Artificial Intelligence Essay -- Essays Papers

Artificial Intelligence Research PaperGenesis, creation, the very beginning from his inception, man has endeavored to control, to name, to create lastly in his own image as he was created from God. Man forges his own destiny from the coals of his imagination and the raw iron of his will to create. His tools have changed as clipping has get throughed, but his desire, his fire to create to change his world has not. Time and technology can temper mans creativity, but the desire burns as strong today as ever. Art, literature, and technology be it paint, paper or steel mans creativity is manifest in everything we do. The crowning jewel for man will be to pass on that spark with which he has been entrusted, robotics, genetic engineering, and their ilk have been try to create new life from the raw tools with which man is so proficient. It can be said that as Prometheus took fire from the heavens to give to man, so shall man give fire of another kind, and be it biological or made fr om the polar steel and silicon gateways through which we now travel man will at last, have his legacy. There is a caveat however, with knowledge comes change, with creation comes distinction, and with difference comes fear, hatred and discrimination. People have forever shunned that which they do not understand, that which is different from the face they see in the mirror in the morning. Since initial forays into the AI ambit in 1950 there have been philosophical as well as technical concerns. As technology advanced and the concept of a machine that thinks became more and more plausible the philosophy became more apparent. The basic problem we are confronted with is Can machines think?In his book entitled Philosophical perspectives in simulated intelligence, Martin Ringle calls for a logical and semantic analysis of the concepts of thought, intelligence, consciousness, and machine, rather than an empirical assessment of computer behaviour (hjhjh,999,2000). Thusly from its nativity AI has been regarded as an unknown, a concept that by its very name challenges nearly every norm and convention we have as individuals and as a edict. Thusly because of its inherent alien nature artificial life will be subject to the same prejudices as race, gender and religion, once it is integrated into society and assumes roles associated with humans. As we vent... ...ealitya paradigm in which both human and computer share a real corporal space within which to make plenty gestures, facial displays, body movements, and real physical objects that can be passed back and forth between the real and virtual worldScholars have long been trying to quantify the actual differences between brain and mind as well as the degree to which psychology can be converted into a physical science. Society as an entity seems unwilling to make leaps of judgment or significant paradigm shifts dealing with such concepts. The realms of the physical and the more nebulous sciences of the mind must for the time being remain separate. Once we begin to mesh technology more closely with ourselves as humans we can begin to accept it as a part of ourselves and as a part of our society. While today we do not possess the technology to achieve a truly animate machine we cannot because of that speculate too deeply as to the results of such an achievement. The image of a cold Terminator style robot or by chance HAL from 2001 is perhaps the exact opposite of the eventual reality. We cannot form opinions without the proper grounding in science, philosophy and indeed, ourselves.

Analysis of Writing Womens Worlds by Lia Adu-Lughod Essay examples --

Analysis of Writing Womens Worlds by Lia Adu-LughodWriting Womens Worlds is some stories on the Bedouin Egyptian people. In this book, thwe writer Lia Adu-Lughods stories differ from the effected ones. While reading, we discover the customs and values of the Bedouin people. We see Migdim, a dominator of the people. Even though her real age is never given, one can assume that she is at the end of her spiritedness, maybe in her mid to late eighties. Migdims life seems to include all the changes inside the Boudin community. Throughout the narrative of her life, we are able to realize the life way and changes within this exclusive society.One of the more in depth stories that Migdim told was how she refused the marriage to the man her father chose for her. It is customary for a woman to get marital to her paternal first cousin. Her female relatives made her the tent she was going to live in and brought her some bridal gifts. Migdim refused to eat as well as cover herself in color in o rder to holdup the wedding. After much objection, Migdim did not marry the man that her father chose. Actuality, her father failed twice trying to apparel a marriage for her.All the way through Migdims incident with arranged marriages, we can understand the old customs that has to do with marriage. It is obvious that, although women were believed to be obedient, they were commensurate to effectively convince men. Yet, today there seems to be a sign toward polygamous marriages that are eventu...

