Interview Essay Example | Topics and Well Written Essays - 500 words - 8

Interview - Essay Example The room itself had an unmistakable aura of grandeur about it. It was old but that wasn’t the only attribute, something struck me when I realized that these walls had been privy to the conversations of some of the greatest minds the world has ever seen.   ‘Oh crap’ I thought, ‘I actually have to answer this question’. While responding, I realized that even though this was an interview to study economics, the professors were interested to see how well I could synthesize different information from a variety of subjects. In this particular case it was perhaps political science and later in the interview, there was a great deal of mathematics involved. Furthermore, this interview was not the usual question and answer session that I had been used to in school; it was an open ended discussion with the experts. I disagreed with some of their opinions on several economic issues and gave some of my own. The dialogue that the interview entailed was nothing like I had previously encountered and therefore provided me with a distinct intellectual challenge.   In comparison with other experiences which have aided my intellectual development, this interview certainly holds the highest rank. My experiences at Model United Nations conferences were full of dialogue and debate regarding critical issues; the topics of discussion were provided beforehand whereas this interview truly tested my thinking skills and ability to adapt to a completely new situation. Alternatively, the training that the school’s Math team and I are currently partaking for the International Schools Mathematics Teacher’s Foundation (Hague, February 2012), has also been a challenge due to the intensity of our work and the fact that we are forced to synthesize information and topics we have previously seen in regular Math classes. However the interview holds more weight as it involved the synthesis of more than one

Dysfunctional Families in Today’s World Essay Example for Free

Dysfunctional Families in Today’s World Essay Children are supposed to grow up together with their parents. A family environment helps them not to feel depress(ed). They learn that their feelings and needs are important and can be expressed. Children growing up in such supportive environments are likely to form healthy, open relationships in adulthood. Family dysfunction can be any condition that interferes with healthy family functioning. In most families, there occurs some periods of time where functioning is impaired by stressful circumstances. These stressful events could be a death in the family or a parents serious illness, among others. But for healthy families, the moment the crisis is over, then normal functioning resumes. In dysfunctional families, however, problems tend to be chronic. The children then do not consistently get their needs. As a result, the negative patterns of parental behavior tend to be dominant in their childrens lives. Healthy families are not families who never argue or have disagreements. These families may have yelling, bickering, misunderstanding, tension, hurt, and anger but not all the time. In healthy families, emotional expression is allowed and accepted. Member of the family can freely ask for and give attention. Rules tend to be made explicit and remain consistent, but with some flexibility to adapt to individual needs and particular situations. Healthy families allow for individuality; each member is encouraged to pursue his or her own interests, and boundaries between individuals are honored. Children from healthy families are consistently treated with respect, and do not fear emotional, verbal, physical, or sexual abuse. Parents can be counted on to provide care for their children. Children are given responsibilities appropriate to their age and are not expected to take on parental responsibilities. Finally, in healthy families, everyone makes mistakes; and most importantly, mistakes are allowed. II. The rise of dysfunctional families and its impact in society There are many types of dysfunction in families. Some parents choose to under-function. They leave their children to fend for themselves. On the other hand there are some parents who over-function. These parents seem to never allow their children to grow up and be on their own. Others are inconsistent or violate basic boundaries of appropriate behavior. Below is a brief description of some types of parental dysfunction along with some common problems associated with each. Deficient Parents Deficient parents hurt their children more by omission than by commission. Frequently, it is because of a chronic mental illness or a disabling physical illness that contributes to parental inadequacy. Parental emotional needs tend to take precedence over the needs of the children. The children, on the other hand, are often asked to be their parents caretakers. Thus, children tend to take on the adult responsibilities at a young age in these families. They are robbed of their own childhood, and they learn to ignore their own needs and feelings. Because of this experience, where these children are simply unable to play an adult role and take care of their parents, they thenoften feel inadequate and guilty. These feelings continue into adulthood. Controlling Parents Unlike the deficient parents described above, controlling parents fail to allow their children to assume responsibilities appropriate for their age. Often, the controlling parents are driven by the fear of becoming unnecessary to their children. These parents then continue dominating and making decisions for their children, even if they are already well beyond the age at which this is necessary. This fear leaves them feeling betrayed and abandoned when their children become independent (Forward, 1989). On the other hand, these children of controlling parents frequently feel resentful, inadequate, and powerless. Transitions into adult roles are quite difficult, as these adults frequently have difficulties making decisions independent from their parents. When they act independently these adults feel very guilty, as if growing up were a serious act of disloyalty. Alcoholic Parents Alcoholic families tend to be chaotic and unpredictable. Rules that apply one day do not apply to another one or to the next instance that you happen to be part of. Promises are neither kept nor remembered. Expectations vary from one day to the next. Parents may be strict at times and indifferent at others. In addition, emotional expression is frequently forbidden and discussion about the alcohol use or related family problems is usually nonexistent. Family members are usually expected to keep problems a secret, thus preventing anyone from seeking help. All of these factors leave children feeling insecure, frustrated, and angry. Children often feel there must be something wrong with them and this is the reason that makes their parents behave this way. Mistrust of others, difficulty with emotional expression, and difficulties with intimate relationships carry over into adulthood. Children of alcoholics are at much higher risk for developing alcoholism than are children of non-alcoholics. Abusive Parents Abuse can be verbal, physical, or sexual. Verbal abuse such as frequent belittling criticism can have lasting effects, particularly when it comes from those entrusted with the childs care. Criticism can be aimed at the childs looks, intelligence, capabilities, or basic value. Some verbal abusers are very direct, while others use subtle put-downs disguised as humor. Both types are just as damaging. Definitions of physical abuse vary widely. Many parents, at one time or another, have felt the urge to strike their child. With physically abusive parents, however, the urge is frequent and little effort is made to control this impulse. Striking a child has much to do with meeting the parents emotional needs and nothing to do with concern for the child. Often, parents erroneously justify the abuse as a process of discipline that is intended to help the child. Physically abusive parents then create an environment of terror for the child. This is particularly worse since the violence is often random and unpredictable. Abused children often feel anger. Children of abusive parents have tremendous difficulties developing feelings of trust and safety even in their adult lives. While parents may justify or rationalize verbal or physical abuse as discipline aimed at somehow helping the child, there is no rationalization for sexual abuse. Sexual abuse is the most blatant