Characteristics of a Real Number

What is a number? Well that depends. There are a variety of different kinds of numbers, each with their own particular properties. One sort of number, upon which statistics, probability, and much of mathematics is based upon, is called a real number. To learn what a real number is, we will first take a brief tour of other kinds of numbers. Types of Numbers We first learn about numbers in order to count. We began with matching the numbers 1, 2, and 3 with our fingers.  Then we and kept going as high as we could, which probably wasnt that high. These counting numbers or natural numbers were the only numbers that we knew about. Later, when dealing with subtraction, negative whole numbers were introduced. The set of positive and negative whole numbers is called the set of integers. Shortly after this, rational numbers, also called fractions were considered. Since every integer can be written as a fraction with 1 in the denominator, we say that the integers form a subset of the rational numbers. The ancient Greeks realized that not all numbers can be formed as a fraction. For example, the square root of 2 cannot be expressed as a fraction. These kinds of numbers are called irrational numbers. Irrational numbers abound, and somewhat surprisingly in a certain sense there are more irrational numbers than rational numbers. Other irrational numbers include pi and e. Decimal Expansions Every real number can be written as a decimal. Different kinds of real numbers have different kinds of decimal expansions. The decimal expansion of a rational number is terminating, such as 2, 3.25, or 1.2342, or repeating, such as .33333. . . Or .123123123. . . In contrast to this, the decimal expansion of an irrational number is nonterminating and nonrepeating. We can see this in the decimal expansion of pi. There is a never ending string of digits for pi, and whats more, there is no string of digits that indefinitely repeats itself. Visualization of Real Numbers The real numbers can be visualized by associating each one of them to one of the infinite number of points along a straight line. The real numbers have an order, meaning that for any two distinct real numbers we can say that one is greater than the other. By convention, moving to the left along on the real number line corresponds to lesser and lesser numbers. Moving to the right along the real number line corresponds to greater and greater numbers. Basic Properties of the Real Numbers The real numbers behave like other numbers that we are used to dealing with. We can add, subtract, multiply and divide them (as long as we dont divide by zero). The order of addition and multiplication is unimportant, as there is a commutative property. A distributive property tells us how multiplication and addition interact with one another. As mentioned before, the real numbers possess an order. Given any two real numbers x and y, we know that one and only one of the following is true: x y, x y or x y. Another Property - Completeness The property that sets the real numbers apart from other sets of numbers, like the rationals, is a property known as completeness. Completeness is a bit technical to explain, but the intuitive notion is that the set of rational numbers has gaps in it. The set of real numbers does not have any gaps, because it is complete. As an illustration, we will look at the sequence of rational numbers 3, 3.1, 3.14, 3.141, 3.1415, . . . Each term of this sequence is an approximation to pi, obtained by truncating the decimal expansion for pi. The terms of this sequence get closer and closer to pi. However, as we have mentioned, pi is not a rational number. We need to use irrational numbers to plug in the holes of the number line that occur by only considering the rational numbers. How Many Real Numbers? It should be no surprise that there are an infinite number of real numbers. This can be seen fairly easily when we consider that whole numbers form a subset of the real numbers. We could also see this by realizing that the number line has an infinite number of points. What is surprising is that the infinity used to count the real numbers is of a different kind than the infinity used to count the whole numbers. Whole numbers, integers and rationals are countably infinite. The set of real numbers is uncountably infinite. Why Call Them Real? Real numbers get their name to set them apart from an even further generalization to the concept of number. The imaginary number i is defined to be the square root of negative one. Any real number multiplied by i is also known as an imaginary number. Imaginary numbers definitely stretch our conception of number, as they are not at all what we thought about when we first learned to count.