The Effect of Racism on the Self-Esteem of African Americans and the He

Throughout the past, scientists have attempted to explain the health disparity amongst African Americans and Whites. With the completion of the human genome project, it was shown that there is very little difference between opposite races on the genetic level. However, African Americans are twice as likely to die from cardiovascular disease as their European counterpart, the question that has arisen is where do these differences stem from (Harell, Floyd, Daniels and Bell). Recently, scientists have begun to believe that racial discrimination could possibly explain these differences (Belgrave &Allison, 2010). racial discrimination has been an issue for African Americans since the early days of slavery. Everywhere they go, they face a possibility of being treated differently because of their race. Within the past fifty years, the racial discrimination that African Americans faced has dropped, however it is still prevalent. According to Belgrave and Allison, racism is defin ed as the negative beliefs, actions, and emotions based on race although there are different types of racism, this definition gives a generalization on what racism is. It seems as though racism is the cause of many of the health problems that are faced by African Americans. Whether it is because African Americans generally do not receive the same health service as Whites or because direct racism causes higher blood pressure, racism has a negative health effect on African Americans (Belgrave &Allison, 2010). PurposeThe purpose of this experiment is to see whether racism has an effect on the self-esteem of African Americans and the health risks associated with racism, mainly high blood pressure. The independent variable is perception of racism and the dependent variables are the self... ...nd blood pressure. Racism is a problem confront many African Americans in the United States. It is affecting both their self-esteem and health. Until they are given the help they need or r acism dissipates from society, they will eternally have these problems. References Belgrave, A. Z., & Allison, K. W. (2010S).African american psychology, from africa to america. (2nd ed., pp. 96-112). Thousand Oaks, California Sage Publications, Inc.Fischer, A. R., & Shaw, C. (1999). African americans mental health and perceptions of racist discrimination The moderating effects of racial socialization experiences and self-esteem. diary of Counseling Psychology,46(3), 395-407. Retrieved from psycnet.apa.org/journals/cou/46/3/395.htmlHarrell, C. P. J., Floyd, L. J., & Bell, S. R. Psychophysiological methods enduring value to research within black psychology.

