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

The Effect of Colonialism in Things Fall Apart by Chinua Achebe Essay

This essay is about the effect of Colonialism seen in the book Things Fall Apart. Through out the whole book you can see different impressions on the tribe, many other people, and the relationships between the white man and the black man.   Ã‚  Ã‚  Ã‚  Ã‚  "Does the white man understand our custom about land?" "How can he when he does not even speak our tongue? But he says that our customs are bad; and our own brothers who have taken up his religion also say that our customs are bad. How do you think we can fight when our own brothers have turned against us? The white man is very clever. He came quietly and peaceably with his religion. We were amused at his foolishness and allowed him to stay. Now he has won our brothers, and our clan can no longer act like one. He has put a knife on the things that held us together and we have fallen apart." (Achebe, 17) This discussion of the white man not understanding the customs and traditions, comes ever since the arrival of the colonialists, Obierika seems to voice Achebe's own thoughts on colonialism. Upset by the fact that the white men have come and completely disregarded the Igbo sense of justice, Obierika points out the impossibility of the colonialists understanding anything about the Umuofians without speaking their language. Then he points out the foolishness of belittling unfamiliar customs.   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Obierika does not lay the whole blame on the white man’s side. He feels also that the Umuofians who have con...

Microsoft And Windows Competitors :: essays research papers

A lot of people today, mostly microsofties, argue that Microsoft should not be split up since it isn’t really a monopoly; Windows has a lot of competitors out there and some of the companies that make them are even bigger than Microsoft. And that is actually quite true: Microsoft has only about 6% of the global software market and only 3% of the global computer market overall. There are several computer companies that make more than Microsoft, like Sun Microsystems and Compaq and there are at least nine other operating systems besides Windows, some of which you could even get for free. After three decades of use, the UNIX computer operating system from Bell Labs is still regarded as one of the most powerful, versatile, and flexible operating systems (OS) in the computer world. Its popularity is due to many factors, including its ability to run a wide variety of machines, from micros to supercomputers, and its portability -- all of which led to its adoption by many manufacturers. The UNIX operating system was designed to let a number of programmers access the computer at the same time and share its resources. While initially meant for medium-sized computers, the system was soon moved to larger, more powerful mainframe computers. As personal computers grew in popularity, versions of UNIX found their way into these boxes, and a number of companies produce UNIX-based machines for the scientific and programming communities. A major contribution of the UNIX system was its portability, permitting it to move from one brand of computer to another with a minimum of code changes. At a time when different computer lines of the same vendor didn't talk to each other -- yet alone machines of multiple vendors -- that meant a great savings in both hardware and software upgrades. It also meant that the operating system could be upgraded without having all the customer's data inputted again. And new versions of UNIX were backward compatible with older versions, making it easier for companies to upgrade in an orderly manner. UNIX comes with hundreds of programs that can be divided into two classes: integral utilities that are absolutely necessary for the operation of the computer, such as the command interpreter, and tools that aren't necessary for the operation of UNIX but provide the user with additional capabilities, such as typesetting capabilities and e-mail; the tools can be added or removed from a UNIX system, depending upon the applications required. Microsoft And Windows Competitors :: essays research papers A lot of people today, mostly microsofties, argue that Microsoft should not be split up since it isn’t really a monopoly; Windows has a lot of competitors out there and some of the companies that make them are even bigger than Microsoft. And that is actually quite true: Microsoft has only about 6% of the global software market and only 3% of the global computer market overall. There are several computer companies that make more than Microsoft, like Sun Microsystems and Compaq and there are at least nine other operating systems besides Windows, some of which you could even get for free. After three decades of use, the UNIX computer operating system from Bell Labs is still regarded as one of the most powerful, versatile, and flexible operating systems (OS) in the computer world. Its popularity is due to many factors, including its ability to run a wide variety of machines, from micros to supercomputers, and its portability -- all of which led to its adoption by many manufacturers. The UNIX operating system was designed to let a number of programmers access the computer at the same time and share its resources. While initially meant for medium-sized computers, the system was soon moved to larger, more powerful mainframe computers. As personal computers grew in popularity, versions of UNIX found their way into these boxes, and a number of companies produce UNIX-based machines for the scientific and programming communities. A major contribution of the UNIX system was its portability, permitting it to move from one brand of computer to another with a minimum of code changes. At a time when different computer lines of the same vendor didn't talk to each other -- yet alone machines of multiple vendors -- that meant a great savings in both hardware and software upgrades. It also meant that the operating system could be upgraded without having all the customer's data inputted again. And new versions of UNIX were backward compatible with older versions, making it easier for companies to upgrade in an orderly manner. UNIX comes with hundreds of programs that can be divided into two classes: integral utilities that are absolutely necessary for the operation of the computer, such as the command interpreter, and tools that aren't necessary for the operation of UNIX but provide the user with additional capabilities, such as typesetting capabilities and e-mail; the tools can be added or removed from a UNIX system, depending upon the applications required.

