The History of Pi Essay Example

The History of Pi Essay Example The History of Pi Essay The History of Pi Essay Before I talk about the history of Pi I want to explain what Pi is. Websters Collegiate Dictionary defines Pi as 1: the 16th letter of the Greek alphabet 2 a: the symbol pi denoting the ratio of the circumference of a circle to its diameter b: the ratio itself: a transcendental number having a value to eight decimal places of 3. 14159265 A number can be placed into several categories based on its properties. Is it prime or composite? Is it imaginary or real? Is it transcendental or algebraic? These questions help define a numbers behavior in different situations. In order to understand where Pi fits in to the world of mathematics, one must understand several of its properties pi is irrational and pi is transcendental. A rational number is one that can be expressed as the fraction of two integers. Rational numbers converted into decimal notation always repeat themselves somewhere in their digits. For example, 3 is a rational number as it can be written as 3/1 and in decimal notation it is expressed with an infinite amount of zeros to the right of the decimal point. 1/7 is also a rational number. Its decimal notation is 0. 142857142857†¦, a repetition of six digits. However, the square root of 2 cannot be written as the fraction of two integers and is therefore irrational. For many centuries prior to the actual proof, mathematicians had thought that pi was an irrational number. The first attempt at a proof was by Johaan Heinrich Lambert in 1761. Through a complex method he proved that if x is rational, tan(x) must be irrational. It follows that if tan(x) is rational, x must be irrational. Since tan(pi/4)=1, pi/4 must be irrational; therefore, pi must be irrational. Many people saw Lamberts proof as too simplified an answer for such a complex and long-lived problem. In 1794, however, A. M. Legendre found another proof which backed Lambert up. This new proof also went as far as to prove that Pi^2 was also irrational. In the long history of the number Pi, there have been many twists and turns, many inconsistencies that reflect the condition of the human race as a whole. Through each major period of world history and in each regional area, the state of intellectual thought, the state of mathematics, and hence the state of Pi, has been dictated by the same socio-economic and geographic forces as every other aspect of civilization. The following is a brief history, organized by period and region, of the development of our understanding of the number Pi. A transcendental number is one that cannot be expressed as a solution of ax^n+bx^(n-1)+ +cx^0=0 where all coefficients are integers and n is finite. For example, x=sqrt(2), which is irrational, can be expressed as x^2-2=0. This shows that the square root of 2 is nontranscendental, or algebraic. It is very easy to prove that a number is not transcendental, but it is extremely difficult to prove that it is transcendental. This feat was finally accomplished for Pi by Ferdinand von Lindemann in 1882. He based his proof on the works of two other mathematicians: Charles Hermite and Euler. In 1873, Hermite proved that the constant e was transcendental. Combining this with Eulers famous equation e^(i*Pi)+1=0, Lindemann proved that since e^x+1=0, x is required to be transcendental. Since it was accepted that i was algebraic, Pi had to be transcendental in order to make i*Pi transcendental. Now that I have explained what Pi is and several of its properties, lets look at its history. In ancient times, Pi was discovered independently by the first civilizations to begin agriculture. Their new sedentary life style first freed up time for mathematical pondering, and the need for permanent shelter necessitated the development of basic engineering skills, which in many instances required a knowledge of the relationship between the square and the circle (usually satisfied by finding a reasonable approximation of Pi). Although there are no surviving records of individual mathematicians from this period, historians today know the values used by some ancient cultures. Here is a sampling of some cultures and the values that they used: Babylonians 3 1/8, Egyptians (16/9)^2, Chinese 3, Hebrews 3 (implied in the Bible, I Kings vii, 23). The first record of an individual mathematician taking on the problem of Pi (often called squaring the circle, and involving the search for a way to cleanly relate either the area or the circumference of a circle to that of a square) occurred in ancient Greece in the 400s B. C. (this attempt was made by Anaxagoras). Based on this fact, it is not surprising that the Greek culture was the first to truly delve into the possibilities of abstract mathematics. The part of the Greek culture centered in Athens made great leaps in the area of geometry, the first branch of mathematics to be thoroughly explored. Antiphon, an Athenian philosopher, first stated the principle of exhaustion (click on Antiphon for more info). Hippias of Elis created a curve called the quadratrix, which actually allowed the theoretical squaring of the circle, though it was not practical. In the late Greek period (300s-200s B. C. ), after Alexander the Great had spread Greek culture from the western borders of India to the Nile Valley of Egypt, Alexandria, Egypt became the intellectual center of the world. Among the many scholars who worked at the University there, by far the most influential to the history of Pi was Euclid. Through the publishing of Elements, he provided countless future mathematicians with the tools with which to attack the Pi problem. The other great thinker of this time, Archimedes, studied in Alexandria but lived his life on the island of Sicily. It was Archimedes who approximated his value of Pi to about 22/7, which is still a common value today. Archimedes was killed in 212 B. C. in the Roman conquest of Syracuse. In the years after his death, the Roman Empire gradually gained control of the known world. Despite their other achievements, the Romans are not known for their mathematical achievements. The dark period after the fall of Rome was even worse for Pi. Little new was discovered about Pi until well into the decline of the Middle Ages, more than a thousand years after Archimedes death.

