Construction Of Real Numbers

Construction Of Real Numbers All mathematicians know (or think they know) all about the real numbers. However usually we just accept the real numbers as being there rather than considering precisely what they are. In this project I will attempts to answer that question. We shall begin with positive integers and then successively construct the rational and finally the real numbers. Also showing how real numbers satisfy the axiom of the upper bound, whilst rational numbers do not. This shows that all real numbers converge towards the Cauchys sequence. 1 Introduction What is real analysis; real analysis is a field in mathematics which is applied in many areas including number theory, probability theory. All mathematicians know (or think they know) all about the real numbers. However usually we just accept the real numbers as being there rather than considering precisely what they are. The aim of this study is to analyse number theory to show the difference between real numbers and rational numbers. Developments in calculus were mainly made in the seventeenth and eighteenth century. Examples from the literature can be given such as the proof that Ï€ cannot be rational by Lambert, 1971. During the development of calculus in the seventeenth century the entire set 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 infinite 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 Definitions Natural numbers Natural 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 numbers Rational 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 example; 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 zero divisor) are possible with all rational numbers. (Stirling, p.2, 1997) Real numbers Real 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 ordered fields and least upper bounds. Ordered fields say that real numbers 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 it is called least upper bound. Sequences A 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 sequence has a function, a, then we can denote the nth term by an. A sequence is commonly denoted by a1, a2, a3, a4†¦ this entire sequences 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 sequences We can also make subsequence from sequences, so if we say that (bn) is a subsequence of (an) if for each n∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ we get; bn = ax for some x ∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ and bn+1 = by for some y ∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ 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 example we can say that a2, a4 is a subsequence if a1, a2, a3, a4. A sequence is increasing if an+1 ≠¥ an ∀ n ∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢. Correspondingly, a sequence is decreasing if an+1 ≠¤ an ∀ n ∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢. 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, look and see sequence. I will be talking about only 2 of the sequences Cauchy and Convergent sequences. Convergent sequences A sequence (an) of real number is called a convergent sequences if an tends to a finite limit as n→∞. If we say that (an) has a limit a∈ F if given any ÃŽ µ > 0, ÃŽ µ ∈ F, k∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ | an a | < ÃŽ µ n ≠¥ k If an has a limit a, then we can write it as liman = a or (an) → a. Cauchy Sequence A 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 ÃŽ µ > 0, ÃŽ µ ∈ F, k∈ à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ | an am | < ÃŽ µ n,m ≠¥ k. Gary Sng Chee Hien, (2001). Bounded sets, Upper Bounds, Least Upper Bounds A 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 lower bounds. This is extendable to subsets of any partially 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 R 3 Real number system Natural Numbers Natural numbers (à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢) can be denoted by 1,2,3†¦ we can define them by their properties in order of relation. So if we consider a set S, if the relation is less than or equal to on S For every x, y ∈ S x ≠¤ y and/or y ≠¤ x If x ≠¤ y and y ≠¤ x then x = y If x ≠¤ y and y ≠¤ z then x ≠¤ z If all 3 properties are met we can call S an ordered set. (Giles, p.1, 