The Colossian Heresy Essay -- essays research papers fc

The urban center of ColossaeLocated on the South bank of the Lycus River in the province of Phrygia stood Colossae. Before the delivererian era, Colossae was a principle city in the Lycus valley . Part of a major trade route in Asia minor from Ephesus to Miletus, the city was most cognize for its production of textiles, especially its purple wool . With separate large cities such as Laodicea and Hierapolis, this was a well-popu modernd and high business area in the Lycus Valley. Yet, with changes in the channel system, Laodicea became a more important trade city than Colossae. And though once a city of great prominence, by A.D. 61, Colossae had suffered a great deal. An earthquake move the city that year, Eusebius writes, and had disappeared from the literature of its day .However, there is oft more to the story of this once prominent city. For instance, the city is written to by the Apostle capital of Minnesota in the late 50s A.D., concerning what was considered to be dange rous pedagogicss that big businessman be infiltrating the perform in that city. What was this false teaching that capital of Minnesota was concerned enough to write to the highest degree? Was there truly a danger? What did this teaching consist of? These questions will attempt to be answered in the following pages. It would be wise to jump dispirit with the church in Colossae.The Church in Colosssae had a problemThe church in Colossae was not planted by Paul. Rather, objet darty believe that one of his students, Epaphras, was the man who built this Gentile church . It is believed that Epaphras is the man who first sent word to Paul about the problems facing the church there. And while it is generally agreed that Paul writes to a specific problem affecting the church in Colossae, it is not agreed upon what exactly the problem was. There are as many possibilities as there are scholars who have written on the subject. For the time being, some of the more likely views will be bri efly examined. The heresyOften referred to as the Colossian Heresy, many debated as to who might be the ones responsible for the false teachings and wrong influences that Paul was so concerned about. Arnold writes that though Paul had probably not visited the church in Colossae before he wrote, he believed the teaching to come from a Pagan and Jewish style of thought and law . This would imply that deuce different groups were responsibl... ...rand Rapids, Michigan 1996.Baird, Cliff. What Was nailed to the Cross. Memphis 1989.Barclay, William. The All Sufficient Christ Studies in Pauls Letter to the Colossians, Westminster Books, Philadelphia 1974.Furnish, Paul Victor. Colossians, Pauls Epistle to the Anchor Bible Dictionary, ed. Freedman, David Noel. Doubleday, in the buff York 1992.Gray, Crete. The Epistle of St. Paul to the Colossians and Philemon, Lutterworth Press, London 1948.Jones, Allen H. Essenes, University Press of America, Lanham, Maryland 1985.Kachelman, put-on L. Jr. Studies in Colossians The Saviors Supremacy, Quality Publications, Abilene, Texas 1985.Lewis, C.S. Colossians, Pauls Epistle to the, The International Standard Bible Encyclopedia, ed. Orr, James, Wm. B. Eerdmans Publishing Co. meter Rapids 1952Lightfoot, J.B. Saint Pauls Epistles to the Colossians and to Philemon, Macmillan and Co., London 1900.Robertson, A.T. Paul and the Intellectuals, Broadman Press, Nashville 1959.Schweizer, Eduard. The Letter to the Colossians, Ausburg Publishing House, Minneapolis 1982.Simon, Marcel. Jewish Sects at the Time of Jesus, Fortress Press, Philadelphia 1967. The Colossian Heresy Essay -- essays explore papers fc The city of ColossaeLocated on the South bank of the Lycus River in the province of Phrygia stood Colossae. Before the Christian era, Colossae was a principle city in the Lycus Valley . Part of a major trade route in Asia minor from Ephesus to Miletus, the city was most known for its production of textiles, especially its purple wool . With other large cities such as Laodicea and Hierapolis, this was a well-populated and high business area in the Lycus Valley. Yet, with changes in the road system, Laodicea became a more important trade city than Colossae. And though once a city of great prominence, by A.D. 61, Colossae had suffered a great deal. An earthquake shook the city that year, Eusebius writes, and had disappeared from the literature of its day .However, there is much more to the story of this once prominent city. For instance, the city is written to by the Apostle Paul in the late 50s A.D., concerning what was considered to be dangerous teachings that might be infiltrating the church in that city. What was this false teaching that Paul was concerned enough to write about? Was there truly a danger? What did this teaching consist of? These questions will attempt to be answered in the following pages. It would be wise to first begin with the church in Colossae.The Church in Colosssae had a prob lemThe church in Colossae was not planted by Paul. Rather, many believe that one of his students, Epaphras, was the man who built this Gentile church . It is believed that Epaphras is the man who first sent word to Paul about the problems facing the church there. And while it is generally agreed that Paul writes to a specific problem affecting the church in Colossae, it is not agreed upon what exactly the problem was. There are as many possibilities as there are scholars who have written on the subject. For the time being, some of the more likely views will be briefly examined. The heresyOften referred to as the Colossian Heresy, many debated as to who might be the ones responsible for the false teachings and wrong influences that Paul was so concerned about. Arnold writes that though Paul had probably not visited the church in Colossae before he wrote, he believed the teaching to come from a Pagan and Jewish style of thought and law . This would imply that two different groups were responsibl... ...rand Rapids, Michigan 1996.Baird, Cliff. What Was nailed to the Cross. Memphis 1989.Barclay, William. The All Sufficient Christ Studies in Pauls Letter to the Colossians, Westminster Books, Philadelphia 1974.Furnish, Paul Victor. Colossians, Pauls Epistle to the Anchor Bible Dictionary, ed. Freedman, David Noel. Doubleday, New York 1992.Gray, Crete. The Epistle of St. Paul to the Colossians and Philemon, Lutterworth Press, London 1948.Jones, Allen H. Essenes, University Press of America, Lanham, Maryland 1985.Kachelman, John L. Jr. Studies in Colossians The Saviors Supremacy, Quality Publications, Abilene, Texas 1985.Lewis, C.S. Colossians, Pauls Epistle to the, The International Standard Bible Encyclopedia, ed. Orr, James, Wm. B. Eerdmans Publishing Co. Grand Rapids 1952Lightfoot, J.B. Saint Pauls Epistles to the Colossians and to Philemon, Macmillan and Co., London 1900.Robertson, A.T. Paul and the Intellectuals, Broadman Press, Nashville 1959.Schweizer, Eduard. The Letter to the Colossians, Ausburg Publishing House, Minneapolis 1982.Simon, Marcel. Jewish Sects at the Time of Jesus, Fortress Press, Philadelphia 1967.