example of an adult abusing a child purely for that adults own gratification. Sexual abuse can be any physical contact between an adult and child wherein that contact must be kept secret. The demonstrations of affection that occurred then such as hugging, kissing, or stroking a childs hair that can be done openly are quite acceptable and even beneficial. When physical contact is shrouded in secrecy then it is most likely inappropriate. Sexual abuse happens to both boys and girls. This act is perpetrated by both men and women. It cuts across lines of race, socioeconomic level, education level, and religious affiliation. In most cases, sexual abuse is part of an overall family pattern of dysfunction, disorganization, and inappropriate role boundaries. Responsibility for the sexual abuse in all cases rests entirely with the adult. No child is responsible for being abused. Most sexually abused children are too frightened and scared of the consequences for themselves and their families. In most cases, they do not risk telling another adult what is happening. As a result of this bottling-up, they grow into adulthood carrying feelings of self-loathing, shame, and worthlessness. They tend to be self-punishing and have considerable difficulties with relationships and with sexuality. When problems and circumstances such as parental alcoholism, mental illness, child abuse, or extreme parental rigidity and control interfere with family functioning, the effects on children can sometimes linger long after these children have grown up and left their problem families. Adults raised in dysfunctional families frequently report difficulties forming and maintaining intimate relationships, maintaining positive self-esteem, and trusting others; they fear a loss of control, and deny their feelings and reality (Vannicelli, 1989). There is a great deal of variability in how often dysfunctional interactions and behaviors occur in families, and in the kinds and the severity of their dysfunction. However, when patterns like the above are the norm rather than the exception, they systematically foster abuse and/or neglect. Abuse and neglect inhibit the development of childrens trust in the world, in others, and in themselves. Later as adults, these people may find it difficult to trust the behaviors and words of others, their own judgement and actions, or their own senses of selfworth. Not surprisingly, they may experience problems in their academic work, their relationships, and in their very identities. In common with other people, abused and neglected family members often struggle to interpret their families as normal. The more they have to accommodate to make the situation seem normal, the greater is their likelihood of misinterpreting themselves and developing negative self-concepts (e. g. , I had it coming; Im a rotten kid). III. The relationship between religion and dysfunctional families Dysfunctional family members have common symptoms and behavior patterns as a result of their common experiences within the family structure. This tends to reinforce the dysfunctional behavior, either through enabling or perpetuation. The dysfunctional family usually suffers from a variety of issues that may require the help of trained professionals. This is in addition to prayer and other spiritual disciplines. Some of those issues include: (1) Individual personality differences and personal issues; (2) Unresolved issues from the past; (3) Marital and family problems due to domestic violence and abuse, infidelity, and poor communication; (5) Financial problems and poverty; (6) Separation/divorce resulting in grief and loss and abandonment issues, depression, and sometimes a lack of financial support; (7) Mental and physical health problems; and (8) Spiritual disconnection. Any of the aforementioned problems can create high levels of distress, and failure to seek help can be lethal. Furthermore, when a major trauma strikes, families are rarely prepared to deal with it, and if they are unaccustomed to seeking outside help, family members may never recover emotionally. Many people stay away from connecting with God in a Christian community because they feel they are not good enough. Maybe they are overly critical of themselves, or maybe they know they have not lived the way God would want them to live. They may be afraid that Christians in church will judge them and reject them, and so they stay away. They say, God couldn’t really love someone like me. God must be angry with me. They could also be very adept at meeting the perceived expectations of others through self-denial. Denial, repression, splitting, and a false sense of self are often well-developed defense mechanisms. The black and white thinking expressed in such conflicting pairs of opposites as God vs. devil, church vs. world, sin vs. righteousness, leads to repression of anything that might possibly be construed as unacceptable. Constant self-monitoring and rigid self control, along with confession of every sin in prayer, are often considered the only means of avoiding divine condemnation. In the literalism characteristic of fundamentalism, an evil thought or feeling is regarded as just as sinful as an evil act. Impulses and feelings may be considered demonic in origin. In truth, religion can play a part in furthering the dysfunction of families. For one who has a dysfunctional family, its teachings of obedience, respect for authority and quiet compliance further enforce the debilitating situation. But when you dig deeper into what religion is, you will see clearly what God wants. Religious Institutions throughout history have used fear, control and manipulation to build kingdoms made by man. They have left many wounded or dead in the battlefield of Satans kingdom. Religion is a powerful tool of the enemy. An extreme example of this is The Taliban and Bin Laden. Religion teaches obedience to law. And Christ teaches grace. But the scripture has confined all under sin, that the promise by faith in Jesus Christ might be given to those who believe. But before faith came, we were kept under guard by the law, kept for the faith which would afterward be revealed. Galatians 3:22-23. Learning to trust wholeheartedly in God, and depend on Him to break free from whatever bondage, will bring emancipation from these bondages, and make an impact in our broken society. The church should be the one institution where the spirit of fear does not exist and coercion and manipulation are not practiced. IV. Healing dysfunctional families Two centuries ago there was relatively little dispute over the existence of God, or the societally beneficial effect of popular belief in a creator. In the twentieth century extensive secularization occurred in western nations, the United States being the only significant exception (Bishop; Bruce; Gill et al. ; Sommerville). Theists often assert that popular belief in a creator is instrumental towards providing the moral, ethical and other foundations necessary for a healthy, cohesive society. Many also contend that widespread acceptance of evolution, and/or denial of a creator, is contrary to these goals. As he helped initiate the American experiment Benjamin Franklin stated; â€Å"religion will be a powerful regulator of our actions, give us peace and tranquility within our minds, and render us benevolent, useful and beneficial to others† (Isaacson: 87-88). Changes in family life have transformed our society in the last thirty years. One of the biggest has been the virtual disappearance of the male-breadwinner lifestyle and the emerging dominance of the dual-earner couple. Working wives and mothers face questions about the effects of their choices on their childrens welfare and their own health under the strain of the second shift, while their husbands confrontand respond to in a variety of waysnew opportunities to construct a masculine identity not focused exclusively on breadwinning. Leaders in education, business, and government debate what policies should be in place to help people manage their work and family lives and who should bear the cost of work-family management. Changing patterns of family formation and disruption have also created widespread concern and vociferous policy debates. High divorce rates raise questions about the effects of divorce on childrens well-being and future success and how to make blended families work. The problems facing single-parent families have become a focus of policy makers, religious leaders, and the national media. Debates about homosexual unions have led to battles over gay and lesbian marriage in a number of states and many