1972) Real numbers Axioms for real numbers can be spilt in to 3 groups; algebraic, order and completeness. Algebraic Axioms For all x, y ∈ à ¢Ã¢â‚¬Å¾Ã‚ , x + y ∈ à ¢Ã¢â‚¬Å¾Ã‚  and xy ∈ à ¢Ã¢â‚¬Å¾Ã‚ . For all x, y, z ∈ à ¢Ã¢â‚¬Å¾Ã‚ , (x + y) + z = x (y + z). For all x, y ∈ à ¢Ã¢â‚¬Å¾Ã‚ , x + y = y + x. There is a number 0 ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x + 0 = x = 0 + x for all x ∈ à ¢Ã¢â‚¬Å¾Ã‚ . For each x ∈ à ¢Ã¢â‚¬Å¾Ã‚ , there exists a corresponding number (-x) ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x + (-x) = 0 = (-x) + x For all x, y, z ∈ à ¢Ã¢â‚¬Å¾Ã‚ , (x y) z = x (y z). For all x, y ∈ à ¢Ã¢â‚¬Å¾Ã‚  x y = y x. There is number 1 ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x x 1 = x = 1 x x, for all x ∈ à ¢Ã¢â‚¬Å¾Ã‚  For each x ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x ≠  0, there is a corresponding number (x-1) ∈ à ¢Ã¢â‚¬Å¾Ã‚  such that x (x-1) = 1 = (x-1) x A10. For all x, y, z ∈ à ¢Ã¢â‚¬Å¾Ã‚ , x (y + z) = x y + x z (Hart, p.11, 2001) Order Axioms Any 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 Axiom If a non-empty set A has an upper bound, it has a least upper bound The thing which distinguishes à ¢Ã¢â‚¬Å¾Ã‚  from is the Completeness Axiom. An upper bound of a non-empty subset 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 if M 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 M A 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 if m 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. John oConnor (2002) axioms of real numbers Rational numbers Axioms for Rational numbers The 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 + x For 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 element (-x) ∈ such that x + (-x) = 0 For 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 x For 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 = 1 For 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 numbers Theorem: The limit of a sequence, if it exists, is unique. Proof Let x and x†² be 2 different limits. We may assume without loss of generality, that x < x†². In particular, take ÃŽ µ = (x†² x)/2 > 0. Since xn→ x, k1 s.t | xn x | < n ≠¥ k1 Since xn→ x k2 s.t | xn x†²| < ÃŽ µ n ≠¥ k2 Take k = max{k1, k2}. Then n ≠¥ k, | xn x | < ÃŽ µ, | xn x†²| < ÃŽ µ | x†² x | = | x†² xn + xn x | ≠¤ | x†² xn | + | xn x | < ÃŽ µ + ÃŽ µ = x†² x, a contradiction! Hence, the limit must be unique. Also all rational number sequences have a limit in real numbers. Gary Sng Chee Hien, (2001). Theorem: Any convergent sequence is bounded. Proof Suppose 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, a3†¦aN} all the terms of the sequence will be bounded by a + 1 and a 1. Showing that an upper bound for the sequence is max{a1, a2, a3†¦aN, a +1}. Using the same method you could alternatively find the lower bound Theorem: Every Cauchy Sequence is bounded. Proof Let (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). Theorem: If (xnx, then any subsequence of (xn) also converges to x. Proof Let (yn) be any subsequence 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). Theorem: If (xn) is Cauchy, then any subsequence of (xn) is also Cauchy. Proof Let (yn) be any subsequence of (xn). Given any s.t | xn xm | . But yn = xi for so we may claim | yn ym | Hence (yn) x Gary Sng Chee Hien, (2001). Theorem Any convergent sequence is a Cauchy sequence. Proof If (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 prove Suppose X ∈ à ¢Ã¢â‚¬Å¾Ã‚ , X2 = 2. Let (an) be a sequence of rational numbers converging to an irrational 12 = 1 1.52 = 2.25 1.42 = 1.96 1.412 = 1.9881 1.41421356237302 = 1.999999999999731161391129 Since (an) is a convergent sequence in à ¢Ã¢â‚¬Å¾Ã‚  it is a Cauchy sequence in à ¢Ã¢â‚¬Å¾Ã‚  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 repeat: 2 =1.4142135623730 John OConnor (2002) Cauchy Sequences. Theorem Prove that is irrational, prove that ≠¤ à ¢Ã¢â‚¬Å¾Ã‚  Proof We 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 | < ÃŽ µ/(2Y) n, m ≠¥ k1 | yn ym | < ÃŽ µ/(2X) n, m ≠¥ k2 Take k = max(k1, k2). Then | xn xm | < ÃŽ µ/(2Y) | yn ym | < ÃŽ µ/(2X) n, m ≠¥ k Hence, | 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(ÃŽ µ/(2Y)) + X(ÃŽ µ/(2X)) n, m ≠¥ k = Hence, (xn yn) is also Cauchy. 5 Conclusion Real 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. First 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. Appendices List of symbols à ¢Ã¢â‚¬Å¾Ã¢â‚¬ ¢ = Natural number à ¢Ã¢â‚¬Å¾Ã‚  = Real number = Rational number ∈ = is an element of = There exists = For all s.t. = Such that