St. Lawrence University Admissions and Acceptance Rate

St. Lawrence University Admissions and Acceptance Rate St. Lawrence University in Canton, New York is a somewhat selective school. Less than half of applicants are accepted each year. Students with strong grades and an impressive application, however, have a decent chance of being admitted. Applicants are not required to submit SAT or ACT scores. They will need to send in high school transcripts, a personal essay, and several letters of recommendation. For help with the application process, feel free to contact the admissions office at St. Lawrence. Calculate your chances of getting in with this free tool from Cappex. Admissions Data (2016) St. Lawrence University Acceptance Rate: 43  percentSt. Lawrence University is test-optionalGPA, SAT and ACT graph for St. LawrenceWhat these SAT numbers meanWhat these ACT numbers mean St. Lawrence University Description St. Lawrence University can boast that it has been coeducational with a progressive teaching philosophy since its founding in 1856. The schools name comes from the nearby St. Lawrence river, the site of both crew practice and academic research. The university is located in Canton, New York, a town not far from Potsdam. Explore the campus with the  St. Lawrence University Photo Tour. The university has a graduate program in education, but SLUs primary focus is at the undergraduate level. With an 12 to 1  student/faculty ratio, students are sure to get a lot of interaction with the faculty. Study abroad, community service, and sustainability are all important parts of St. Lawrences identity, and starting in the fall of 2012, some first-year students got the opportunity to spend their first college semester in London, and the university has had a second-semester program in France for several years. On the athletic front, the St. Lawrence Saints compete in the NCAA Division III Liberty League for most sports. The university has 32 intercollegiate teams and made my list of  top equestrian colleges. Enrollment (2016) Total Enrollment: 2,464  (2,377 undergraduates)Gender Breakdown: 45 percent male; 55 percent female99 percent full-time Costs (2016-17) Tuition and Fees: $51,200Books: $750 (why so much?)Room and Board: $13,190Other Expenses: $900Total Cost: $66,040 St. Lawrence University Financial Aid (2015 -16) Percentage of New Students Receiving Aid: 100  percentPercentage of New Students Receiving Types of AidGrants: 100 percentLoans: 55 percentAverage Amount of AidGrants: $32,191Loans: $7,775 Academic Programs Most Popular Majors:  Art, Biology, Economics, English, Fine Arts, History, Mathematics, Political Science, Psychology, Social Science, Sociology What major is right for you?  Sign up to take the free My Careers and Majors Quiz at Cappex. Graduation and Retention Rates First Year Student Retention (full-time students): 90  percentTransfer-out Rate: 13 percent4-Year Graduation Rate: 81  percent6-Year Graduation Rate: 84  percent Intercollegiate Athletic Programs Mens Sports:  Skiing, Football, Golf, Ice Hockey, Riding, Rowing, Lacrosse, Basketball, Baseball, Cross CountryWomens Sports:  Skiing, Riding, Soccer, Softball, Basketball, Field Hockey, Ice Hockey, Cross Country If You Like St. Lawrence University, You May Also Like These Schools Ithaca College: Profile | GPA-SAT-ACT GraphUniversity of Vermont: Profile | GPA-SAT-ACT GraphSkidmore College: Profile | GPA-SAT-ACT GraphSyracuse University: Profile | GPA-SAT-ACT GraphTrinity College: Profile | GPA-SAT-ACT GraphUniversity of Rochester: Profile | GPA-SAT-ACT GraphAlfred University: Profile | GPA-SAT-ACT GraphBoston University: Profile | GPA-SAT-ACT GraphCornell University: Profile | GPA-SAT-ACT GraphBowdoin College: Profile | GPA-SAT-ACT GraphBates College: Profile | GPA-SAT-ACT GraphHobart William Smith Colleges: Profile | GPA-SAT-ACT Graph Data Source: National Center for Educational Statistics

Factionalism in American Politics Essay Example | Topics and Well Written Essays - 500 words