local controversies over what legal rights should be extended to homosexual partners. Delayed marriage and childbearing mean that more American households comprise single adults and childless couples, and remaining childless throughout life has become much more common, fueling concern among some about the decline of the family. All of these changes have led to an increasing pluralism in family life and a new consensus that there are many kinds of loving, caring families. Most people spend some portion of their adult lives outside of a nuclear family, forming and reforming family-like connections periodically over the course of their lives, causing many to rethink long-held assumptions about the necessity of marriage and parenting for adults happiness, security, and well-being. But this pluralism is intensely contested and debated for both moral and social philosophical reasons. Not everyone agrees about what constitutes the good family and what kinds of families are morally legitimate. Many people see the family as the bellwether of our society and find the rapid and numerous changes in family life over the last few decades to be troubling. Some even argue that a devaluing of family life, and especially of lifelong, heterosexual marriage, inevitably leads to a decline of the nation. These debates also focus on questions of resources and inequality. Who has access to the rights that marriage confers? Why does divorce lead to a reduction in women and childrens standard of living, and what can be done to change that? Making Changes Changes in family life have been a central concern for religious leaders, activists, and local communities of faith. Throughout American history, religion and family have been intertwined and interdependent institutions. Congregations, parishes, and synagogues have provided an important context for families to spend time together and have shaped the religious education and moral development of children and youths. Sociologists have long noted that marriage and parenthood make religion more important to people and increase their participation in local congregations. Sometimes, children who come from dysfunctional families continue in their roles because they are waiting for their parents to give their permission; to change. Like most people, parents in dysfunctional families often feel threatened by changes in their children. As a result, they may thwart efforts to change and insist that the children change back. Some specific changes one can do are: (1) Identify painful or difficult experiences that happened during your childhood; (2) Make a list of your behaviors, beliefs, etc. that you would like to change; (3) Next to each item on the list, write down the behavior, belief, etc. that you would like to do/have instead; (4) Pick one item on your list and begin practicing the alternate behavior or belief. Choose the easiest item first; (5) Once you are able to do the alternate behavior more often than the original, pick another item on the list and practice changing it, too; (6) Believe in God’s grace that he will heal all the painful experiences in your list (in no. 1) and pray His love will help you change all the behaviors listed and free you from bondage. References: Bishop, George. (1999). â€Å"What Americans Really Believe, and Why Faith Isnt as Universal as They Think. † Free Inquiry 19(3): 38-42 Bruce, Steve. (2001). â€Å"Christianity in Britain, R. I. P. † Sociology of Religion 61: 191-203. Forward, S. (1989). Toxic parents: Overcoming their hurtful legacy and reclaiming your life. New York: Bantam Books. Gill, R, C. Hadaway and P. Marler 1998 â€Å"Is Religious Belief Declining in Britain? Journal for the Scientific Study of Religion 37: 507-16 Isaacson, Walter. (2003) Benjamin Franklin: An American Life. New York, Simon Schuster. Meeting the Needs of Dysfunctional Families Catechist, January, (1993) Sommerville, C. (2002) â€Å"Stark’s Age of Faith Argument and the Secularization of Things: A. Commentary. † Sociology of Religion 63: 361-72 Vannicelli, M. (1989). Group psychotherapy with adult children of alcoholics: treatment techniques and countertransference. New York: Guilford Press.

Euthanasia Essay - Religious Views on Assisted Suicide -- Euthanasia P

Official Religious Views on Euthanasia and Assisted Suicide      Ã‚  Ã‚  Ã‚   This essay is dedicated to the expression of the various official views of religious bodies within our nation. Most major denominations are represented. These religions have long been the custodians of the truth, serving to check the erratic and unpredictable tendencies of political, judicial and social bodies which would have Americans killing off their elderly and handicapped.    The National Association of Evangelicals believe that human beings are made in the image of God and are, therefore, of inestimable worth. God has given people the highest dignity of all creation. Such human dignity prohibits euthanasia, that is actively causing a person's death.    Is it moral to withdraw a life-support system which is believed to be an inappropriate extension of the dying process? The National Association of Evangelicals (NAE) believes that in cases where patients are terminally ill, death appears imminent and treatment offers no medical hope for a cure, it is morally appropriate to request the withdrawal of life-support systems, allowing natural death to occur. In such cases, every effort should be made to keep the patient free of pain and suffering, with emotional and spiritual support being provided until the patient dies. The National Association of evangelicals believes that in cases where extensive brain injury has occurred and there is clear medical indication that the patient has suffered brain death (permanent unconscious state), no medical treatment can reverse the process. Removal of any extraordinary life-support system at this time is morally appropriate and allows the dying process to proceed.(National)    The Union of Orthodox... ...rson cannot be legitimately interrupted. The use of painkillers to alleviate the sufferings of the dying, even at the risk of shortening their days, can be morally in conformity with human dignity if death is not willed as either an end or a means, but only foreseen and tolerated as inevitable. Palliative care is a special form of disinterested charity. As such it should be encouraged. (Catechism) WORKS CITED: National Association of Evangelicals of America http://www.euthanasia.com/evangel.html Union of Orthodox Jewish Congregations of America http://www.ou.org/public/statements/1999/nate43.htm   Lutheran Church - Missouri Synod http://www.euthanasia.com/missouri.html Reformed Church in America http://www.euthanasia.com/lakes.html Catechism of the Catholic Church http://www.euthanasia.com/catech.html

The Effect of the Automobile on American Life

The automobile is one of the most important inventions in our countries history. It has totally affected the American way of life. Four main areas it affected are other transportation, roadways, rural life, and urban life. Automobiles affected other ways of transportation. Horse-drawn transportation was almost completely replaced. Trains and trolleys lost rides as cars became more affordable and popular. Roadways were then created. More than 400,000 miles of new road connected people and places. The construction of the new roadways created many jobs. Along the roadways structures such as filling stations popped up. Soon after drive-in restaurants, banks, movies, and convenience stores became popular. Tourist cabins were made for travelers passing through the country. Billboards advertising restaurants and products soon lined the streets. The automobile affected the farmers’ way of life. Farmers got their good to the markets faster, giving them a better income. It created greater contact between neighbors in the country, also. People in the country also gained access to shopping and leisure activity within the cities. The urban life was also affected by the automobile. They gained access to the country. Although most affects were for the better, the cities got hectic. Between traffic snarls and pollution the cities changed. The automobile also caused depopulation of inner cities to the suburbs. As the cities depopulated and automobiles became more popular there became a lack of parking of all the people driving to work. Parking in the city was very high. As a result the automobile changed the American way of life. The way the automobile affected the way of life still continues today. It’s still a major effect on people as in how and where they live and work.