Absolutism and Constitutionalism

1. Why did monarchs in the late 16th/early 17th centuries need new sources of income? Why did monarchs wish to get their income without the permission of the nobility? 2.Explain the role that each of the following played in the failure of England achieving absolutism, as well as the success of the French: England France  · Religion – Religion  · Parliament/Tradition – Estates General  · Personalities of Leaders – Personalities of Leaders  · Trust of Nobility – Trust of nobility 3. Henry IV came to the throne and helped to end the French Wars of Religion.After over 30 years of civil war, France was a wreck politically and economically. To set the stage for absolutism, he and his finance minister, the duke of Sully, needed to strengthen certain aspects of the nation and weaken others. Explain where the following fell into their plans, using the terms in parenthesis in your answer:  · power of nobility (parlements)  · increased money for crown (m onopolies)  · increased opportunity for economic success (canals, corvee) 4. After Henry IV’s assassination, his 9 year old son Louis XIII became king.Too young to rule, his mother named a brilliant official to serve as his regent. Explain how the following policies show the nature of Richelieu’s â€Å"raison d’etat†:  · Use of intendants  · Actions in the 30 Years War (1618-1648)  · Treatment of nobles  · Treatment of Huguenots 5. Describe the factors that led to the revolt (known as the Fronde) of the French nobility against the young Louis XIV and Cardinal Mazarin (1649-1652). What lesson did the Fronde teach Louis XIV? How did it help prepare the French people for absolutist rule?Helpful Hints for Reading this Section: o The first part of this reading provides an overview of the two systems of government that developed, and offers you hints as to why they developed as they did. Read this carefully so that many of the details listed in future sections will mean more to you. o There is a big difference between the English Parliament and the French parlements (note difference in spellings). The Parliament (-ia) historically had the ability to declare whether taxes were just (fair) or not, and so the kings/queens always had to go before Parliament whenever they needed money.Parliament, located in London, had the ability to make laws. French parlements (-e) were different in that there were many regional parlements as opposed to one central one, and they lacked the ability to make laws. Rather, the French parlements just had the authority to accept or refuse policies proposed by the monarchy. The French also had the Estates General to serve as its legislature, but it was a medieval creation that never really caught on among the French nobles. It could only meet when it was called into session by the king, and that was very rare (met once between 1618 and 1788).Chapter 13 – England 1. The Policy of Circumvention refer s to the various English kings’ efforts to go around, or circumvent, Parliament in their gathering of money for the crown. Explain how each of the following was an effort to raise or save money, and why it angered nobles/members of Parliament:  · impositions- These additional custom duties were seen as taxation on imports and exports by Parliament and as requiring parliamentary consent. They angered nobles and members of Parliament because they viewed it as taxation without consent. selling titles of nobility (think supply and demand– why would nobles feel like they each had less power if there were more nobles in the nation? )- people did anything they could to get a higher position so of course they would buy titles of nobility and since there were many nobles now, previous nobles felt less power because there weren’t as many â€Å"important† titles for everyone.  · James I’s role as peacemaker- 2. Summarize the many religious complaints that were raised against King James I. In defending the episcopacy, what did James mean when he retorted â€Å"No bishops, no king (pg. 452). – James viewed the proposal to replace bishops with presbyteries as an attempt to diminish his power in the church so he quoted that. 3. What actions led Parliament force Charles I to accept the Petition of Right? How would Charles I have attempted to defend himself and his actions? What freedoms did the Petition guarantee? -disputes between Parliament and King Charles I over the execution of the Thirty Years’ War, Parliament refused to grant subsidies to support the war effort, leading to Charles gathering â€Å"forced loans† without Parliamentary approval and arbitrarily imprisoning those who refused to pay.The Petition guaranteed restrictions on non-Parliamentary taxation, forced billeting of soldiers, imprisonment without cause, and restricts the use of martial law. 4. Why did Parliament not meet between 1629 and 1640? De scribe the circumstances that required their meeting in 1640. – The Triennial Act was intended to prevent kings from ruling without Parliament, as Charles had done between 1629-1640. The act required that Parliament meet for at least a fifty-day session once every three years. 