Fences by August Wilson Essay - 977 Words

Fences by August Wilson We all lead lives filled with anxiety over certain issues, and with dread of the inevitable day of our death. In this play, Fences which was written by the well known playwright, August Wilson, we have the story of Troy Maxson and his family. Fences is about Troy Maxson, an aggressive man who has on going, imaginary battle with death. His life is based on supporting his family well and making sure they have the comforts that he did not have in his own childhood. Also, influenced by his own abusive childhood, he becomes an abusive father who rules his younger son, Cory?s life based on his own past experiences. When the issue comes up of Cory having a bright future ahead of him if he joins the football team,†¦show more content†¦The main character of this play, Troy Maxson, considers death to be a part of the baseball game of life. In one part of the book he gives a description of how death came to him when he was in the hospital, and he told it to return another time. Death is one of his main insecurities. As he gets older he feels death coming closer and closer. It seems as if, just to hold death in abeyance by trying to feel young again, he has an affair with this other woman named Alberta. Death strikes closer to home base for Troy, when his mistress dies while giving birth to a baby girl. At this point in the book he makes a deal with death. He says to it, ? Ain?t nobody else got nothing to do with this. This is between you and me. Man to man. You stay on the other side of that fence until you ready for me.? (page 77). The fence is very symbolic in this situation. It symbolizes his insecurity about death and how he has put up these imaginary fences for protection. It leaves us with a very clear image of this agressive man who fears nothing in life, except death. Rose?s childhood was also filled with uncertainity. Her father was a drifter who went from place to place starting a new family every where he went, so that it got to the point where each of her step brothers and sisters had a different mother. This was the basis for her insecurity and possessiveness of her own family. In her case also,Show MoreRelatedFences : Fences By August Wilson1541 Words   |  7 PagesName: Alejandro Ahmed Date: 12/6/16 Unit: American Drama Title: Fences Author: August Wilson Year of Publication: 1986 Fences Study Guide Descriptive i. Overview: â€Å"Fences†, was a play written by August Wilson in 1986. The play communicates the story of the son of an unsuccessful sharecropper, Troy Maxson, who has lived most of his life independently due to his father’s abuse during his childhood. The play focuses on Troy’s interactions and conflicts with other characters, and the complicated relationshipsRead MoreFences by August Wilson791 Words   |  3 Pageshave built fences to keep enemies out and keep those they want to protect inside. In society today, people create metaphorical fences in order to fence in their feelings, while others create literal fences in order to keep the unwanted away. In the play Fences, the Maxon family lives in 1950’s America whose love for sports and one another are questioned at times when they need to be together the most. In the play Fences by August Wilson, two main characters Troy and Cory Maxon build a fence, literallyRead MoreFences, By August Wilson1104 Wor ds   |  5 PagesAfrican-Americans during the middle of the twentieth century were treated differently than those of the white population. Fences, a play by August Wilson, demonstrates the frustration of white dominance during a time when African-Americans were secluded from society. The Maxson family are the main characters of the play, showing the life they lived in their black tenement in Pittsburg in the 1950s. The setting demonstrates the drama of their struggle, frustration, rebellion, and the predicamentsRead MoreFences, By August Wilson807 Words   |  4 PagesFences â€Å"Some people build fences to keep people out...and other people build fences to keep people in† (61). In the play, Fences, by August Wilson who displays how fences symbolize different situations to represent different characters. The story takes place in Pittsburgh, Pennsylvania in the 1950’s, and in the play fences gives an outlook of providing an obstacle or barrier that is intended to keep something out or in. It shows through the protagonist character, named Troy Maxson and his wifeRead MoreThe Fences By August Wilson1813 Words   |  8 PagesIn the Fences, by August Wilson shows that life of African Americans in the U.S. in the 1950s with the story of Troy and his family. Wilson uses the symbol of the fence to