John Steinbeck’s use of Realism Essay

John Ernst Steinbeck has written many award winning novels, some of which has even been produced as plays that captured audiences everywhere. Steinbeck wrote about real life experiences using realism, characterization, and dreams to emphasize his points and make an impact on his readers in order to reform or change society. The realism used in Steinbeck’s works is not only effective in informing the reader of circumstances that should be changed, but this nineteenth century literary style also creates great feelings of empathy toward the characters and their dreams. Steinbeck used realism to convey his points for a purpose, and his main purpose was that he wanted something to be made known to the public. Of Mice and Men and The Grapes of Wrath both tell of the hardships people went through and also the harsh conditions of their situations. The characters in both of these novels play and important role in personalizing the occurring events for the reader, making the novel more effective in getting the writers’ messages across to the audience. Steinbeck’s use of the American Dream and the will for the characters to succeed is also evident throughout the two novels. These dreams help the reader to relate to the characters, therefore making a bigger impact on those who read the novel. John Steinbeck’s novel â€Å"The Grapes of Wrath†, tells of a very harsh journey to California that the Joads, like many other families in the 1930s Depression era, embarked upon in order to find work and escape their dying farms in Oklahoma. During the depression a severe drought covered the plains, called the Dust Bowl. This natural disaster destroyed any chance that the farmers had of making a living, and they were forced to travel west and leave their homes in hopes of finding a job. Part of the novel’s sense of realism comes from the setting. The Joads head to California from Oklahoma and their journey is characterized by poverty, starvation, death, and suffering (Stegener 405). Steinbeck not only utilizes the setting as a sense of realism but also uses vivid description and specific details to draw attention to the numerous hardships encountered by the families forced to travel west in search of opportunity (Jackson 316). The first evening after the Joads leave their home, they stop on the side of the road to help another family. This family  is very thankful for the Joads help, and offers their tent for the Joads’ sickly grandfather, but unfortunately Grandpa passes away that night. The family is troubled from the very start of their journey and Steinbeck creates feelings of sympathy toward the Joads as well as the other families in the same situation. As the novel progresses, the family’s disparity becomes more and more evident. Grandma Joad dies shortly after Grandpa does and since the Joads are indigent and can barely meet ends to survive, they are forced to beg for money to bury Grandma Joad. Despite family’s disappointments, they just keep going (Britch and Lewis). As they continue to travel the Joads encounter many more obstacles, but there is one incident as the novel comes to a close. Rose of Sharon, the oldest daughter, goes into labor and her baby is stillborn. Rose of Sharon notices a dying old man who is malnourished and offers him her breast milk so that he might be able to survive. As the novel closes, the reader is left with despair and a sense of loneliness. Steinbeck’s uses of realism makes a strong impact on the reader, which makes this one of the greatest American novels ever written (Jackson 316). Of Mice and Men was also written during the time of the 1930s, the depression era. This novel is a short story of two men a small, short, and smart guy named George and a big, tall, mentally retarded man named Lennie. The novel is based around these two main characters and their journey to fulfill their dream to find true happiness on a farm that they can one day call their own (Hearle) In â€Å"Of Mice and Men† the use of realism is not only seen through the depressing life of George and Lennie, but it also shows the life that Crooks, a black stable hand on the farm, had. The character of Crooks is used to symbolize the social standing of the black community occurring during the time at which the novel is set. Crooks is a lonely African American on the farm that feels out of place. As George and Lennie explain their dream to Crooks he brushes them off and says that no one around here can implement their dreams. This realism gives the reader an impression that Crooks has absolutely no hope. However, Crooks may be pessimistic, but yet even he has a dream, which is the hope of one day experiencing the joys of  his childhood again. Crooks’ character is portrayed as very lonely in the novel, this is evident when Crooks explains, â€Å"A guy needs someone.† (Mice 77). Crooks is telling the reader the need of human interaction. This realism that Steinbeck uses shows the reader the harsh realities of the black community during the time of the depression in the 1930’s Steinbeck not only uses realism to make an impact on his readers but he also uses strong characterization to his advantage as well (Howard). By using characters such as Ma Joad and Curley’s wife, he is able to personalize the novel for the reader. This technique that Steinbeck uses makes the reader feel like he/she can relate with the characters’ struggles, therefore, creating a greater sense of understanding toward the characters’ situations (Horn). These two characters can best be described as strong, courageous, and they show great amounts of integrity as well as faith. In order to write a great book that will make an impact on the reader, the author must not only use realism but also have a strong sense