5. Create a timeline using the following terms, explaining what each is and how the terms relate to one-another.  · Short Parliament Scottish Invasion  · Long Parliament’s New Laws (1640-41)  · Grand Remonstrance  · Invasion of Parliament (Roundhead/Cavaliers) 1639-1640 – Scottish Invasion= Breakdown of Charles's government of Scotland and two attempts to impose his will by force. Scots rose in 1639 against Charles' introduction of the English Prayer Book into Scotland, the anti-royalist London merchants encouraged the invading Scots to capture Newcastle. This they did in 1640, totally disrupting the export of coal. The Scottish army remained in Newcastle for a year and charged the Corporation a regular fee for billeting its troops. 640 – Short Parliament= sat from 13 April to 5 May 1640 during the reign of King Charles 1 of England and called â€Å"short† because it only lasted 3 weeks. He was forced to call the Short Parliament primarily to obtain money to finance his military struggle with Scotland in the Bishops’ War. -Long Parliament= established to pass financial bills. It received its name from the fact that through an Act of Parliament, it could be dissolved only with the agreement of the members and those members did not agree to its dissolution until after the English Civil War and at the end of interregnum in 1660. 641 -Grand Remonstrance= a list of grievances presented to King Charles I by English Parliament on 1 December 1641, but passed by the House of Commons on the 22nd of November 1641, during the Long Parliament; it was one of the chief events which were to precipitate the English Civil War. 1642-1651 -Invasion of Parlia ment (Roundhead/Cavaliers)= was a series of armed conflicts and political machinations between Parliamentarians (Roundheads) and Royalists (Cavaliers).The first (1642–46) and second (1648–49) civil wars pitted the supporters of King Charles I against the supporters of the Long Parliament, while the third war (1649–51) saw fighting between supporters of King Charles II and supporters of the Rump Parliament. The Civil War ended with the Parliamentary victory at the Battle of Worcester on 3 September 1651. 6. Explain how the â€Å"Rump Parliament† and, more appropriately, Oliver Cromwell, ruled England during the period between Charles I and Charles II. – ruled first England, and then Ireland and Scotland from 1649 to 1660.After the English Civil War and the execution of Charles I, the republic's existence was initially declared by â€Å"An Act declaring England to be a Commonwealth† adopted by the Rump Parliament, on 19 May 1649. The governmen t took the form of direct personal rule by Oliver Cromwell. Just before and after the execution of King Charles I on 30 January 1649, the Rump passed a number of acts of Parliament creating the legal basis for the republic. Helpful Hints for Reading this Section: o To help remember the order of the English monarchs, try to remember the â€Å"Cromwell Sandwich. As with any good sandwich, it is named after the meat, which goes in the middle. Surrounding the meat is usually CHeese. In the Cromwell Sandwich then, the buns equal James (James I on top, or first, and James II bottom, or last) and the CHeese equals CHarles (Charles I on top of the meat or first, and then Charles II below the meat or second). Thus the order goes Bun (James I), Cheese (Charles I), Meat (Oliver Cromwell), Cheese (Charles II), and Bun (James II). I don’t know, it helps me. : ) 1. Describe England under the Restoration of the Monarchy (what powers did King have?Religion? )- began in 1660 when the English , Scottish and Irish monarchies were all restored under Charles II. 2. Religion and the monarchy became an increasingly touchy subject in Restoration-era England. Discuss how the following acts/events display the conflict developing between monarch and Parliament:  · Clarendon Code= The Clarendon Code was a series of four legal statutes passed between 1661-1665 which effectively re-established the supremacy of the Anglican Church after the interlude of Cromwell's Commonwealth, and ended toleration for dissenting religions. Declaration of Indulgence= Charles II of England's attempt to extend religious liberty to Protestant nonconformists and Roman Catholics in his realms, by suspending the execution of the penal laws that punished recusants from the Church of England. Charles issued the Declaration on 15 March 1672.  · Test Act (note who this one was aimed at)= were a series of English penal laws that served as a religious test for public office and imposed various civil disabili ties on Roman Catholics and Nonconformists.The principle was that none but persons professing the Established Church were eligible for public employment, and the severe penalties pronounced against recusants, whether Catholic or Nonconformist, were affirmations of this principle. In practice nonconformists were often exempted from some of these laws through the regular passage of Acts of Indemnity 3. Both politics and religion played a large role in the forced removal of James II as King of England. Summarize the role of each, and note what served as the immediate cause of the Glorious Revolution. . Describe the political philosophy of John Locke found in his Two Treatises on Government. 