Love and Basketball: An Overview

Here’s the run-down. Love & Basketball is deceivingly simple in its structure. The movie is divided into the quarters of a basketball game and tells the story of a boy and a girl. Meeting at about the age 11, the film traces their lives as they run parallel and run apart from childhood, to high school, to college, and just after. Monica and Quincy each have their hopes and their dreams. They both want to play basketball on a professional level. For Quincy, it is easier and expected since he is the son of a professional player. It is harder for Monica, both being a woman and as a daughter whose mother cannot understand why she does not want to grow up to be a pretty stay at home wife. Through the whole film the constant between the two is their love for each other and for the game of basketball. The movie is full of honest moments, laughs, tears and some awesome basketball scenes. There are a lot of positives to this movie. This movie shows that no matter what race, gender, or where you came from you can be a successful athlete. Monica is a black female basketball player with an attitude of a male who makes it to the pros. Going into her senior year of high school, Monica was afraid she wasn’t getting any looks by colleges and at the games she was getting looked at she was riding the bench because of her attitude, but the movie showed that it is important to have a strong support system at home. Her parents recognized it and put her in her place! A good athlete has to be all around good. They have to be focused in the classroom and respectable on and off the court! The movie showed how important a healthy home life is needed in more ways than just at Monica’s home. Quincy’s father was a professional athlete that was cheating on his mother. This unhealthy home life affected Quincy and his athletics. Quincy didn’t finish college because of it and entered the draft. After he entered the draft he hurt his knee; consequently he thought his basketball career was over. That was also another positive aspect of the movie, showing the importance of education! If Quincy had finished college and received a college degree he would have had something to fall back on. The main plot line of the movie is very positive in and of itself! A story based on two individuals whom are childhood sweethearts trying to balance following their dreams while trying to keep their love alive is ultimately the hardest thing to do in the eyes of a student athlete of any age! Watching this movie gives you hope that it can actually happen. I know people who try to live this life. Truth be told, it can only happen in a fairy tale though! I don’t believe it. A little girl finds herself in a new neighborhood and having to make new friends. She stumbles upon some boys playing basketball. Being the tomboy she is, she assumes they will let her play. She ends up in a fight with one boy, Quincy. She goes home only to hear her mom go on and on about how she needs to be more girly and quit trying to be one of the boys. Monica has heard this bit her whole life. The young boy is fascinated by Monica; he has probably never had a girl ever stand up to him in that way. He asks her to be his girlfriend and they share their first kiss together. Throughout the years they maintain their strong friendship, living so close together they comfort each other during family problems. They live window to window. They get to high school and Quincy is, of course, quite the ladies’ man; being the best basketball player in the state, they tend to have that effect. Monica plays too, but in high school her anger problems are out of control on the court. Little did they know their romantic lives were about to cross paths again at their very last hooray of high school; senior prom. Quincy of course took one of his random hoes to the prom; whereas Monica just to please her mom. She went with a college guy who her sister set her up with.