show the desires of each character like Rose’s desire is to keep her family together, Troy’s desire is to keep death out and to be not bound forever, and Bono’s desire is to follow Troy, his best friend, as an example of the rig ht way to live and to be with Rose and Troy who are basically his family. Rose and the other seen charactersRead MoreAugust Wilson Fences984 Words   |  4 Pagesbecause in order to get what you want you have to fight for it. Two literary works that reflect this quote are Fences, a play about the struggles of African American before and during the Civil Rights Era in the 1950s and 60s, by August Wilson and â€Å"The Yellow Wallpaper† a short story that shows the difference between women and men during that time, by Charlotte Perkins Gilman. ‘Fences’ shows the revolution the characters face, their actions and what is the result of those choices. Whereas, ‘TheRead MoreFences by August Wilson883 Words   |  4 PagesFences, written by August Wilson, is a play about a man, named Troy, struggling to support his family during the late 1950’s. In this play, we see that Troy hurts the people closest to him. He has been uncaring towards his wife, Rose, his brother Gabriel and his son, Cory. This is because Troy had nothing to go on but the harsh example set by his father. In Fences, Troy has felt like he has been fenced in all of his life, which causes him to fence others in. Troy has felt fenced in all of his lifeRead MoreFences, By August Wilson895 Words   |  4 PagesFences written by August Wilson is an award winning drama that depicts an African-America family who lives in Pittsburg, Pennsylvania during the 1950’s. During this time, the Mason’s reveal the struggles working as a garbage man, providing for his family and excepting life as is. The end of segregation began, more opportunities for African American people were accessible. Troy, who’s the father the Cory and husband of Rose has shoes fill as a working African America man. He is the family breadwinnerRead MoreFences, August Wilson1835 Words   |  8 PagesAs illustrative of the kind of analysis I would bring to Fences, by August Wilson, if my bid to direct is successful, O prose to take direction for a part of Act 1, Scene 3 of the play. This will include possible blocking, camera work, music, and what the actor should be feeling and experiencing while acting the par t. I will examine how crucial it is that the actors portray their characters effectively, and I will offer commentary to assure just that. On the basis of these findings I will determineRead MoreSymbolism In Fences By August Wilson1460 Words   |  6 PagesKeep Love in or Lock it Out?: An Analysis of Symbolism in Fences Symbolism is defined as an artistic and poetic movement or style using symbolic images and indirect suggestion to express mystical ideas, emotions, and states of mind. In Fences by August Wilson, symbolism is used heavily throughout the play in order to represent deeper meanings and add to the emotion of the storyline. In order for the play to have so much depth and emotion, symbolism is crucial to the work itself and the heavy topics

Mini-Case - Finance free essay sample

In this scenario, we will continue the company’s growth rate of 5%, with no change in plowback or dividends. In this scenario, price per share is determined by the current dividends, divided by (r-g) The value of the company will be equal to the present value of all future cash flows ( i. e. dividend payments) that investors expect to receive. Constant growth scenario: EPS 2013 = $ 12,000,000 / 400,000 shares = $ 30. 00 Book equity per share in 2013 = $80,000,000 / 400,000 shares = $200. 00 per share Dividends paid out per share in 2013 = $ 8,000,000 / 400,000 shares = $ 20. 00 per share Payout ratio in 2013 = $ 20. 0 (DIV2013) / $ 30 (EPS 2013) = 0. 67 Plowback ratio 2013 = $10. 00 (RE per share 2013) / $ 30. 00 (EPS 2013) = 0. 33 Sustainable growth rate = 0. 15 (rate of return) x 0. 33 (plowback ratio) = 5 % Price per share 2012 = DIV2013/(r-g) = $20/(11%-5% ) = $ 333. 33 $ 333. 33 price per share x 400,000 shares = $ 133,333,333 value of the company in 2012 P/E ratio = $ 3 33. 33( price per share) / 30 (EPS) = 11. 11 Rapid Growth Scenario: Since Price = DIV / r-g, and there are no dividends paid in the years 2013 – 2016, we can calculate the value of the company in 2016 and discount it to obtain the Present value in 2012. EPS 2017 = $21,000,000 / 400,000 shares = $52. 50 Book equity per share 2017 = $139,900,000 / 400,000 shares = $349. 75 Dividends paid out per share 2017 = $14,000,000 / 400,000 shares = $35. 00 Payout ratio in 2017 = $ 35. 00 (DIV per share 2017) / $ 52. 50 (EPS 2017) = 0. 