5. How could one point to the Glorious Revolution and the English Bill of Rights as early successes for the history of Democracy? Helpful Hints for Reading this Section: o When trying to keep straight Thomas Hobbes and John Locke, think about what they stood for in regards to man’s â€Å"nat ural state. † Hobbes, who believed man was horrible, awful, wicked nasty and cruel, was a â€Å"Hater. † Thus, the first letter of his name and how he felt both match up (Hobbes; Hater).Locke, who said man was naturally good, was all about the Love. Thus, his name and beliefs are also connected by a first-letter principle (Locke; Love). Chapter 13 – France 1. Explain how the theory of Divine Right strengthened Louis XIV’s power as king. How did his saying â€Å"L’etat, c’est moi† reflect the teachings of Bishop Bossuet? – The Divine Right strengthened Louis XIV’s by saying that dictators, nobles, and parliament don’t have power over the people. â€Å"L’etat, c’est moi† means â€Å"I am state† reflected the teaching of Bishop Bossuet by saying God has the almighty power. 2.The Palace at Versailles is an integral part of Louis XIV’s strong reign. Answer the following with regard to life in Versailles:  · To â€Å"domesticate† something means to make it tame, or to train it to be useful to humans. In what ways did Louis XIV â€Å"domesticate the nobility? †  · Why did Louis XIV order nobles to follow such trivial social rules and elaborate social functions at Versailles? Louis ordered nobles to follow trivial social rules and elaborate social functions at Versailles because he wanted them to have less power, so they won’t be a threat to him. What types of people did Louis choose to head his government agencies? Why did he prefer to use them instead of the nobles, as was the case in other times and other countries? Louis XIV replaced the princes who had previously held positions as ministers with new aristocrats who feared him more. This gave Louis XIV, the King of France more power. 3. Explain the ways in which Jean-Baptiste Colbert made France’s economy superior to any other nations’ in the 17th century. How did the marqu is of Louvois dramatically increase the effectiveness of France’s military? 4.Louis XIV’s France became so powerful that no one nation could likely stand up and defeat the French. For that reason, Louis chose to involve himself in a series of wars that would eventually break his nation’s finances. Include each of Louis XIV’s war in a timeline that shows the following:  · Years fought  · Reasons for War  · How foreign nations worked to Balance out French power  · Outcomes of War 5. Why did Louis XIV feel it was necessary to revoke the Edict of Nantes? What impact did this move have on his nation? Huguenots as a threat to his power.Huguenots through his reign were hostile to the crown and launched revolts. The revocation of the Edict of Nantes caused large numbers of them to flee to other protestant countries and establish themselves there, weakening the French economy. 6. Explain why the War of Spanish Succession (and the subsequent Treaty of Utrec ht) makes a fitting culmination to a. the grand wishes of Louis XIV’s plans for domination of Europe, and; b. the principle of Balance of Power used to stop Louis XIV and contain French power 7. Consider Louis XIV’s Legacy. Create a list of positive and negative aspects of Louis XIV’s legacy.Then, create a thesis statement that evaluates the impact of Louis XIV on French history. Be sure your thesis 1) takes a side/has direction, and 2) does not simply list 3 things Louis did good and/or bad. o Intendants were government officials (royal civil servants,’ according to the text) in charge of oversight duties across all areas of French society. They were responsible for making sure that all tax money collected by regional tax officials went directly to the king (as opposed to some going to the collector’s pocket), and to oversee the training and discipline of the French military.They studied efficiency in production and carried their lessons to emergi ng French industries. Possibly the most important aspect of the intendants was the fact that they were not of strong noble birth. Richelieu and Louis XIV realized that using nobles to do the most important jobs of government ran counter to the idea of centralizing full power in the hands of the crown. For that reason, the intendants typically came from middle class backgrounds, people who did not possess large lands of their own. Thus, these peoples’ success in life was fully-dependent upon the king.If they became corrupt, another intendant would call them out and they would lose their job and what was likely their only chance to be very successful in life. This created an incredibly driven, obedient and loyal bureaucracy for the French kings. o During the Counter-Reformation, several different groups of Catholics came up with various ways they saw as correct in regards to Catholicism. The Jesuits were