Howards End :: essays research papers

Young, pretty Helen has go away her London home to visit the Wilcox family estate, Howards End. (Helen and hersister Margaret met Mr. Wilcox and his wife while traveling in Germany.) Margaret was also invited toHowards End, merely stayed home to care for their 16-year-old brother Tibby who has hay fever. From Howards End, Helen sends Margaret several letters describing the beautiful estate and the energetic, materialistic Wilcoxes. Her last letter sends a shock through Margaret when she reads it Helen has locomote in love with Paul the youngest Wilcox son.When Mrs. Wilcox dies not long afterward, she leaves a handwritten note behind asking that Howards End be given to Margaret. But her practical husband,Henry, a prominent businessman, and her greedy son Charles, astruggling businessman, refuse to act on the matter and never mention it to Margaret. One night, Margaret andHelen run into Henry, and they wrangle the case of Leonard Bast Henry warns them that Leonards insurance comp any is doomed to failure, and they advise him to find a new job. But poor Leonard, who associates theSchlegels with all things cultural and romantic--he reads constantly, hoping to mend himself--resents this intrusion into his business life and accuses them of trying to profit from his knowledge of the insurance industry.Margaret and Henry develop a halting, gradual friendship. When the lease expires at Wickham Place,the Schlegels begin expression for another endure (their landlord wants to follow the general trend and replace theirhouse with a more profitable apartment building). Henry offers to rent them a house he owns in London, andwhen he shows it to Margaret, he suddenly proposes to her. She is surprised by her happiness, and after considering the proposal, she accepts.Shortly before Margaret and Henry are scheduled to be married, Henrys girlfriend Eviemarries a man named Percy Cahill the wedding is held at a Wilcox estate near Wales. After the party, which Margaret finds quite unpleasant, Helen arrives in a disheveled state, with the Basts in tow. She declares indignantly that Leonardhas left his old company, found a new job, and been summarily fired he is now without an income. Helen angrily blames Henry for his ill-considered advice. Margaret asks Henry to give Leonard a job, yet when he seesJacky Bast, he realizes that he had an affair with her 10 years ago, when she was a prostitute in Cyprus. Margaretforgives him for the indiscretion--it was before they even met--but she writes to Helen that there will be no job

Howards End :: essays research papers

Young, pretty Helen has left her London home to visit the Wilcox family estate, Howards give up. (Helen and hersister Margaret met Mr. Wilcox and his wife musical composition traveling in Germany.) Margaret was also invited toHowards End, but stayed home to care for their 16-year-old brother Tibby who has hay fever. From Howards End, Helen sends Margaret several earns describing the beautiful estate and the energetic, materialistic Wilcoxes. Her last letter sends a shock through Margaret when she reads it Helen has fallen in love with Paul the youngest Wilcox son.When Mrs. Wilcox dies not long afterward, she leaves a handwritten note behind asking that Howards End be given to Margaret. that her pragmatic husband, total heat, a prominent businessman, and her greedy son Charles, astruggling businessman, refuse to act on the matter and never mention it to Margaret. One night, Margaret andHelen run into Henry, and they discuss the case of Leonard Bast Henry warns them that Leonards insurance company is doomed to failure, and they advise him to find a new job. But poor Leonard, who associates theSchlegels with all things cultural and romantic--he reads constantly, hoping to better himself--resents this intrusion into his business life and accuses them of trying to profit from his knowledge of the insurance industry.Margaret and Henry develop a halting, piecemeal friendship. When the lease expires at Wickham Place,the Schlegels begin looking for another house (their landlord wants to follow the general trend and replace theirhouse with a more profitable apartment building). Henry offers to rent them a house he owns in London, andwhen he shows it to Margaret, he suddenly proposes to her. She is surprised by her happiness, and after considering the proposal, she accepts.Shortly before Margaret and Henry are scheduled to be married, Henrys daughter Eviemarries a man named Percy Cahill the wedding is held at a Wilcox estate near Wales. After the party, which Marga ret finds quite unpleasant, Helen arrives in a disheveled state, with the Basts in tow. She declares indignantly that Leonardhas left his old company, found a new job, and been summarily fired he is now without an income. Helen angrily blames Henry for his foolish advice. Margaret asks Henry to give Leonard a job, but when he seesJacky Bast, he realizes that he had an affair with her 10 years ago, when she was a prostitute in Cyprus. Margaretforgives him for the indiscretion--it was before they eventide met--but she writes to Helen that there will be no job