67 Plowback ratio in 2017 = $ 17. 50 (RE in 2017) / 52. 50 (EPS in 2017) = 0. 33 Sustainable growth rate = 0. 15 (rate of return) x 0. 33 (plowback ratio) = 5 % Price per share in 2016= $35. 00 (DIV 2017) / 0. 06 (r g)= $583. 33 Let’s discount it to 2012 value: Financial calculator: FV = 583. 33 N = 4, I/Yr = 11% PV = 384. 5 – price per share in 2012 384. 25 x 400,000 shares = 153,700,000 – value of the company in 2012 under rapid growth Conclusion: Rapid growth scenario promises higher stock price, so it should be chosen. PVGO between the previous example and this one: 153,700,000 – 133,333,333 = 20,366,667 Under both scenarios, current price per share is more than $200. Now here†™s my calculations: Constant growth scenario: Assuming a 15% required return: P0 = DIV1 / (r-g) = $20 / (. 15 . 05) = $20/. 1 = $200 Assuming an 11% required return, we’ll have: P0 = DIV1 / (r-g) = $20 / (. 11 . 05) = $20/. 06 = $333. 33 In the constant growth scenario, the stock is valued at $200 if we assume a 15% expected return, and $333. 33 if we assume 11% expected return. Now, in the rapid growth scenario, things get even more exciting. I think that 2017/2020 is the horizon year, because it’s AFTER that point when the growth goes down to 5%. In paragraph 6, the problem states â€Å" would require reinvestment of all of Prairie Home’s earnings from 2016 to 2019. After that the company could resume its normal dividend payout and growth. † your book’s years:20122013201420152016201720182019 my book’s years:20152016201720182019202020212022 year #01234567 arnings growth from previous year4. 6%15%15%15%15%5%5% dividend0000$35$36. 75$38. 59 todayH NB: neither book shows 2019 or 2022, but we know that the beginning of the year figures are the same as the end of year figures for the previous year, so that’s where I got those. Ultimately, it doesn’t really matter Iâ€℠¢m just reinforcing the point that we turn into a constant growth scenario beginning with year 6. Our non-constant growth model says this: PV = D1/(1+r)1 + D2 / (1+r)2 + †¦ + DH / (1+r)H + PH / (1+r)H and we get PH with this formula: PH = Dt+1 / (r-g) The dividends for the foreseeable future (years 1 4) will all be 0, so hose first numbers will add up to 0. We know that the dividend at the horizon year year 5 is $35. The expected future price of the stock at year 5 will be: P5 = D6 / (r-g) Plugging in numbers there, we have: P5 = $36. 75 / (. 15 . 05) = $36. 75/. 1 = $367. 50 Again, that’s assuming a 15% required return. Then the third part of the process is to add up all of those numbers, discounting them to the present value: P0 = D1 + D2 + D3 + D4 + D5 / (1. 15)5 + P5 / (1. 15)5 = 0 + (35 + 367. 5) / (1. 15)5 = 402. 5/(1. 15)5 Or on the calculator: FV = 402. 50, I/YR = 15, N=5, PV = $200. 11 Then we go to the 11% required return. There, we’ll see that: P5 = D6 / (r-g) = $36. 75 / (. 11 . 05) = $612. 50 And then: P0 = D1 + D2 + D3 + D4 + D5 / (1. 11)5 + P5 / (1. 11)5 = 0 + (35 + 612. 50) / (1. 11)5 = †¦ (Calculator: FV = 647. 50, I/YR = 11, N = 5, PV = $384. 26) In the rapid growth scenario, the stock is valued at $200. 11 if we assume a 15% expected return, and $384. 26 if we assume 11% expected return. This is the point where I defer to you, or we can talk about this more tomorrow. Our math says to price the stock somewhere between $200 and $384, but how do we choose? I get the sense that you understand that better than I do, so I can use your input for sure. We believe that Prairie Home Stores should value the stock at $384. xx because we should choose the We recommend choosing the rapid growth scenario, plowback more earnings into growing the company, and set the IPO price as $384. whatever. Our prospectus will show that we intend to invest more of our earnings into growing the company over the next 4 years, and as a result investors and the market will support a price of $384. We chose to use 11% as our expected rate of return, because this is the rate shown in the Journal of Finance as being the rate offered by other, equally risky stocks in the same industry as Prairie Home Stores. The PVGO is $153,700,000 – 133,333,333 = $20,366,667. This indicates that the company has room to grow, which will be attractive to investors. Investors believe that under the rapid growth scenario. According to our calculations, Mr. Breezeway was wise to counsel his son( ) to not sell the stock for $200, as we believe that the company is worth more than current BOOK VALUE PER SHARE include something about this. $200 per the current values (this is what the whoever dude offered the son), but our calculations show that the company is more valuable than the $200 price indicates.