likely the most famous and most successful, as they traveled to every coas tline and set up schools and monasteries in most all prominent nations.One of the Jesuits’ key messages to people was, â€Å"we can help save you. † By this, the Jesuits promoted the fact that people who were Catholic could be guaranteed salvation, so long as they followed the 7 sacraments and did as their priest told them. Another group, the Jansenists, saw this as untrue. Much like Luther and Calvin, the Jansenists believed that there was nothing people could do on earth to guarantee their salvation. They still believed in following all 7 of the Sacraments and other areas of Catholic doctrine, but they said that getting into heaven had to be a â€Å"gift† of God’s grace.A big religious dispute broke out within the Catholic church, and the influential Jesuits led an out-cry of opposition against the Jansenists. The point the text is trying to make with the Jansenists is that their group offered a form of Catholicism that included aspects of many Protesta nt religions (role of faith/grace as gift for salvation etc. ), which potentially could have kept French Huguenots (French Protestants) within the Catholic faith and kept them within France. When Louis XIV outlawed Jansenists, he made legal only the strongly anti-Protestant Jesuit Catholics, who began pushing for strong laws against Protestants.

Studying: Then and Now

Studying: Then and Now One June day, I staggered into a high school classroom to take my final exam in United States History IV. I had made my usual desperate effort to cram the night before, with the usual dismal results—I had gotten only to page 75 of a 400-page textbook. My study habits in high school, obviously, were a mess, but in college, I've made an attempt to reform my note-taking, studying, and test-taking skills. The first thing I tried to improve in college was note-taking. I took notes in high school classes but I often lost interest as compared to my note- taking at college.Note taking in high school classes often began with doodling, drawing Martians, or seeing what my signature would look like if I married the guy in the second row. Now, however, I try not to let my mind wander, and I pull my thoughts back into focus when they begin to go fuzzy. In high school, my notes often looked like something written in Arabic. In college, I've learned to use a semi-print writing style that makes my notes understandable. When I would look over my high school notes, I couldn't understand them.There would be a word like â€Å"Reconstruction,† then a big blank, then the word â€Å"important. † Weeks later, I had no idea what reconstruction was or why it was important. I've since learned to write down connecting ideas, even if I have to take the time to do it after class. Taking notes is one thing I've really learned to do better since high school days. Another area that I needed to concentrate on was my method of studying. Ordinary studying during the term is another area in which I've made changes. In high school, I let reading assignments go.I told myself that I'd have no trouble catching up on 200 pages during a 15-minute ride to school. College courses have taught me to keep pace with the work. Otherwise, I feel as though I'm sinking into a quicksand of unread material. When I finally read the high school assignment, my eyes would run over the words, but my brain would be plotting how to get the car for Saturday night. Now, I use several techniques that force me to really concentrate on my reading. I also changed the way of study for taking testsIn addition to learning how to cope with daily work, I've also learned to handle study sessions for big tests. My all-night study sessions in high school were experiments in self-torture. Around 2:00 A. M. , my mind, like a soaked sponge, simply stopped absorbing things. Now, I space out exam study sessions over several days. That way, the night before can be devoted to an overall review rather than raw memorizing. Most important, though, I've changed my attitude toward tests. In high school, I thought tests were mysterious things with completely unpredictable questions.Now, I ask instructors about the kinds of questions that will be on the exam, and I try to â€Å"psych out† the areas or facts instructors are likely to ask about. These practices really work, and f or me they've taken much of the fear and mystery out of tests. Since I've reformed, note-taking and studying are not as tough as they once were, and I am beginning to reap the benefits. As time goes on, my college test sheets are going to look much different from the red-marked tests of my high school days. Outline Introduction:Thesis Statement: My study habits in high-school, obviously were a mess, but in college, I’ve made an attempt to reform my note-taking, studying and test-taking skills. Topic sentence 1: I took notes in high school classes but I often lost interest as compared to my note- taking at college 1. Interest 2. Note writing 3. Connection Topic Sentence 2: Ordinary studying during the term is another area in which I've made changes. 1. Pace 2. Concentration Topic Sentence 3: In addition to learning how to cope with daily work, I've also learned to handle study sessions for big tests. 1. Spacing 2. Attitude Conclusion