African American and Their Rights Essay

Since slavery, African Americans dedicate gone through a lot to reach their current state. In the early 20th century, African Americans face up discrimination, isolation, and were discriminated according to their skin color. It started when Europeans brought the first Africans to America, and continued throughout the Civil War. The American political sympathies do some changes in policies. A variety of leaders shaped the lucky struggle toward swarthy tintity in America (Bowles, 2011). Ever since slavery begun, African Americans have been impelled to end segregation, discrimination, and isolation.Activists such as, Martin Luther King Jr. and Malcolm X, and early(a)s, joined to filmher to put an end to segregation, discrimination, and isolation to attain civil rights and equality. Slavery had changed dramatically in the new 1600s. About this time the slave trade to American colonies also began increasing to meet the demand for cheap labor. Traders exchange slaves to the No rthern colonies, but English and other European immigrants satisfied the demand for labor there (Echerd, 2009). Slaves in America came from western and central Africa.African tribes some measure enslaved those defeated in intertribal wars and sold their captives to European slave traders. The tribes raided villages to obtain slaves to trade for European goods. Slave traders had even offered the Africans guns and other goods for the slaves. Slaves lived a rough, laboured life. Cheap labor was a huge percent of their lives. They had to work from sunrise to sunset. The work consisted of clearing land, tended to fields of tobacco, rice, and vegetables. They also per word formed mevery other tasks that had helped make plantations al nigh completely self-sufficient.No slaves saw any gold for their tasks that they had performed, but they did receive food, clothing, and shelter. The slaves had resided in small one-room huts, which had nowindows and the floors were all dirt. Most slaves accepted their living condition, however, they knew no other way of life (Koehler, 2009). However, white Southerners regained project of state goernments in the South during the late 1870s, however, and reversed most of the previous(prenominal) gains made by source slaves. For example segregation. What is segregation?According to Websters Dictionary, to segregate is define as to make out or set apart from others isolate or to require, often with squash, the separation of a specific racial, religious, or other free radical from the body of society. Segregation has been a part of our American heritage, almost from the moment slaves arrived on the shores of the New World (Bowles, 2011). In 17th century Virginia, the theocratic government feared that racial mixing between freed and enslaved blacks and white indentured servants would become a means to usurp government power.They passed laws in which the color line was clearly defined in any criminal punishments. By treating whites and blacks separately and unequally, these Virginian leaders set up a system of white supremacy that would become an essential parcel of American slavery. Separation and segregation was the order of the day, with African Americans being forced to ride in separate railroad cars, have their own hotels and courthouses, and even get water out of their own drinking fountains. Their children could non attend the same schools with the light children.To further push the color-line, they then added in segregation with the Jim Crow Laws. This is mainly because the albumins were considered to be superior, and hence were thought to deserve bankrupt schools with better facilities. African Americans on the other hand were considered inferior, and hence their children attended low-quality schools that lacked adequate facilities (Sitkoff & Franklin, 2008). The Northern States, which had grew and prospered during the war, believed the origin slaves to be equal as any other person.The Souther n States, still angry over the loss of the war and their firm belief in White superiority, took a different nest. They created and enforced what were known as the Black Codes. These were legislations passed in Southern states to ascendancy labor, migration and other activities of the freed slaves. Black Codes allowed profound marriage, property ownership and express access to the court systems. It prohibited them from testifying against whites, serving on juries or militias, voting and publicly expressing any form of legal concerns ( www. history. com).Any former slave that did not sign yearly labor contract with the plantation owners could be arrested and hired out. The Black codes in short allowed for the continued and legal discrimination against the former slaves (www. history. com). copulation quickly responded to these laws in 1866 and seized the initiative in remaking the south. Republicans cherished to ensure that with the remaking the south, freed blacks were made via ble members of society. But the strong southern legislatures finally gave in in 1868 they repealed most of the laws that discriminated against blacks. Things were starting to look up.But by 1877 Democratic parties regained their power of the south and ended reconstruction. In 1882, southern states passed Jim Crow laws that enforced strict segregation between blacks and whites and limited African-American civil rights. This was devastating to the blacks. After all the strides