Factionalism in American Politics - Essay Example Madison underscores the importance of breaking or controlling the factions and the ‘factious spirit’ that make public administration a complex affair. Factions exist in the American politics and continue to pose a problem especially in the development of public policies. The factions are mainly caused by the inequality that exists in the distribution of wealth and property.   The passing of The Healthcare Reform Bill brought to the fore the various factions at the various stages of the passing of the bill. The healthcare bill while designed for the benefit of the majority of the Americans faced major hurdles during the legislative process. The factions that opposed the bills include those who were opposed to the Stupak-Pitts Amendment provision in the bill. The pro-choice faction in the house argued that the Stupak-Pitts Amendment imposes restrictions on the right of women to access reproductive health services. Women should be guaranteed these rights by law. The other faction in the healthcare debate is those that were in support of the Stupak-Pitts Amendment. This faction argues that abortion should be restricted, and proposes a further restriction on funding on abortion. In the process of enacting the legislation there was a faction that was against the passing of the healthcare reform bill as they had some concerns about impact that the bill will have on the illegal immigrants’ attempts to access healthcare services (‘Obama Hails Vote on Health Care as Answering ‘the Call of History’).The factions that were at play in the development of the healthcare reform bills have brought about a situation in which the proponents, as well as the opponents of healthcare reforms policies, give different explanations to their positions on healthcare reform.

Abuse of the Elderly Research Paper Example | Topics and Well Written Essays - 1500 words

Abuse of the Elderly - Research Paper Example Abuse and neglect are too often visited upon older individuals who have lost some degree of their independence, and many areas do not have the programs necessary to effectively counter this threat on the community level. There is even abuse and neglect that goes on with healthcare facilities, nursing homes, and home environments, and this is perhaps the most insidious sort of abuse. In some cultures, the elderly are prized and honored above all other citizens and groups, but unfortunately this is not the case in the present culture. Elder abuse can happen in healthcare or nursing home situations, or it could happen at the hands of the elder’s own family and support system. In the US today, older individuals are more likely to be seen as being in the way of the young than as role models who should be exalted because of their aged wisdom. Presently, however, many individuals are treated harshly by healthcare facilities and even their own kin, making elder abuse a significant pro blem in society. Also, in terms of economic scales, older individuals who are of a lower socio-economic class are more likely to be abused or mistreated. These people may lack a stable caregiver. There are those who say the aged should be used as models who should be exalted because of their aged wisdom. Presently, however, many individuals are treated harshly by healthcare facilities and even, as the proposed report will show, their own families and support systems, making elder abuse a significant problem in society.

The film of Step Mom by Gigi Levangie Essay Example for Free

The film of Step Mom by Gigi Levangie Essay A Comparison between my improvisation of Generation gap by us and the film of Step Mom by Gigi Levangie The two pieces of improvisation are good for a comparison because our improvisation is about two single parents one who lives with their mother moving in and how the children react to each other. Step mom is however slightly different as it is about parents who have just got divorced and are introducing the fathers new girlfriend in to the childrens lifes too. So our play is about the reaction towards the children and the film is about the reaction between the step mom. It is also very good from comparison as the family with the Grandmother in Generation gap previous relationship ended by death and in Step mom it ends with a death by the mother of the children dying from cancer. They are similar, as they are both sets in present day and shows how people react when introducing new members in to their lives. They may react snappy or just miserable around them and whether it is possible for them to change their feelings as in step mom the girl Anna was snappy and miserable at different times but by the end of the play she was able to accept the step mom. In our improvisation the teenager had similar symptoms but in each others presence they just argued and the younger boy would just wind the older sister but was upset as the sister was able to be mean to. So in this play the relations seem to worsen between the characters and it even affected the way the parents felt towards each other to some aspects as there children are more important to them and are likely to take their childs side and end up arguing. This also happened in the movie, as the mother would get annoyed when the step mom arrived late or did things wrong as she hasnt as much experience as her in being a mother. They both showed signs that the children meant more to them as they stick up for their child always and end up in argument in our play In Step Mom The father talks to the children and says relationships can change. The way he acts around his girlfriend suggests that it would be hard to make a choice as he loves both his girlfriend and children but you see that he would probably put his children first but this isnt certain. But they could be equal you dont know whether he means this or whether hes mocking the child in saying its impossible to stop loving your child. This is similar in both plays showing children is most important in keeping familys together.

Sakis Stories and the Impression they Create about Adult-Child Relatio

Saki's Stories and the Impression they Create about Adult-Child Relationships In order to answer this question I am going to be focusing on three of Saki?s stories; the story-teller, the lumber room and the open window. Children at the time Hector Munro (Saki) was writing these stories would have had very vivid imagination; this is shown in Saki's story 'The Open Window' when the niece makes up the saga of Mr Sappleton's death, illustrates how imaginative children can be, and that adults are very easily fooled. Fooling adults is a key theme in Saki's stories; another time when it occurs is in the story 'The Lumber Room' when Nicholas dupes his aunt into believing that he is somehow trying to get into the gooseberry garden, but instead has no intention of doing so but does sn...