Research Paper on Twitter Essay Example

Research Paper on Twitter Essay Twitter is a social networking service and microblogging where people can leave their messages, called tweets, with a limit size of 140 characters. These are published on the user profile page. Users can subscribe to other users’ message flow, which is called â€Å"to follow,† subscribers are called â€Å"followers.† All users can post (tweet) and read tweets via the Twitter website, but many choose instead to use third party applications, as these are designed to work on a cell phone, or as a standalone application that gives better view than the Web browser does. It costs nothing to use Twitter, but mobile operators may also send you the bill for using text massage service for reading and posting your tweets. Twitter’s headquarters is in San Francisco, but offices and server rooms are also available in San Antonio, Texas, and Boston. We will write a custom essay sample on Research Paper on Twitter specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Research Paper on Twitter specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Research Paper on Twitter specifically for you FOR ONLY $16.38 $13.9/page Hire Writer It is known that Twitter was created by Jack Dorsey in March 2006 and launched in July of that year. The service had 200 million registered users worldwide (September 2011). Of these, half (100 million) who log into their account at least once a month, while 50 million do so daily (September 2011). In 2010, they sent over 25 billion tweets. In addition to individual users, there also are many companies, services, artists, and organizations presenting themselves on Twitter as a way to reach out to their users, customers, and fans. The service received much attention during the presidential elections in the U.S. in 2008, when the presidential candidate Barack Obama’s campaign used Twitter and other social media to reach out to their constituents. Twitter is also a tool for dissidents in totalitarian regimes to communicate with the outside world, as well as during natural disasters, conflicts and the like when the individuals in place are immediately able to report on events. People who use Twitter use some special expressions. One of these conventions is to type the pound sign ) in front of a word or acronym to make it easy to find messages about this. Such words are called a â€Å"hashtagg.† Large organizations, events, and social movements often have their own hashtagg to gather information about them. Twitter accounts are referred to in a similar manner with an at sign (@) before the username. Disseminating or forwarding someone else’s tweet is called â€Å"to re-tweet.† An early way to â€Å"re-tweet† was made by typing RT followed by an at sign and the original author’s account name. The current built-in way to â€Å"re-tweet† was introduced in November 2009. Students, who are looking for the help in writing their research proposal on twitter, can use free research papers on different topics. These free example papers can be an excellent source of relevant data, if properly written. Are you looking for a top-notch custom research paper on Twitter topics? Is confidentiality as important to you as the high quality of the product? Try our writing service at EssayLib.com! We can offer you professional assistance at affordable rates. Our experienced PhD and Master’s writers are ready to take into account your smallest demands. We guarantee you 100% authenticity of your paper and assure you of dead on time delivery. Proceed with the order form: Please, feel free to visit us at EssayLib.com and learn more about our service!