural numbers are the fundamental numbers which we use to count. We can add and multiply two natural numbers and the result would be another natural number, these operations obey various rules.(Stirling, p.2, 1997)Rational numbersRational numbers consists of all numbers of the form a/b where a and b are integers and that b 0, rational numbers are usually called fractions. The use of rational numbers permits us to solve equations. For subject a + b = c, ad = e, for a where b, c, d, e are all rational numbers and a 0. Operations of subtraction and division (with non cryptograph divisor) are possible with all rational numbers.(Stirling, p.2, 1997)Real numbersReal numbers can also be called irrational numbers as they are not rational numbers like pi, square root of 2, e (the base of natural log). Real numbers can be given by an infinite number of decimals real numbers are used to measure continuous quantities. There are two basic properties that are involved with real numbers secerned fields and least upper bounds. Ordered fields say that real numb ers comprises a field with addition, multiplication and division by non zero number. For the least upper bound if a non empty set of real numbers has an upper bound then(prenominal) it is called least upper bound. placesA Sequence is a set of numbers arranged in a particular order so that we know which number is first, second, third etc and that at any positive natural number at n we know that the number will be in nth place. If a ecological succession has a function, a, then we can denote the nth term by an. A duration is commonly denoted by a1, a2, a3, a4 this entire grades can be written as or (an). You can use any letter to denote the sequence like x, y, z etc. so giving (xn), (yn), (zn) as sequencesWe can also make sequel from sequences, so if we say that (bn) is a subsequence of (an) if for apiece n we getbn = ax for some x and bn+1 = by for some y and x y.We can alternatively imagine a subsequence of a sequence being a sequence that has had terms missing from the original sequence for illustration we can say that a2, a4 is a subsequence if a1, a2, a3, a4.A sequence is increasing if an+1 an n . Correspondingly, a sequence is decreasing ifan+1 an n . If the sequence is either increasing or decreasing it is called a monotone sequence.There are several different types of sequences such as Cauchy sequence, convergent sequence, monotonic sequence, Fibonacci sequence, reflexion and see sequence. I will be talking about only 2 of the sequences Cauchy and Convergent sequences.Convergent sequencesA sequence (an) of real number is called a convergent sequences if an tends to a finite marge as n. If we say that (an) has a limit a F if given any 0, F, k an a n kIf an has a limit a, then we can write it as liman = a or (an) a.Cauchy SequenceA Cauchy sequence is a sequence in which numbers become closer to each other as the sequence progresses. If we say that (an) is a Cauchy sequence if given any 0, F, k an am n,m k.Gary Sn g Chee Hien, (2001).Bounded sets, Upper Bounds, Least Upper BoundsA set is called bounded if there is a certain sense of finite size. A set R of real numbers is called bounded of there is a real number Q such that Q r for all r in R. the number M is called the upper bound of R. A set is bounded if it has both upper and humble bounds. This is extendable to subsets of any part ordered set. A subset Q of a partially ordered set R is called bounded above. If there is an element of Q r for all r in R, the element Q is called an upper bound of R3 Real number systemNatural NumbersNatural numbers () can be denoted by 1,2,3 we can define them by their properties in order of relative. So if we consider a set S, if the relation is less than or equal to on SFor every(prenominal) x, y S x y and/or y xIf x y and y x then x = yIf x y and y z then x zIf all 3 properties are met we can call S an ordered set.(Giles, p.1, 1972)Real numbersAxioms for real numbers can be spilt in to 3 group s algebraic, order and completeness.Algebraic AxiomsFor all x, y , x + y and xy .For all x, y, z , (x + y) + z = x (y + z).For all x, y , x + y = y + x.There is a number 0 such that x + 0 = x = 0 + x for all x .For each x , there exists a corresponding number (-x) such that x + (-x) = 0 = (-x) + xFor all x, y, z , (x y) z = x (y z).For all x, y x y = y x.There is number 1 such that x x 1 = x = 1 x x, for all x For each x such that x 0, there is a corresponding number (x-1) such that x (x-1) = 1 = (x-1) xA10. For all x, y, z , x (y + z) = x y + x z(Hart, p.11, 2001)Order AxiomsAny pair x, y of real numbers satisfies precisely one of the following relations (a) x y (b) x = y (c) y x.If x y and y z then x z.If x y then x + z y +z.If x y and z 0 then x z y z(Hart, p.12, 2001)Completeness AxiomIf a non-empty set A has an upper bound, it has a least upper boundThe thing which distinguishes from is the Completeness Axiom.An upper bound of a non-empty subs et A of R is an element b R with b a for all a A.An element M R is a least upper bound or supremum of A ifM is an upper bound of A and if b is an upper bound of A then b M.That is, if M is a least upper bound of A then (b R)(x A)(b x) b MA lower bound of a non-empty subset A of