Between 1820 and 1860 there were a few fundamental differences between the economies of the North and South Essay

?Between 1820 and 1860 there were a few fundamental differences between the economies of the North and South. How far do you agree? Between 1820 and 1860 there were problems in America that resulted in the civil war in 1861, the economy affected the North and South during the civil war, and some major differences helped win the civil war. However this does not mean that the economy had many fundamental differences. For example the methods of earning money were different, but overall both earned roughly the same amount of money. Firstly, the main difference was that the North relied much more on money from industry and manufacturing, as they had more connections with foreign countries, and more factories were built. In the North in 1860 there were 74,000 factories producing two thirds, of goods for America. Whereas in the South there were only 2 major factors: Textiles factory in South Carolina and an Iron works in Virginia set up in 1840. Although they had fewer factories, the Iron works was vital, as it supplied the North with weaponry during the Civil war. In 1850 the South only produced 10% of the nations manufactured output. However the South did lack the industrialisation because their agricultural methods were so effective they had no need for change. The South was also very traditional and disliked change, whereas seven out of eight immigrants had chosen to settle in the North, and they had seen new ideas in other places in Europe so were welcome to change. This is shown by the South’s labour force being reduced from 82% to 81% over 60 years, on the other hand, the North’s labour force was reduced from 68% to 40% over the 60 years. This is however not a reflection on the North rapidly reducing agriculturally, because they still relied on agriculture, and a lot of the North was still rural. Around the urban areas, the population was increasing as towns and cities were developing swiftly. As the North had smaller but still prosperous farmers known as yeomen, they did not earn as much money through agriculture, because they did not have many or any slaves at all, so they could not produce the high quantities that the south did. Farmers that had lots of land were called plantation owners; there were many more plantations in the south, and when the cotton gin was created in 1793, there was a boom in cotton, and the benefit of having slaves increased. This made Southern plantation owners invest their capital in the Slaves, because they produced so much cotton, which was very lucrative. 55% of slaves worked in cotton which shows how much money was made out of it. The South needed to transport their goods to the North so that it could be transported abroad, but there was a lack of transport in South, as they only had 35% of the train tracks, which affected them in the Civil war. This meant it was fairly expensive to transport the cotton, but they still received a good amount of money if they lived near a train track. When the cotton arrived in the North the Lowell factory system meant the manufacturing of it was very cheap, which meant the North earnt a large margin on it. Although the danger of the South investing all their money in Slaves was that when slavery got banned, they lost a lot of money, which was partly why they were so keen to keep slavery. However only one quarter of the population owned a slave, as they were expensive (in 1860 they were $1800 which was double the price in 1820). The North’s economy was based on free labour whereas the South’s was based on slave labour, which meant that the North’s economy was more stimulated by the presence of workers with money who could buy products, which would make money for the company, which in turn would increase wages or hire more workers. The South’s economy was largely devoid of this benefit. Essentially, the economy in the South was weaker and only based on a few important exports (cotton, tobacco, and sugar). This made the South less diversified, as they focused their sales on cotton. Another difference is tariffs; in 1828 congress passes a controversial high protective tax. This majorly benefitted the North as they were the main distributors. However the South were very worried about this because they believed that if they stopped buying the expensive foreign goods, then foreign countries would stop buying their goods, as they would have tax on them too. This affected the economy until the tax was promised to be reduced after the law in 1833. In contrast the North and South didn’t have many fundamental differences. They both created a lot of money, and were not lacking at all, the only difference was the method in which was used to make the money. The economic diversification was similar too, as 10% of Northerners owned 68% of the wealth, and the wealth was dominated by the plantation owners, called oligarchies, as there were few free-soilers. 12% of the plantation owners owned half the slaves. This shows that there were few farmers, but the big plantations owned many slaves. Even though the North seems to be more industrial, the South also made efforts to modernise industrially, but they just lacked because of the closed opinions of the southerners. Also not many people in the North and the South owned slaves, it was more of a rare thing, and the main owners, owned a large amount of slaves. However there was a larger need for slaves in the South, because there was cheap labour force from Europe settling in the North which meant slaves were more expensive so were not deemed as important. Overall I think there were many differences between the North and South between 1820 and 1860. Some affected the outcome of the Civil war such as the transport boom, and the industrialisation in the North, but a fundamental point is that numerically they have equal amounts of income; there were just different methods to earning the money. However the differences were major; the amount of industrialisation, and amount of agriculture, the manufactured output, and the need for slaves, was all very different for the North and South. The main similarity was the economic diversification.