they made were reversed. From holding political offices, the right to vote, and participating as equal members of society was changed. The south gradually reinstated the racially discriminatory laws. The two main goals they wanted these laws to achieve disenfranchisement and segregation.To take away the power that the blacks had gained, the Democratic Party began to substantiation Blacks from voting. There were many another(prenominal) ways to stop blacks from voting. Some of these things were poll tax, which were fees were charged at voting booths and were expensive for most blacks, and the literacy test. Since teaching blacks were illegal, most adult blacks were former slaves and illiterate. And the other goal, segregation, causes the democrats to create laws that segregated the schools and public facilities. The Northern States, which had grew and prospered during the war, believed the former slaves to be equal as any other person.The Southern States, still angry over the loss of the war and their firm belief in White superiority, took a different approach. They created and enforced what were known as the Black Codes. These were legislations passed in Southern states to control labor, migration and other activities of the freed slaves. Black Codes allowed legal marriage, property ownership and limited access to the court systems. It prohibited them from testifying against whites, serving on juries or militias, voting and publicly expressing any form of legal concerns.Any former slave that did not sign yearly labor contract with the plantation owners could be arrested and hired out. The Black codes in short allowed for the continued and legal discrimination against the former slaves. Just like some African Americans activists fought this segregation, some Whites had some groups of their own to carry the segregation on and on. The Ku Klux Klan was one of them. The Ku Klux Klan, Knights of White Camellia, and other terrorists murdered thousands of blacks and some whites to prevent them from voting and participating in public life.The KKK was founded in 1865 to 1866. They directed their violence towards black landowners, politicians, and lodge leaders. They also did this to people who supported Republicans or racial equalities (Anti-Defamation League, 2012). After the abolishment of slavery in the U. S. the KKK formed. They hated blacks and would commit crimes against them. Murders, hangings, and lynches are just some of the crimes against the blacks (www. kkk. bz, n. d. ). The Ku Klux Klan claims to be just defending their people like other races do. What is a lynching?Lynching is a form of punishment with no legal permission. Most times lynching occurred against African Americans by hanging them. This was very popular during the Gilded Age after the American Civil War when African Americans were freed from slavery. Many White men would use lynching against Black men for being in a mixed relationship with a White woman. However, because lynching had no legal basis, it was thought to have been a tool that was used against freed slaves that had achieved financial stability and authority in order to remain a White-dominated nation.Lynching was most likely performed by White Supremacy groups like the KKK. Lynching was done by hanging or shooting, or both. However, many were of a more hideous nature. hot at the stake, maiming, dismemberment, castration, and other brutal methods of physical torture are all part of a lynching. Lynching therefore was a cruel combination of racialism and sadism, which was utilized primarily to sustain the caste system in the South. Many white people believed that Negroes could only be controlled by fear. To them, lynching was seen as the most effective means of control.Defending your people is one thing, but to torture another human being is inhuman. The KKK has several stories out there today on how they treated the blacks, whether they did anything wrong or not. For instance, a Louisiana woman is in critical condition after she was set on fire, resulting in burns on roughly 60 percent of her body, and her car appears to have had racial slurs written on it at the time of her attack (Mach, n. d. ). They had even gone as cold as church bombings. The KKK launched a bomb into a church during a Sunday service, which left four innocent teenage girls dead.The men responsible hid hind end the cloak of secrecy, intimidation and the white robes of the oldest terrorist organization in the world, the Ku Klux Klan (Gado, n. d. ). Therefore, until the Civil Rights Act of 1964, racial discrimination is an issue that was not seriously tackled. The act was a successful result of most wide-ranging civil rights legislation and Civil Rights Movements for close to a century (Finkelman, 2009). The act declared discrimination on the basis of color, race, ethnicity, religion, and many other aspects as unconstitutional.During the critical years from 1954 to 1963, a variety of leaders with different backgrounds, such as lawyers from the NAACP, women sitting on batches, ministers from southern black churches, militants from black power organizations, and youth from colleges had shaped the successful struggle toward black equality in America (Bowles, 2011). In 1896, the Supreme Courts Plessy v. Ferguson decision establish that separate but equal facilities for whites and blacks were allowable under the U. S. Constitution. Local governmental officials could designate separate