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.

Definition and Examples of Irony (Figure of Speech)

Definition and Examples of Irony (Figure of Speech) Irony is the use of words to convey the opposite of their literal meaning. Similarly, irony may be a statement or situation where the meaning is contradicted by the appearance or presentation of the idea. Adjective: ironic or ironical. Also known as  eironeia, illusio, and the dry mock. Three kinds of irony are commonly recognized: Verbal irony is a trope in which the intended meaning of a statement differs from the meaning that the words appear to express.Situational irony involves an incongruity between what is expected or intended and what actually occurs.Dramatic irony is an effect produced by a narrative in which the audience knows more about the present or future circumstances than a character in the story. In light of these different varieties of irony, Jonathan Tittler has concluded that irony has meant and means so many different things to different people that rarely is there a meeting of minds as to its particular sense on a given occasion (quoted by Frank Stringfellow in The Meaning of Irony, 1994). Etymology From the Greek, feigned ignorance Examples and Observations EarthA planet doesnt explode of itself, said drilyThe Martian astronomer, gazing off into the air- That they were able to do it is proof that highlyIntelligent beings must have been living there.(John Hall Wheelock, Earth)Kampenfeldt: This is a grave matter, a very grave matter. It has just been reported to me that youve been expressing sentiments hostile to the Fatherland.Schwab: What, me sir?Kampenfeldt: I warn you, Schwab, such treasonable conduct will lead you to a concentration camp.Schwab: But sir, what did I say?Kampenfeldt: You were distinctly heard to remark, This is a fine country to live in.Schwab: Oh, no, sir. Theres some mistake. No, what I said was, This is a fine country to live in.Kampenfeldt: Huh? You sure?Schwab: Yes sir.Kampenfeldt: I see. Well, in future dont make remarks that can be taken two ways.(Raymond Huntley and Eliot Makeham in Night Train to Munich, 1940)Gentlemen, you cant fight in here! This is the War Room.(Peter Sellers as President Merkin Muffley in Dr. Strangelove, 1964) It is a fitting irony that under Richard Nixon, launder became a dirty word.(William Zinsser)Irony in Mark Twains Novel Puddnhead WilsonDavid Wilson, the title character of Puddnhead Wilson, is a master of irony. In fact, his use of irony permanently marks him. When he first arrives in Dawsons Landing in 1830, he makes an ironic remark that the villagers cannot understand. Distracted by the annoying yelping of an unseen dog, he says, I wished I owned half of that dog. When asked why, he replies, Because I would kill my half. He does not really want to own half the dog, and he probably does not really want to kill it; he merely wants to silence it and knows killing half the dog would kill the whole animal and achieve the desired effect. His remark is a simple example of irony, and the failure of the villagers to understand it causes them immediately to brand Wilson a fool and nickname him puddnhead. The very title of the novel is, therefore, based on irony, and that irony is compounde d by the fact that Wilson is anything but a fool.(R. Kent Rasmussen, Blooms How to Write About Mark Twain. Infobase, 2008) Irony in Shakespeares Play Julius CaesarA classic example of irony is Mark Antonys speech in Shakespeares Julius Caesar. Although Antony declares, I come to bury Caesar, not to praise him, and declares that the assassins are honorable men, he means just the opposite.(Bryan Garner, Garners Modern American Usage. Oxford University Press, 2009)Uses and Characteristics of IronyIrony may be used as a rhetorical device to enforce ones meaning. It may be used . . . as a satiric device to attack a point of view or to expose folly, hypocrisy, or vanity. It may be used as a heuristic device to lead ones readers to see that things are not so simple or certain as they seem, or perhaps not so complex or doubtful as they seem. It is probable that most irony is rhetorical, satirical, or heuristic. . . .In the first place irony is a double-layered or two-story phenomenon. . . . In the second place, there is always some kind of opposition that may take the form of contradiction, incongruity, or incom patibility. . . . In the third place, there is in irony an element of innocence.(D.C. Muecke, The Compass of Irony. Methuen, 1969) An Age of IronyIt is sometimes said that we live in an age of irony. Irony in this sense may be found, for example, all throughout The Daily Show with Jon Stewart. Suppose you hear a political candidate give a terribly long speech, one that rambles on and on without end. Afterward, you might turn to a friend sitting next to you, roll your eyes, and say, Well, that was short and to the point, wasnt it? You are being ironic. You are counting on your friend to turn the literal meaning of your expression, to read it as exactly the opposite of what your words actually mean. . . .When irony works, it helps to cement social bonds and mutual understanding because the speaker and hearer of irony both know to turn the utterance, and they know that the other one knows they will turn the utterance. . . .Irony is a kind of winking at each other, as we all understand the game of meaning reversal that is being played.