Business Axe Commercial Research Essays - 1132 Words

Introduction with Background Information Company Axe is one of the 400 brands which is belonged to Unilever Company. The portfolio of this multinational company focuses on health and wellbeing mainly, including food, beverages, cleaning agents and personal care products. Many world-leading brands including Axe, Lipton, Knorr, Dove, Hellmann’s and Omo are some of these brands (unilever.com). Market Axe Brand, which was named Lynx in Europe, was first launched in France in 1983 as a teenage boys’ grooming category (theguardian.com). For its first 19 years since 1983, AXE brand has developed its market merely in Europe. Later on, in 2002, AXE was introduced to another big market-- the U.S and increased its target customers†¦show more content†¦However, according to most of the commercials they did before, their advertisements are well known for sexual humor and exaggerated scenarios and exaggerated scenarios that play into male fantasies. Significant changes about the campaign has changed in the year of 2014. The way how the company promoted the Axe Peace fragrance line gave a new idea to the public with its content and meaning behind. The new global campaign and commercial advertisement is related to peace. The concept behind the advertisement and its marketing campaign was developed with the Peace fragrance because Axe often visits college campuses an d they finds peace and harmony are the topics and themes that are supported by students (time.com). Thus, it aims to generate the awareness of the peace message, as well as encouraging people to take simple but meaningful actions that will have a positive impact on the future of their world. Also, partnered with international non-profit organization, Peace One Day, Axe brand set a Peace Day on 21st September, 2013 (unilever.com). $250,000 from Axe brand was donated to the organization and is promoting it in its advertisements and on Facebook page (businessinsider.com). Besides these, the promotional campaign also includes a Kiss-For-Peace Twitter hashtag. Users can submit photos of themselves kissing to Axe, and the best photos are being broadcast on electronicShow MoreRelatedEthical Policies Vs. Corporate Social Responsibilities1238 Words   |  5 Pagesand senses make use of right or off-base. Presently apply this as business definition, the ultimate goal of the company is t o make profits and there can be either positive or negative Impact by the company on operation of business. Simply business ethics is the behavior of the business in accordance with the society or community [1]. Unilever Company Code of Practice Paul Polman (CEO) of Unilever Company reported that its business earned reputation based on integrity and interests in accordanceRead MoreNew Product Entry Strategies1678 Words   |  7 Pagestruly profitable business growth in an otherwise depressing economic environment. 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What Is the Largest Jellyfish

Question: What Is the Largest Jellyfish? What is the largest jellyfish, and where is it found? And most importantly, is it dangerous to humans? Find out below. Answer: The largest jellyfish is the lions mane jellyfish (Cyanea capillata). Although most are much smaller, the bell of a lions mane jellyfish can be over 8 feet across. As huge as their bell is in diameter, thats not even the biggest part of the lions mane jellyfish. Their long, slender tentacles can reach over 100 feet, and they have many of them - the lions mane jellyfish has eight groups of tentacles, and there are 70-150 tentacles in each group. The tentacles hang down underneath the jellyfishs bell, along with its much-folded lips and gonads. All these structures together in a mass resemble a lions mane. Interestingly, the lions mane jellyfish changes in color as it ages. They start out pink and yellow, and then once the bell grows to 5 inches, the jellyfish is reddish to reddish brown. As the bell grows over 18 inches, the jellyfish deepens in color. Where Are Lions Mane Jellyfish Found? Lions mane jellyfish have a relatively wide distribution - they are found in both the Atlantic and the Pacific Oceans, but in cooler water that is less than 68 degrees F. What Do Lions Mane Jellyfish Eat? Lions mane jellyfish eat plankton, fish, crustaceans and other jellyfish. They have an interesting feeding strategy in which they rise into the water column, then spread out their tentacles in a wide net and descend, trapping prey as they fall into the water column. This page shows a beautiful image of a lions mane jellyfish with its tentacles spread out. Are Lions Mane Jellyfish Dangerous? Lions mane jellyfish stings are rarely fatal, but their stings can be painful, although the pain is generally temporary and causes redness in the area. According to this site, more severe reactions can include muscle cramps, breathing difficulty and skin burning and blistering. What If I Get Stung? First, rinse the area with sea water (not fresh water, which can cause more severe stinging), and neutralize the sting using vinegar. Scrape off any remaining stingers using something stiff like a credit card, or by making a paste using sea water and talcum powder or baking soda, and letting it dry. Covering the area with shaving cream or meat tenderizer and letting it dry before scraping it off may also help reduce the sensation and remove stingers. How to Avoid a Lions Mane Jellyfish Sting Lions mane jellyfish may be large, with a mass of long tentacles, so always give them a wide berth. And remember, the stingers may still work even after the jellyfish has died, so dont assume its safe to touch a jellyfish, even if its dead on a beach.