of characterization for the reader to relate to. Steinbeck successfully achieved both of these aspects of a great novel. John Steinbeck’s character, Ma Joad in â€Å"The Grapes of Wrath†, is the epitome of a strong character. She is a supportive, loving woman, and surprisingly the family’s center of strength (Britch and Lewis). She is intent on keeping the family whole and together. The family looks to Ma Joad for their source of hope and encouragement. She is the emotional and physical backbone of the family, and it is very important to her to provide the family with comfort, nourishment, and support. â€Å"Without warning Grandpa began to cry. His chin wavered and his old lips tightened over his mouth and he sobbed hoarsely. Ma rushed over to him and put her arms around him. She lifted him to his feet,her broad back straining, and she half lifted, half helped him to the tent† (Wrath 185). Steinbeck makes it clear to the reader in this passage that Ma is the first to step up in times of need, and she puts others before herself no matter what. â€Å"Under the spread of the tarpaulin Grandma lay on a mattress, and Ma sat beside her. The air was stifling hot, and the flies buzzed in the shade of the canvas. Grandma was naked under a long piece of pink curtain. She turned her old head restlessly from side to side, and she  muttered and choked. Ma sat on the ground beside her, and with a piece of cardboard drove the flies away and fanned a stream of moving hot air over the tight old face. Rose of Sharon sat on the other side and watched her mother† (Wrath 285). The reader notices several times throughout the novel that Ma will care for anyone in need no matter what the circumstance. â€Å"In show of her fundamental spirit she accepts Casey into the family because the Joads just do not refuse ‘food an’ shelter or lift on the road to anybody that ask[s]'† (Britch and Lewis). She takes it upon herself to look out for to avoid any discouragement to the other family. Grandma dies in the back of the wagon with Ma by her side, but trying to avoid any discouragement to the other family members, Ma does not share the news until they have reached their stopping point. The strength that Ma shows in this situation is far great that any exhibited by the other family members. â€Å"Her capacity to care marks the measure of her self-respect. â€Å"As the action progresses her caring does not change in kind but rather grows in breadth and intensity† (Britch and Lewis). By creating such a noble character, Steinbeck greatly influences his readers, and his efforts in sharing the hardships these families in the 1930s went through are successful. The character of Curley’s wife is a very vivid and unique one. She represented the way that women were viewed by society as a whole. Steinbeck portrayed Curley’s wife as a temptress, when actually the woman is just craving for attention. Steinbeck made Curley’s wife stand out by the fact that she was the only woman on the ranch, she was the only one who dressed to impress, and the only person without a full name (Thesing). Steinbeck often used very colorful statements like â€Å"She had full, rouged lips and wide-spaced eyes, heavily made up. Her fingernails were red. Her hair hung in little rolled clusters, like sausages. She wore a cotton dress and red mules, and the insteps of which were little bouquets of red ostrich feathers† (Mice 31). These vivid and wild descriptions that Steinbeck created for Curley’s wife is just one of the many characters that Steinbeck created, in order for the reader to feel or relate to what the character is going through. In Curley’s wife’s case she is an attention deprived woman that is just one of the many lonely people on the ranch. â€Å"Of Mice and Men† and â€Å"The Grapes of Wrath† both examine the morality and necessary actions the characters choose as they pursue their dreams. Steinbeck uses the concept that dreams are better that reality, but that the dream is really hard to grasp. This concept is also used by three other famous authors who are Crane, Norris, and Dreiser (Benson 256-257). What is the American dream? The American dream is the faith held by many Americans, that through hard work, courage, and determination one can adhere financial prosperity and also some believe that it is led to an emphasis of material wealth as a measure of success and/or happiness. Like many people today the characters George, Lennie, Candy, Curley’s wife, and the Joads all wanted the American dream. Most of their dreams of success, wealth and prosperity were depleted by the harsh realities of life. Most of the characters in the novel Of Mice and Men have a dream of something. George and Lennie are best friends that go everywhere together. Their dream is to someday own a farm of their own and have rabbits to tend to. The dream to Lennie is an antidote to disappointment and loneliness, and he often asks George to recite the description of the farm to him. This dream is ultimately lost when Lennie kills Curley’s wife and George kills Lennie towards the end of the novel. This dream was later shared by two other friends on the farm, Candy and Crooks. As the reader can see these dreams are one way that the characters can cover up the loneliness and hopelessness of their existence. One other character in the novel that had a dream is Curley’s wife. Curley’s wife’s dream is her fantasies in becoming a part in movies and having a life of luxury. This dream is shattered and never fulfilled, because of the dissatisfaction of her life and when she is accidentally killed by Lennie. As the reader can see the dreamers in the novel are undermined by the hard facts of reality (Astro 43). In the 1930s the country suffered ecological devastation called the depression. During this