(Barry Brummett, Techniques of Close Reading. Sage, 2010)Irony as Mass TherapyIron y has always been a primary tool the under-powered use to tear at the over-powered in our culture. But now irony has become the bait that media corporations use to appeal to educated consumers. . . . Its almost an ultimate irony that those who say they dont like TV will sit and watch TV as long as the hosts of their favorite shows act like they dont like TV, either. Somewhere in this swirl of droll poses and pseudo-insights, irony itself becomes a kind of mass therapy for a politically confused culture. It offers a comfortable space where complicity doesnt feel like complicity. It makes you feel like you are counter-cultural while never requiring you to leave the mainstream culture it has so much fun teasing. We are happy enough with this therapy that we feel no need to enact social change.(Dan French, review of The Daily Show, 2001) Alanis Morissettes IronicAlanis Morissettes Ironic, in which situations purporting to be ironic are merely sad, random, or annoying (a traffic jam when youre late, a no-smoking sign on your cigarette break) perpetuates widespread misuse of the word and outrages irony prescriptivists. It is, of course, ironic that Ironic is an unironic song about irony. Bonus irony: Ironic is widely cited as an example of how Americans dont get irony, despite the fact that Alanis Morissette is Canadian.(Jon Winokur, The Big Book of Irony. St. Martins, 2007)Direct expression, with no tricks, gimmickry, or irony, has come to be interpreted ironically because the default interpretive apparatus says, He cant really mean THAT! When a culture becomes ironic about itself en masse, simple statements of brutal fact, simple judgments of hate or dislike become humorous because they unveil the absurdity, friendliness, and caution of normal public expression. Its funny because its true. Honestly. Were all upside d own now.(R. Jay Magill, Jr., Chic Ironic Bitterness. University of Michigan Press, 2007) Alan Bennett on IronyWere conceived in irony. We float in it from the womb. Its the amniotic fluid. Its the silver sea. Its the waters at their priest-like task, washing away guilt and purpose and responsibility. Joking but not joking. Caring but not caring. Serious but not serious.(Hilary in The Old Country by Alan Bennett, 1977)Thomas Carlyle on IronyAn ironic man, with his sly stillness, and ambuscading ways, more especially an ironic young man, from whom it is least expected, may be viewed as a pest to society.(Thomas Carlyle, Sartor Resartus: The Life and Opinions of Herr Teufelsdrockh, 1833-34)Irony DeficiencyIrony deficiency  is an  informal term for the inability to recognize, comprehend, and/or utilize  ironythat is,  a tendency to interpret  figurative language  in a  literal  way.-  Mobsters are reputedly huge fans of  The Godfather. They don’t see it as a tale of individual moral corruption. They see it as a nostalgia trip to better days for the mob.(Jonah Goldberg, The Irony of Irony.  National Review, April 28, 1999)-  Irony deficiency  is directly proportional to the strength of the political commitment or religious fervor. True believers of all persuasions are irony deficient. . . .Brutal dictators are irony deficienttake Hitler, Stalin, Kim Jong-il, and Saddam Hussein, a world-class vulgarian whose art collection consisted of kitsch paintings displayed unironically.(Jon Winokur,  The Big Book of Irony. Macmillan, 2007)-  Here is something ironic: We live at a time when our diets are richer in irony than ever before in human history, yet millions of us suffer from that silent crippler,  irony deficiency  . . . not so much a deficiency in irony itself, but an inability to utilize the abundance of irony all around us.(Swami Beyondananda,  Duck Soup for the Soul. Hysteria, 1999)-  Will people who detect a lack of irony in other cultures never stop to consider that this may be a sign of their own  irony deficiency? Maybe its defensible when the apes detect a lack of irony in Charlton Heston in  Planet of the Apes, but not when, say, Brits detect it in, say, Americans as a race . . .. The point of irony, after all, is to say things behind peoples backs to their faces. If you look around the poker table and cant tell who the pigeon is, its you.(Roy Blount, Jr., How to Talk Southern.  The New York Times, Nov. 21, 2004) The Lighter Side of IronyRachel Berry: Mr. Schuester, do you have any idea how ridiculous it is to give the lead solo in Sit Down, Youre Rocking the Boat to a boy in a wheelchair?Artie Abrams: I think Mr. Schue is using irony to enhance the performance.Rachel Berry: Theres nothing ironic about show choir!(Pilot episode of Glee, 2009)​Woman: I started riding these trains in the forties. Those days a man would give up his seat for a woman. Now were liberated and we have to stand.Elaine: Its ironic.Woman: Whats ironic?Elaine: This, that weve come all this way, we have made all this progress, but you know weve lost the little things, the niceties.Woman: No, I mean what does ironic mean?Elaine: Oh.​(The Subway, Seinfeld, Jan. 8 1992)Im aware of the irony of appearing on TV in order to decry it.(Sideshow Bob, The Simpsons)Math was my worst subject because I could never persuade the teacher that my answers were meant ironically.(Calvin Trillin)Lyn Cassady: Its okay, you can at tack me.Bob Wilton: Whats with the quotation fingers? Its like saying Im only capable of ironic attacking or something.​(The Men Who Stare at Goats, 2009) Pronunciation: I-ruh-nee