Madness in Hamlet

DRAFT- NOT MLA-LACKS FORMATTING/REFERENCES Hamlet-Melancholy, Madness and Sanity Hamlet, a play by William Shakespeare, is as much a mystery as a tale about depression, madness and sanity. Shakespeare reveals how the scourge of corruption and decay rapidly spread; and the emotional consequences that follow. Insanity, madness and depression are as intolerable as corruption and deceit; and just as intertwined. The play makes one ponder if it is possible to be sane in an insane world full of treachery, revenge, incest, and moral corruption?By examining the themes of melancholy, madness and sanity in Hamlet, Shakespeare details his character’s descent from depression to madness. Additionally, Hamlet’s psychological state can be analyzed by utilizing modern psychological diagnoses, in order to understand his mental state. Throughout the story, Hamlet exists in a melancholy state, â€Å"essentially not in madness, / But mad in craft† (3. 4. 204-205). Hamlet states to H oratio â€Å"as I perchance hereafter shall think meet/ To put an antic disposition on† in order to deceive the king that he is insane (Act I, Scene V, Line 190).However, was Hamlet acting or was he already mentally disturbed? Did Hamlet go mad in the end, or was Hamlet insane from the start of the play, and his mental condition only worsened as the play unfolded? The world in which Hamlet existed appears hostile. The king is a murderer; his mother the queen lusts after her deceased husband’s brother; friends spy and deceive one another; and Hamlet’s lover Ophelia literally loses touch with reality. Hamlet believes that only suicide can free him from his misery. Hamlet is not the only person struggling with depression.From the beginning, Francisco says, â€Å"’Tis bitter cold, / And I am sick at heart†. Marcellus states that â€Å"Something is rotten in the state of Denmark† (). Fear is spread by the ghost of King Claudius wandering the city streets. Reoccurring themes of corruption result in Hamlet stating â€Å"The dram of evil / Doth all the noble substance of a doubt / To his own scandal† (Shakespeare 51). One evil person can contaminate an entire kingdom. As the kingdom decays, emotional trauma increases. Hamlet concludes that the world â€Å"’Tis an unweeded garden / that grows to seed.Things rank and gross in nature / Possess it merely ( ). The weeds represent decay in a world of evil and sin. The â€Å"things† are symbolic of man and his temporary dominance over himself, his fellow man and nature. Pessimism permeates the screenplay and the reader is led to connect the dysfunction with the resulting mental states of depression that infect Hamlet. Hamlet’s psychological status can also be analyzed from a modern perspective. Today we have the diagnostic tools to identify and treat the disorders that afflicted Hamlet.Though Hamlet presents as melancholy throughout the play, there are th ree additional disorders that afflict Hamlet including post-traumatic stress disorder, schizophrenia, and bipolar depression. Three traumatic events contribute to Hamlet’s descent into madness: the death of his father, the incestuous marriage between his uncle and mother, and the discovery of his father’s murder by Claudius. These emotional traumas contributed to Hamlet suffering from what we would now diagnose as Post-traumatic stress disorder.Post-traumatic stress disorder (PTSD) is characterized by the inability to properly construct emotional appropriate responses to situations and an emotional blunting. Throughout the play, Hamlet had issues with people in his social circle. Hamlet becomes enraged with Laertes because of his dramatic sorrow over the death of Ophelia (Act V, Scene I). Soon thereafter, Hamlet realizes his reaction to Laertes was inappropriate and later apologizes to Laertes. Hamlet demonstrates impulsive behavior and lashes out at people without con sidering the ramifications.In addition to Laertes, Ophelia, Hamlet’s beloved, is also not immune to his rage. Hamlet is susceptible to highly reactive emotional responses, and suicidal ideation is mentioned throughout the play. Symptoms of PTSD often include changes in self-perception, relationship stressors, and frequently revenge fantasies. Hamlet’s emotional state deteriorates over the duration of the play. These changes include increased helplessness and the inability to make decisions as he becomes more ill. Hamlet exhibits other PTSD symptoms. Hamlet feels inferior to Fortinbras and Horatio, and in his attempts to kill Claudius.Hamlet also isolates himself and becomes more paranoid as the play progresses. Not that being paranoid would be an inappropriate emotional response to the environment he is in, but his paranoia takes over. Eventually Hamlet becomes obsessed with revenge after he is compelled by his father’s ghost to avenge his death. Symptoms of sch izophrenia also present in Hamlet. The symptoms of schizophrenia can include hallucinations, isolation