economy downfall the mid-west prairie state families suffer from a large drought that leaves them with nothing to survive. This terrible phenomenon is called the Dust Bowl. In attempt to escape this unforgettable time, a family called the Joads dream of wealth and success in  the great state of California. In order to pursue their dream they have to leave their belongings behind. This journey carries death and suffering with it when Grandma and Grandpa Joad dies and many of the family members leave the family. The family left Oklahoma in order to find jobs in California (Hearle). The Joad’s dream, which is well known as the American dream, is shattered by certain events and conditions that the Joads faced. In conclusion, John Steinbeck’s style of writing impacted society just as he had hoped. He wrote about real life experiences and hardships to show the reader what people went through, like the Joads and the many characters in â€Å"Of Mice and Men†. Steinbeck informed the reader of what families had to overcome, which ultimately impacted whoever read his novels. The use of realism, characterization, and dreams in his two novels made them the greatest American novels in history. Works Cited Astro, Richard Dictionary of Literary Biography, volume 9: American Novelists, 1910-1915. A Bruccoli Clark Layman Book. Ed. James J. Martine. Saint Bonaventure University. Gale Research 1981. 413-68 Benson, Jackson J., â€Å"John Steinbeck: Novelist as a Scientist,† in Novel: A Forum on Fiction. Spring, 1977, 228-264. Britch, Carroll and Cliff Lewis, â€Å"Growth of the family in The Grapes of Wrath,† in Critical Essays on Steinbeck’s The Grapes of Wrath, Ed. John Ditsky, G.K. Hall. 1989, 97-108. Hearle, Kevin Dictionary of Literary Biography, volume 212: Twentieth-century American western writers. Second series. A Bruccoli Clark Layman Book. Ed. Richard H. Carcoft Brigham Young University. Gale Research Group. 1999, 278-294. Horn, Jason G. Dictionary of Literary Biography: volume 275. Twentieth century American Nature Writers: Prose A Bruccoli Clark Layman Book. Ed. Roger Thompson, Virginia Military Institute, and J. Scott Bryson. St. Mary’s College. Gale Group 2003. 314-323 Jackson, Joseph Henrey. â€Å"The Finest Book John Steinbeck has written.† Contemporary Literary Criticism, volume 59. Ed. Roger Matuz. Detroit, MI: Gale Research Inc. 1990. 316-317. Levant, Howard. â€Å"The Novels of John Steinbeck: A Critical Study.† Contemporary Literary Criticism, volume 75. Ed. Thomas Volteler. Detroit, MI: Gale Research Inc. 1993, 356-351. Steinbeck, John. The Grapes of Wrath New York, New York: Penguin Books U.S.A Inc. 1992. Steinbeck, John. Of Mice and Men, New York, New York: Penguin Putnam Inc. 1993. Stegner, Wallace, â€Å"The True Adventures of John Steinbeck, Writer† Contemporary Literary Criticism, volume 34. Ed. Sharon K. Hall. Detroit, MI: Gale Research Company, 1985. 405. Thesing, William B. Dictionary of Literary Biography, volume 7: Twentieth-century American Dramatists, first series. A Bruccoli Clark Layman Book. Ed. John MacNicholas University of South Carolina, Gale Research. 1981. 271-276.

Beyond Pythagoras Essays

Beyond Pythagoras   Essays Beyond Pythagoras   Essay Beyond Pythagoras   Essay I have been asked to investigate Pythagorean triplets where the shortest side is an odd number and all the three sides are positive integers. A pythagorean triple is a set of integers (a,b,c) that specifies the lengths of a right angle triangle aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in which a is the shortest side b is the middle side and c is the hypotenuse. The first set of triples (3,4,5) which has already been proved to satisfy Pythagorass theory. I have also been given two other pythagorean triples (5,12,13) and (7,24,25) I will now prove these to satisfy Pythagorass theory aà ¯Ã‚ ¿Ã‚ ½ = 5à ¯Ã‚ ¿Ã‚ ½ =25 bà ¯Ã‚ ¿Ã‚ ½ = 12à ¯Ã‚ ¿Ã‚ ½ =144 cà ¯Ã‚ ¿Ã‚ ½ = 13à ¯Ã‚ ¿Ã‚ ½ =169 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =25+144 =169 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (5,12,13) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. aà ¯Ã‚ ¿Ã‚ ½ = 7à ¯Ã‚ ¿Ã‚ ½ =49 bà ¯Ã‚ ¿Ã‚ ½ = 24à ¯Ã‚ ¿Ã‚ ½ =576 cà ¯Ã‚ ¿Ã‚ ½ = 25à ¯Ã‚ ¿Ã‚ ½ =625 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =49+576 =625 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (7,24,25) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. I am now going to put my results in to a table so that I can predict more values: a b c 3 4 5 5 12 13 7 24 25 I will now predict the next two values in the table so I can work out a general formula for this pythagorean family. By using the differencing method I can see that a has a difference of 2 between each pythagorean triplet I predict the next two a values to be 9 and 11 From the table I can also see that there is a quadratic sequence for the b value so I predict the next two values to be 40 and 60 Also from my table, I can see the c value is b+1 so my two next predictions Im going to prove are: a b c 9 40 41 11 60 61 aà ¯Ã‚ ¿Ã‚ ½ = 9à ¯Ã‚ ¿Ã‚ ½ =81 bà ¯Ã‚ ¿Ã‚ ½ = 40à ¯Ã‚ ¿Ã‚ ½ =1600 cà ¯Ã‚ ¿Ã‚ ½ = 41à ¯Ã‚ ¿Ã‚ ½ =1681 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =81+1600 =1681 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (9,40,41) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. aà ¯Ã‚ ¿Ã‚ ½ = 11à ¯Ã‚ ¿Ã‚ ½ =121 bà ¯Ã‚ ¿Ã‚ ½ = 60à ¯Ã‚ ¿Ã‚ ½ =3600 cà ¯Ã‚ ¿Ã‚ ½ = 61à ¯Ã‚ ¿Ã‚ ½ =3721 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =121+3600 =3721 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (11,60,61) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. My table now looks like this: a b c 3 4 5 5 12 13 7 24 25 9 40 41 11 60 61 When looking at the table I found a pattern so I am going to prove if it will work. 3à ¯Ã‚ ¿Ã‚ ½ =9 -4-5 (3,4,5) 