public facilities like drinking fountains, restrooms, and schools.Even courthouses often had separate Bibles according to the defendants race. The problem was that separate usually meant unequal, and segregation subverted the freedom of every African American (Sundquist, 1993). Now, it is time for the African- Americans to fight back. The incident that made them want to make a difference was the Rosa Park bus ride. After a long day of work on December 1, 1955, Parks, feet hurt, looked before to sitting on the bus for her ride home. At the time, there was a city ordinance stating that African Americans had to give up their seats on a train or bus if a white man asked for them.When a white man approached Parks and told her that he wanted her seat, she simply said no. Although she acted as a private citizen, her response was as an informed, committed member of the NAACP movement. The bus driver had asked Parks to move. When she did not, the bus driver said, Look, woman, I told you I wanted the seat. A re you going to stand up? When Parks again said no, the driver threatened, If you dont stand up, Im going to have you arrested. She gave no reply but at the next stop, Rosa was arrested (Garrow, 2004).A pastor known as Martin Luther King Jr., organized a boycott, the Montgomery bus boycott. King Jr. took this to a higher level and maintained and organized the Southern Christian Leadership Conference (SCLC), which coordinated similar bus boycotts in other cities. Shortly after the boycott, King had found a bomb on his porch. King went to Birmingham, Alabama, where he continued his nonviolent withstands and marches. However, the police authorized force to disband Kings followers by using electric cattle prods, tear gas, and fire hoses (Bowles, 2011).King was arrested with the others, but upon his release from jail he went to Washington, D. C., where he and demonstrators met at the National Mall and addressed them with his famous We Shall Overcome speech on August 23, 1963. Kings wo rds at the cap that day were a defining moment of the Civil Rights movement. Other demonstrations and civil disobedience campaigns sought to increase African-American voter registration and win better jobs. Malcolm X actively promoted the Black Muslim cause. Even after speaking about non-violence and wanting peace, Martin Luther King Jr. was assassinated. The civil rights movement dramatically change magnitude participation of African- American voters in both the South and the North today.By the mid-70s some 4000 African-Americans have been elected to political office at all levels of government. certified African-Americans now have a wider range of opportunities than ever before. Whether you are White or African-American, each group has faced its own peculiar challenges on its approach to democracy (Rappaport, 2001). This racism is wrong and unconstitutional. The 13th Amendment is ratified, abolishing slavery, which some people still went against it. The 14th Amendment granted ci tizenship to the former slaves and forbade states from denying any person life, liberty, or property without collectable process of the law.The 14th Amendment also guaranteed equal protection of the law for all citizens. The 15th Amendment barred states from denying citizens the right to vote based on race, color, or previous servitude (Hertz, 2009). In a perfect world, everyone would be equal. The color of ones skin, religious beliefs or sexual preference would mean nothing. We would accept everyone for whom and what they are. We would rejoice in the differences between each other instead of belittling, hating and discriminating against those differences. We dont however live in a perfect world.We live in a world filled with distrust and hate. If we dont know or understand it in our society, then it is wrong. It will be discriminated against in one form or another. We as a country have made major strides in overcoming racism, however we still have far to go. In conclusion, African Americans faced isolation, discrimination, and segregation during the post-construction period. Racial discrimination was also prevalent in the military where back soldiers were considered inferior to white soldiers and hence poorly trained and equipped.The issue of racial discrimination, isolation and segregation was not seriously tackled until the Civil Rights Act of 1964 was enacted. Civil rights activists such as Malcolm X and Martin Luther King Jr. organized the famous 1963 protest in Washington that eventually forced President John Kennedy to pass the Act. It is therefore, clear that the journey to end isolation, discrimination, and segregation to attain equality and civil rights has been hard but worthwhile. ? References Bowles, M. (2011). American History 1865- Present End of Isolation. San Diego, CA Bridgepoint. Retrieved at https//content.ashford. edu/books/AUHIS204. 11. 2 Finkelman, P. (2009) Encyclopedia of African American history, 1896-present, Madison Avenue, New Yor k Oxford University Press Rappaport, D. (2001). Martins Big Words The Life of Dr. Martin Luther King, Jr. Sitkoff, H. , & Franklin, J. (2008) The Struggle for Black equality. Hill and Wang Publication http//www. adl. org/learn/ext_us/kkk/default. asp? LEARN_Cat=Extremism&LEARN_SubCat=Extremism_in_America&xpicked=4&item=kkk http//www. history. com/topics/black-codes Civil Rights Act of 1964 http//www. our atomic number 101uments. gov/doc. php? doc=97&page=transcript.