from others, a loss of reality and paranoia. While berating his mother about her rapid remarriage, suddenly Hamlet begins addressing an invisible specter. â€Å"What would your gracious figure? [QUEEN] Alas, he’s mad! † (Act III, Scene IV, Line 116-117). Hamlet has visions, or hallucinations, of his dead father’s ghost. The ghost was seen once by Horatio and some guards at the start of the play, but not by anyone else. The appearance of the ghost may have been reality or a hallucination. However, the second ghost to appear, that only Hamlet could see and hear, was confirmation of Hamlet’s insanity. Schizophrenia is often symptomatic with visual, but more commonly auditory hallucinations. After the death of Hamlet’s father, Hamlet began to withdraw and became reclusive in nature.He didn’t combat be forced to leave to boarding school, he spen t less time with Ophelia, and he became very isolated. Hamlet appears to have been in the initial stages of schizophrenia where the real and unreal become blurred, but he is not yet incapacitated by his disorder prior to his murder. Perhaps if he didn’t die young the disease would have progressed. Even Polonius questioned Hamlet’s state-of-mind when he said, â€Å"How pregnant sometimes his replies are – a happiness that often madness hits on† (Act II, Scene II, Lines 213-215). Throughout the play, evidence of bipolar disorder (also known as manic depression) is identified.It is hard to distinguish if Hamlet is bipolar or having a schizophrenic break since both diseases have common behavioral features. Bipolar disorder is known for manic or escalated behaviors coupled with severe depression when the mania recedes. Manic behaviors include pressured (or rapid) speech, grandiose beliefs, insomnia, and hyper or frantic behavior, followed by periods of extreme anergic depression that is emotionally and physically debilitating. Hamlet escalates between periods of excitement and kinetic activity to states of absolute misery and unproductivity.Hamlet procrastinates, but it may be due to the depressive period he cycles in and out of. Hamlet’s soliloquies all contain themes of suicide and hopelessness (depression), but his manic states are more difficult to define. Hamlet creates a plan to prove Claudius’s guilt while justifying his own rage and revenge. When Hamlet discovers that the players are coming to the castle, he rapidly goes from melancholy to a state of over-excitement. He becomes highly focused and implements a detailed plan to kill Claudius. Hamlet quickly abandons his plan and slips back into his depression (sparing Claudius).If Hamlet was able to execute all of the schemes and plans he devises, the play would be chaotic. Another incident suggests another bipolar episode when Hamlet, on the boat to England, facilita tes the execution of Rosencrantz and Guildenstern; and also plans an attack on a pirate ship for revenge. Grandiose thoughts are a common feature while in a manic phase. Hamlet quickly forgets or fails to follow through with his plan and falls back into a depressive state. Hamlet’s manic depression is referred to as rapid cycling because the mania and depressive states change quickly.Ultimately, Hamlet may not be procrastinating at all like is assumed throughout the play. Hamlet may be incapable of fulfilling his grandiose plans while in a depressive state. In Hamlet’s time people had no concept of mental illness or mood disorders. People believed that behavior was deliberate and labeled unacceptable behaviors as character defects. Hamlet was a deeply disturbed individual. Author Gertrude Morin suggests that a cognitive approach be used to understand Hamlet. Hamlet, Morin says, is â€Å"a portrayal of a tortured, depressed young man who loses his way in the labyrinth of his negative thoughts. He not only suffered from negativity, mania and depression; the environment in which he lived was highly dysfunctional for anyone’s mental health, let alone an ill man. At the beginning of the play, Hamlet is depressed but the events that develop during the play exacerbate his descent into madness. Not many people in the Kingdom of Denmark escape the toxicity of the world in which they lived. Ophelia may have suffered the most, and was also tormented by Hamlet’s illness, and eventually by saw no other option but to end her life. Depression and despair seemingly became as contagious as the small pox- and just as untreatable.Hamlet’s mental state and the events in his life may have finally drove him to madness. The reader may first believe that Hamlet is faking his dementia, when in fact, his mental dysfunctions were both organic (of biological origin)and very real. Among the mentally ill it is common to see a mentally vulnerable individu al succumb to the pressures of his environment. That person may have been able to keep their illness in check in a stable environment, but when external stressors mount- they may not have the coping behaviors to cope and they escalate out of control.Suicide, in our modern world, is often a result of life stressors the vulnerable person could no longer manage. Had