5à ¯Ã‚ ¿Ã‚ ½ =25 -12-13 (5,12,13) 7à ¯Ã‚ ¿Ã‚ ½ =49 -24-25 (7,24,25) 9à ¯Ã‚ ¿Ã‚ ½ =81 -40-41 (9,40,41) 11à ¯Ã‚ ¿Ã‚ ½ =121 -60-61 (11,60,61) aà ¯Ã‚ ¿Ã‚ ½ = b+c aà ¯Ã‚ ¿Ã‚ ½ = b+b+1 aà ¯Ã‚ ¿Ã‚ ½ + 2b+1 Now to prove if it will work for Pythagoras aà ¯Ã‚ ¿Ã‚ ½ aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ = (b+1) à ¯Ã‚ ¿Ã‚ ½ bà ¯Ã‚ ¿Ã‚ ½ aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ = bà ¯Ã‚ ¿Ã‚ ½=2b+1 b+1à ¯Ã‚ ¿Ã‚ ½ aà ¯Ã‚ ¿Ã‚ ½ = 2b + 1 aà ¯Ã‚ ¿Ã‚ ½-1=b 2 Pythagoras theory is satisfied So now with 5 triplets that I have proved I will now find a general formula to find other pythagorean triplets in this family. By looking at the table its obvious that the formulae for a is a=2n+1 Im now going to find the general formulae for the b and c value. (b+1) à ¯Ã‚ ¿Ã‚ ½ = (2n+1)à ¯Ã‚ ¿Ã‚ ½ + b bà ¯Ã‚ ¿Ã‚ ½+2b+1 = 4nà ¯Ã‚ ¿Ã‚ ½+4n+1+bà ¯Ã‚ ¿Ã‚ ½ bà ¯Ã‚ ¿Ã‚ ½+2b-bà ¯Ã‚ ¿Ã‚ ½ = 4nà ¯Ã‚ ¿Ã‚ ½+4n+1-1 2b = 4nà ¯Ã‚ ¿Ã‚ ½+4n b = 2nà ¯Ã‚ ¿Ã‚ ½+2n c = 2nà ¯Ã‚ ¿Ã‚ ½+2n+1 I now have the formulae: a =2n+1 b = 2nà ¯Ã‚ ¿Ã‚ ½+2n c = 2nà ¯Ã‚ ¿Ã‚ ½+2n+1 I am now going to prove that these formulae work aà ¯Ã‚ ¿Ã‚ ½ = (2n+1)à ¯Ã‚ ¿Ã‚ ½ ) bà ¯Ã‚ ¿Ã‚ ½ = (2nà ¯Ã‚ ¿Ã‚ ½+2n)à ¯Ã‚ ¿Ã‚ ½ ) cà ¯Ã‚ ¿Ã‚ ½ = (2nà ¯Ã‚ ¿Ã‚ ½+2n+1)à ¯Ã‚ ¿Ã‚ ½ = aà ¯Ã‚ ¿Ã‚ ½ = (2n+1) (2n+1) 4nà ¯Ã‚ ¿Ã‚ ½+2n+2n+1 4nà ¯Ã‚ ¿Ã‚ ½+4n+1 bà ¯Ã‚ ¿Ã‚ ½ = (2nà ¯Ã‚ ¿Ã‚ ½+2n) (2nà ¯Ã‚ ¿Ã‚ ½+2n) 4n +4nà ¯Ã‚ ¿Ã‚ ½+4nà ¯Ã‚ ¿Ã‚ ½+4nà ¯Ã‚ ¿Ã‚ ½ aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ = 4n + 8nà ¯Ã‚ ¿Ã‚ ½+8nà ¯Ã‚ ¿Ã‚ ½+2n+1 cà ¯Ã‚ ¿Ã‚ ½ = (2nà ¯Ã‚ ¿Ã‚ ½+2n+1) (2nà ¯Ã‚ ¿Ã‚ ½+2n+1) 4n + 4nà ¯Ã‚ ¿Ã‚ ½+2nà ¯Ã‚ ¿Ã‚ ½+ 4nà ¯Ã‚ ¿Ã‚ ½+4nà ¯Ã‚ ¿Ã‚ ½+2n+2nà ¯Ã‚ ¿Ã‚ ½+2n+1 4n + 8nà ¯Ã‚ ¿Ã‚ ½+8nà ¯Ã‚ ¿Ã‚ ½+2n+1 I will give an example of how to use the formulae for the 10th as the nth term: a =2n+1 =21 b = 2nà ¯Ã‚ ¿Ã‚ ½+2n = 220 c = 2nà ¯Ã‚ ¿Ã‚ ½+2n+1 = 221 the 10th pythagorean triple in this pythagorean family is (21,220,221) I am now going to investigate the 2nd pythagorean family. I will get this by doubling the values of the first family. (3,4,5) = (6,8,10) (5,12,13) = (10,24,26) (7,24,25) = (14,48,50) (9,40,41) = (18,80,82) (11,60,61) = (22,120,122) Using the internet for research I found out that there are in-between values so these are the pythagorean triples I have for this family now: (6,8,10) (8,15,17) (10,24,26) (12,35,37) (14,48,50) (16,63,65) (18,80,82) (20,99,101) (22,120,122) I will now prove that Pythagoras theorem holds for some of these pythagorean triplets. aà ¯Ã‚ ¿Ã‚ ½ = 6à ¯Ã‚ ¿Ã‚ ½ =36 bà ¯Ã‚ ¿Ã‚ ½ = 8à ¯Ã‚ ¿Ã‚ ½ =64 cà ¯Ã‚ ¿Ã‚ ½ = 10à ¯Ã‚ ¿Ã‚ ½ =100 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =36+64 =100 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (6,8,10) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. aà ¯Ã‚ ¿Ã‚ ½ = 8à ¯Ã‚ ¿Ã‚ ½ =64 bà ¯Ã‚ ¿Ã‚ ½ = 15à ¯Ã‚ ¿Ã‚ ½ =225 cà ¯Ã‚ ¿Ã‚ ½ = 17à ¯Ã‚ ¿Ã‚ ½ =289 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =64+225 =289 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (8,15,17) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. aà ¯Ã‚ ¿Ã‚ ½ = 10à ¯Ã‚ ¿Ã‚ ½ =100 bà ¯Ã‚ ¿Ã‚ ½ = 24à ¯Ã‚ ¿Ã‚ ½ =576 cà ¯Ã‚ ¿Ã‚ ½ = 26à ¯Ã‚ ¿Ã‚ ½ =676 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =100+576 =676 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (10,24,26) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. I am now going to find a general formula for the 2nd pythagorean family that I will name the b+2 family. From the values I have I know a = 2n+4 by using the differencing method I will now work out the b and c values. (b+2) à ¯Ã‚ ¿Ã‚ ½ = (2n+4)à ¯Ã‚ ¿Ã‚ ½ + b bà ¯Ã‚ ¿Ã‚ ½+4b+1 = 4nà ¯Ã‚ ¿Ã‚ ½+16n+16+bà ¯Ã‚ ¿Ã‚ ½ bà ¯Ã‚ ¿Ã‚ ½+4b-bà ¯Ã‚ ¿Ã‚ ½ = 4nà ¯Ã‚ ¿Ã‚ ½+16n+16-4 4b = 4nà ¯Ã‚ ¿Ã‚ ½+16n+12 b = nà ¯Ã‚ ¿Ã‚ ½+4n+3 c = nà ¯Ã‚ ¿Ã‚ ½+4n+5 I now have the formulae: a =2n+4 b = nà ¯Ã‚ ¿Ã‚ ½+4n+3 c = nà ¯Ã‚ ¿Ã‚ ½+4n+5 I am now going to prove that these formulae work aà ¯Ã‚ ¿Ã‚ ½ = (2n+4)à ¯Ã‚ ¿Ã‚ ½ ) bà ¯Ã‚ ¿Ã‚ ½ = (nà ¯Ã‚ ¿Ã‚ ½+4n+3)à ¯Ã‚ ¿Ã‚ ½ ) cà ¯Ã‚ ¿Ã‚ ½ = (nà ¯Ã‚ ¿Ã‚ ½+4n+5)à ¯Ã‚ ¿Ã‚ ½ = aà ¯Ã‚ ¿Ã‚ ½ = (2n+4) (2n+4) 4nà ¯Ã‚ ¿Ã‚ ½+8n+8n+16 4nà ¯Ã‚ ¿Ã‚ ½+16n+16 bà ¯Ã‚ ¿Ã‚ ½ = (nà ¯Ã‚ ¿Ã‚ ½+4n+3) (nà ¯Ã‚ ¿Ã‚ ½+4n+3) n +4nà ¯Ã‚ ¿Ã‚ ½+3nà ¯Ã‚ ¿Ã‚ ½+4nà ¯Ã‚ ¿Ã‚ ½+16nà ¯Ã‚ ¿Ã‚ ½+4n+12n+3nà ¯Ã‚ ¿Ã‚ ½+12n+9 n +8nà ¯Ã‚ ¿Ã‚ ½+22nà ¯Ã‚ ¿Ã‚ ½+24n+9 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ = n + 8nà ¯Ã‚ ¿Ã‚ ½+26nà ¯Ã‚ ¿Ã‚ ½+40n+25 cà ¯Ã‚ ¿Ã‚ ½ = (nà ¯Ã‚ ¿Ã‚ ½+4n+5) (nà ¯Ã‚ ¿Ã‚ ½+4n+5) n + 4nà ¯Ã‚ ¿Ã‚ ½+5nà ¯Ã‚ ¿Ã‚ ½+ 4nà ¯Ã‚ ¿Ã‚ ½+16nà ¯Ã‚ ¿Ã‚ ½+20n+5nà ¯Ã‚ ¿Ã‚ ½+20n+25 n + 8nà ¯Ã‚ ¿Ã‚ ½+26nà ¯Ã‚ ¿Ã‚ ½+40n+25 I will give an example of how to use the formulae for the 10th as the nth term: a =2n+4 =24 b = nà ¯Ã‚ ¿Ã‚ ½+4n+3= 143 c = nà ¯Ã‚ ¿Ã‚ ½+4n+5= 145 the 10th pythagorean triple in this pythagorean family is (21,220,221) I am now going to investigate the 3rd pythagorean family. I will get this by tripling the values of the first family. (3,4,5) = (9,12,15) (5,12,13) = (15,36,39) (7,24,25) = (21,72,75) (9,40,41) = (27,120,123) (11,60,61) = (33,180,183) I will now prove that Pythagoras theorem holds for these pythagorean triplets. aà ¯Ã‚ ¿Ã‚ ½ = 9à ¯Ã‚ ¿Ã‚ ½ =81 bà ¯Ã‚ ¿Ã‚ ½ = 12à ¯Ã‚ ¿Ã‚ ½ =144 cà ¯Ã‚ ¿Ã‚ ½ = 15à ¯Ã‚ ¿Ã‚ ½ =100 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =81+144 =225 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (9,12,15) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. aà ¯Ã‚ ¿Ã‚ ½ = 15à ¯Ã‚ ¿Ã‚ ½ =225 bà ¯Ã‚ ¿Ã‚ ½ = 36à ¯Ã‚ ¿Ã‚ ½ =1296 cà ¯Ã‚ ¿Ã‚ ½ = 39à ¯Ã‚ ¿Ã‚ ½ =1521 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =225+1296 =1521 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (15,36,39) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. aà ¯Ã‚ ¿Ã‚ ½ = 21à ¯Ã‚ ¿Ã‚ ½ =441 bà ¯Ã‚ ¿Ã‚ ½ = 72à ¯Ã‚ ¿Ã‚ ½ =5184 cà ¯Ã‚ ¿Ã‚ ½ = 35à ¯Ã‚ ¿Ã‚ ½ =5625 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =441+5184 =5625 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (21,72,35) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. aà ¯Ã‚ ¿Ã‚ ½ = 27à ¯Ã‚ ¿Ã‚ ½ =729 bà ¯Ã‚ ¿Ã‚ ½ = 120à ¯Ã‚ ¿Ã‚ ½ =14,400 cà ¯Ã‚ ¿Ã‚ ½ = 123à ¯Ã‚ ¿Ã‚ ½ =15,129 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =729+14,400 =15,129 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (27,120,123) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. aà ¯Ã‚ ¿Ã‚ ½ = 33à ¯Ã‚ ¿Ã‚ ½ =1089 bà ¯Ã‚ ¿Ã‚ ½ = 180à ¯Ã‚ ¿Ã‚ ½ =32,400 cà ¯Ã‚ ¿Ã‚ ½ = 183à ¯Ã‚ ¿Ã‚ ½ =33,489 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ =441+5184 =5625 aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ so Pythagorass theory holds for (33,180,183) because they satisfy the condition of aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½=cà ¯Ã‚ ¿Ã‚ ½ in a right angled triangle. I am now going to find a general formula for the 3rd pythagorean family that I will name the b+3 family. From the values I have I have used the differencing method to find out that a = 6n+3 I will now work out the b and c values. (b+3) à ¯Ã‚ ¿Ã‚ ½ = (6n+3)à ¯Ã‚ ¿Ã‚ ½ + b bà ¯Ã‚ ¿Ã‚ ½+6b+1 = 36nà ¯Ã‚ ¿Ã‚ ½+36n+9+bà ¯Ã‚ ¿Ã‚ ½ bà ¯Ã‚ ¿Ã‚ ½+6b-bà ¯Ã‚ ¿Ã‚ ½ = 36nà ¯Ã‚ ¿Ã‚ ½+36n+9-9 6b = 36nà ¯Ã‚ ¿Ã‚ ½+36n b = 6nà ¯Ã‚ ¿Ã‚ ½+6n c = 6nà ¯Ã‚ ¿Ã‚ ½+6n+3 I now have the formulae: a = 6n+3 b = 6nà ¯Ã‚ ¿Ã‚ ½+6n c = 6nà ¯Ã‚ ¿Ã‚ ½+6n+3 I am now going to prove that these formulae work: aà ¯Ã‚ ¿Ã‚ ½ = (6n+3)à ¯Ã‚ ¿Ã‚ ½ ) bà ¯Ã‚ ¿Ã‚ ½ = (6nà ¯Ã‚ ¿Ã‚ ½+6n)à ¯Ã‚ ¿Ã‚ ½ ) cà ¯Ã‚ ¿Ã‚ ½ = (6nà ¯Ã‚ ¿Ã‚ ½+6n+3)à ¯Ã‚ ¿Ã‚ ½ = aà ¯Ã‚ ¿Ã‚ ½ = (6n+3) (6n+3) 36nà ¯Ã‚ ¿Ã‚ ½+18n+18n+9 36nà ¯Ã‚ ¿Ã‚ ½+36n+9 bà ¯Ã‚ ¿Ã‚ ½ = (6nà ¯Ã‚ ¿Ã‚ ½+6n) (6nà ¯Ã‚ ¿Ã‚ ½+6n) 36n +36nà ¯Ã‚ ¿Ã‚ ½+36nà ¯Ã‚ ¿Ã‚ ½+36nà ¯Ã‚ ¿Ã‚ ½ 36n +72nà ¯Ã‚ ¿Ã‚ ½+36nà ¯Ã‚ ¿Ã‚ ½ aà ¯Ã‚ ¿Ã‚ ½+bà ¯Ã‚ ¿Ã‚ ½ = 36n +72nà ¯Ã‚ ¿Ã‚ ½+72nà ¯Ã‚ ¿Ã‚ ½+36n+9 cà ¯Ã‚ ¿Ã‚ ½ = (6nà ¯Ã‚ ¿Ã‚ ½+6n+3) (6nà ¯Ã‚ ¿Ã‚ ½+6n+3) 36n + 36nà ¯Ã‚ ¿Ã‚ ½+18nà ¯Ã‚ ¿Ã‚ ½+36nà ¯Ã‚ ¿Ã‚ ½+36nà ¯Ã‚ ¿Ã‚ ½+18n+18nà ¯Ã‚ ¿Ã‚ ½+18n+9 36n + 72nà ¯Ã‚ ¿Ã‚ ½+82nà ¯Ã‚ ¿Ã‚ ½+36n+9 I will give an example of how to use the formulae for the 10th as the nth term: a = 6n+3 =63 b = 6nà ¯Ã‚ ¿Ã‚ ½+6n = 3660 c = 6nà ¯Ã‚ ¿Ã‚ ½+6n+3= 3663 the 10th pythagorean triple in this pythagorean family is (21,220,221)

Leslie Marmon Silko, “Lullaby” Essays

Leslie Marmon Silko, â€Å"Lullaby† Essays Leslie Marmon Silko, â€Å"Lullaby† Paper Leslie Marmon Silko, â€Å"Lullaby† Paper American Mosaic, July 2011 FOCUS: Leslie Marmon Silko, â€Å"Lullaby† â€Å"Lullaby† is a short story that first appeared in a book entitled Storyteller in 1981. This was a book written by Leslie M. Silko that uses short stories, memories, poetry, family pictures, and songs to present her message. The book is concerned, in general, with the tradition of story-telling as it pertains to the Native American culture. Lullaby seems to be a story of tradition, change, death, loss and the tensions fostered as a result of them between the old couple in the story and the Anglo-American authorities of the time. Throughout the story there are quite a few conflicts. Some are internal between Ayah and herself and others are external ones through Ayah, the white man, and Chato, her husband. The story is told by the main character, Ayah. She’s an old woman retracing tragic memories of life occurrences like the death of her son, Jimmie, in a helicopter crash during a war. She was not sure about what happened to him until a man in khakis drove up in a blue sedan and told her that he was dead and how he died. Jimmie was the one that taught Ayah to sign her name. She regrets this greatly as she relays the loss of her other two children who were taken by white doctors because they were thought to have a disease, allegedly given to them by their grandmother. They were taken because, in fear of the white men who were yelling and pointing for her signature, she â€Å"signed† the children away. Later on, when they were brought to visit, it was apparent the children were forgetting their customs and language; further evidence of the completeness of her loss. These events seem to have severely alienated Ayah towards Chato as well. Especially those specifically related to the children as indicated by, â€Å"She slept alone on the hill until the middle of November until the first snows came. Then she made a bed for herself where the children slept. She did not lie down next to Chato again until many years later when he was sick and shivering and only her body could keep him warm. † Ayah also speaks of her husband’s work (Chato) as a fence mender for a nearby rancher. She took offense at the exploitation Chato endured at the hands of the rancher that employed im, and let him go without hesitation when Chato gets too old to work. As a result they lost their home when the rancher told Chato he [and â€Å"his old woman†] had to be out of the shack [they lived in] by the next afternoon. Despite Ayah’s immense sense of devotion to Chato it seems apparent that she sees him as a weak husband and resents him deeply for it. Though much of the story is of Ayah’s reminiscences, its present tense has Ayah searching for Chato. She finds him walking along the road late on a very cold night [seemingly] in a daze brought on by illness [and wine]. While resting together beside the road she wraps Chato in the army blanket Jimmie sent her thereby eliciting comfort from a symbol of one of her greatest losses. The lullaby she sings to him at the end of the story, as they lie together in the snow, is one that her grand-mother and mother sang to her as a child and seems to provide a sense of closure for her as she sings it. It is one of the last pieces of tradition she can cling to from her own culture as she waits for death to take her and her husband from under the cold, clear winter sky.