R is an element d R with d a for all a A.An element m R is a greatest lower bound or infimum of A ifm is a lower bound of A and if d is an upper bound of A then m d.If all 3 axioms are satisfied it is called a complete ordered field. put-on oConnor (2002) axioms of real numbersRational numbersAxioms for Rational numbersThe axiom of rational numbers operate with +, x and the relation , they can be defined on corresponding to what we know on N.For on +(add) has the following properties.For every x,y , there is a unique element x + y For every x,y , x + y = y + xFor every x,y,z , (x + y) + z = x + (y + z)There exists a unique element 0 such that x + 0 = x for all x To every x there exists a unique elem ent (-x) such that x + (-x) = 0For on x(multiplication) has the following properties.To every x,y , there is a unique element x x y For every x,y , x x y = y x xFor every x,y,z , (x x y) x z = x x (y x z)There exists a unique element 1 such that x x 1 = x for all x To every x , x 0 there exists a unique element such that x x = 1For both add and multiplication properties there is a closer, commutative, associative, identity and inverse on + and x, both properties can be related by.For every x,y,z , x x (y + z) = (x x y) + (x x z)For with an order relation of , the relation property is a. we can claim that b. if not then since a and b we would have b a.John OConnor (2002) axioms of real numbersTheoremThe limit of a sequence, if it exists, is unique. substantiation allow x and x be 2 different limits. We may assume without loss of generality, thatx x. In particular, take = (x x)/2 0.Since xn x, k1 s.t xn x n k1Since xn x k2 s.t xn x n k2Take k = maxk1, k2. The n n k, xn x , xn x x x = x xn + xn x x xn + xn x + = x x, a contradictionHence, the limit must be unique. Also all rational number sequences have a limit in real numbers.Gary Sng Chee Hien, (2001).TheoremAny convergent sequence is bounded.ProofSuppose the sequence (an)a. take = 1. Then choose N so that whatever n N we have an within 1 of a. apart from the finite set a1, a2, a3aN all the terms of the sequence will be bounded by a + 1 and a 1. Showing that an upper bound for the sequence is maxa1, a2, a3aN, a +1. Using the same method you could alternatively find the lower boundTheoremEvery Cauchy Sequence is bounded.ProofLet (xn) be a Cauchy sequence. Then for xn xm 1 n, m k.Hence, for n k, we have xn = xn xk + xk xn xk + xk 1 + xk Let M = max x1 , x2 , , xk-1, 1 + xk and it is clear that xn M n, i.e. (xn) is bounded.Gary Sng Chee Hien, (2001).TheoremIf (xnx, then any subsequence of (xn) also converges to x.ProofLet (yn) be any subseque nce of (xn). Given any 0, s.t xn x n N.But yn = xi for some so we may claim yn x also.Hence, (Gary Sng Chee Hien, (2001).TheoremIf (xn) is Cauchy, then any subsequence of (xn) is also Cauchy.ProofLet (yn) be any subsequence of (xn). Given any s.t xn xm .But yn = xi for so we may claim yn ym Hence (yn) xGary Sng Chee Hien, (2001).TheoremAny convergent sequence is a Cauchy sequence.ProofIf (an) a then given 0 choose N so that if n N we have an- a . Then if m, n N we have am- an = (am- a) (am- a) am- a + am- a 2.We use completeness Axiom to proveSuppose X , X2 = 2. Let (an) be a sequence of rational numbers converging to an irrational12 = 11.52 = 2.251.42 = 1.961.412 = 1.98811.41421356237302 = 1.999999999999731161391129Since (an) is a convergent sequence in it is a Cauchy sequence in and hence also a Cauchy sequence in . But it has no limit in.An irrational number like 2 has a decimal expansion which does not repeat2 =1.4142135623730John OConnor (2002) Cauchy Sequen ces.TheoremProve that is irrational, prove that ProofWe will get 2 as the least upper bound of the set A = q Q q2 2. We know that a is bounded above and so its least upper bound b does not exists.Suppose x , x2 0 be given. Then k1, k2 s.t xn xm /(2Y) n, m k1 yn ym /(2X) n, m k2Take k = max(k1, k2). Then xn xm /(2Y) yn ym /(2X) n, m kHence, xn yn xm ym = (xn yn xm yn) + (xm yn xm ym) xn yn xm yn + xm yn xm ym = yn xn xm + xm yn ym Y xn xm + X yn ym Y(/(2Y)) + X(/(2X)) n, m k=Hence, (xn yn) is also Cauchy.5 ConclusionReal numbers are infinite number of decimals used to measure continuous quantities. On the other hand, rational numbers are defined to be fractions formed from real numbers. Axioms of each number system are examined to determine the difference between real numbers and rational numbers. Conclusion of the analysis of axioms resulted to be both real numbers and rational numbers contain the same properties. The properties being addition, multiplication and there exist a relationship of zero and one.The four fundamental results are obtained from this study. low gear concept is that the property of real number system being unique and following the complete ordered field. Second is that if any real number satisfies the axioms then it is upper bound, whilst rational numbers are not upper bound. The third being that all Cauchy sequences are converges towards the real numbers. Finally found out that all real numbers are equivalence classes of the Cauchy sequence.AppendicesList of symbols = Natural number = Real number= Rational number = is an element of= There exists= For alls.t. = much(prenominal) that