Hamlet not been murdered, his suicidal impulses may have caused him to self destruct. Psychiatry was just emerging as a science during the time Shakespeare wrote Hamlet. Robert Burton, the most famous author on melancholy from the Renaissance, first published his Anatomy of Melancholy in 1621. Burton, who was afflicted with melancholy himself suggested treatments â€Å"from taking hellebore to boring a hole in the skull to let out the ‘fuliginous vapours,’† while adding his consolations (Burton ).Had Hamlet received the services of Burton, it is probable that if the hellebore didn’t work, the hole in the skull would have left Hamlet disabled or dead. In conclusion, the question persists: Is it possible to be sane in an insane world full of treachery, revenge, incest, and moral corruption? It appears that the majority of people is somewhat resilient and can cope with adversity to a certain degree. However, there are people like Ophelia and Hamlet who are vulnerable to chaotic conditions. These people tend to internalize the trauma and don’t have the coping mechanisms to deal with intense emotional situations.It is fascinating that Shakespeare had the insight to connect intolerable conditions with states of mental vulnerability. ‘Madness’ was a catch-all term for all mental disorders in Shakespeare’s time, but through his dialogue and scenery, he was able to identify numerous mental states of dysfunction- centuries before we had terminology to describe these illnesses in diagnostic terms. Shakespeare’s acuity in describing a mentally-ill princ e is why Hamlet has endured over time. The issues, diseases, desires, fears and dreams remain constant over time among humanity. To be, or not to be’ will always be relevant in more ways than life and death. Our views on mental illness and physical ailments that relate to Melancholia are endless, and therefore so is our application of each possibility to Shakespeare’s work of Hamlet. Sources: Diagnostic Manual DMV-IV Burton, Robert. The Anatomy of Melancholy. Ed. Floyd Dell and Paul JordanSmith, NewYork: Tudor Publishing Co. , 1941. Morin, Gertrude. â€Å"Depression and Negative Thinking: A Cognitive Approach to Hamlet. † Mosaic: A Journal for the Interdisciplinary Study of Literature 25. 1 (1992): 112.

Prior Knowledge Deficit Essay

Prior knowledge comes from visual experiences, seeing those mental pictures of a subject. Prior knowledge comes before understanding what readers read, or understanding the subject. A deficit in prior knowledge inhibits student learning by taking away their educational experiences and replacing them with standardized testing. To begin with, prior knowledge plays an important role in enabling students to learn to read, and to read to learn. First, learning to read starts in kindergarten and continues through third grade, where teachers use books called primers. Primers teach students how to read because they tell stories about subjects that students already have visual experience with, such as family, friends, food, pets, or games. Since students know the basic information about such subjects, when they are being taught how to read, they understand the primers. Finally, when students enter fourth grade, they begin reading textbooks to learn about science and social studies. These students developed prior knowledge of science and social studies from kindergarten through third grade, when they attended field trips to zoos and museums, watched educational films, and completed science projects. If they learned this basic information about the natural world of science, and about the human world of social studie s, students are able to understand their textbooks. As a result, they can add new knowledge to further their education. Additionally, many students struggle with the prior knowledge deficit because of standardized testing. First of all, standardized testing takes time away from educational experience by using all the time and money on the tests. Teachers use the time on teaching students how to become test takers, instead of educating them in different experiences of the world. In the past, schools would take students on field trips to teach them mostly about science, and social studies. However, such field trips cost a lot of money; money that is currently being used on standardized tests. On one hand, traditional education consisted of students reading and writing to understand and explain the knowledge of the world. On the other hand, now day’s students are test takers, and they use their reading and writing skills for the test, just like knowledge is used to memorize all the information without understanding it. Teachers need to focus on this method because they are evaluated by the student s test scores. To summarize, prior knowledge deficit hinders students learning by taking away their educational experiences and focusing their time on standardized testing. A solution to the prior knowledge deficit is for teachers to dedicate more time on teaching students about subjects that will be used later on in further education, rather than teaching students how to become test takers.