Body Piercing and Management

Of the many growing trends, body penetrating has become very common. With this practice becoming more popular everyday, many race ar frightened away because of sanitary and health reasons. To most peoples surprise the business of body piercing is a safe and impudent procedure. Body piercing is a form of self-expression, such as tattoos and hair styles. Piercing are more widely accepted among business today than a couple years ago. Even-though they are becoming more acceptable in society today there is still a misconception of the safety, sterility, and health issues involved.Many people fear the sterility of the piercing process, ergo they dont issue forth the piercing. In actuality piercing is very safe. The piercer has to follow many steps in ensuring the safety of the piercer and the patient. After make full out the proper paperwork, the patient has to decide of the location of the piercing. There are many spots over the human body where one can get pierced. Some of the com mon spots to get pierced are the inner and outer ear, the nose, the bridge of you nose, the cheek, lips, vernacular, eyebrows, nipples, naval, and the genitalia.If one was to get their tongue pierced, the piercer must decide whether or not it is piercable. If the tongue has a large under-webbing it cannot be pierced. Once the tongue is deemed piercable, the piercer sterilizes all of his equipment. A common set-up for a piercing is as follows two pairs of working(a) gloves, many gauze, a needle, cork, rubber-band, surgical clamp, toothpick, and the jewelry. All of these are placed in a metal cartridge and placed inside a sterilizing machine.The machine heats and compresses distilled irrigate and wherefore blows steam into the cartage sterilizing everything inside. During this time the patient is instructed on the procedure and washing his babble out with bioclean. Bioclean is antibacterial mouth cleaner that destroys 99% of all mouth bacteria. During this time the piercer scrubs his hands with an antibacterial soap, and places one pair of gloves on. The patient sticks out his tongue and the piercer makes a horizontal and vertical make on the tongue if a mild antiseptic dye.The clamps are placed on the top and direct bottom of the tongue and double checked, to ensure that the needle will not pierce a vein. At this time the piercer discards his current pair of gloves and dons the second. He consequently places the needle on the tongue and pushes it through. Once the needle has passed through, he then places a cork on the end so no one is harmed. The needle is push the rest of the way through with the jewelry. The needle and cork are placed in a sealed biohazard container to await proper disposal.The clamp is removed and placed in an antibacterial solution. The other half is then screwed on and the piercing is over. All the gausses and swabs with no blood are placed in the trash, and any items with blood on them are placed in a biohazard bag. At this time th e piercer informs the patient on the proper care and maintenance of the piercing. In an interview with Richard, a piercer at Factor V in Charleston, SC, he states that the most unsanitary and dangerous time for a piercing is seven days afterward.People dont follow directions and end-up with an infection. Most piercers pass out a pamphlet with the proper care directions on it. The piercing process is safe when done by a professional. The customer is responsible for the piercing once the piercing process is over. The procedure is so safe that one doesnt even lose taste due to the piercing. Some swelling may occur and pain in very minimal because no nerves were hit. Piercing can be a healthy and fun way express oneself, when done in a clean and experienced environment. But one has to make sure that proper care is given to the piercing.

Cry, The Beloved Country Commentary Essay

And now for all the people of Africa, the beloved country. Nkosi Sikelel iAfrika, God save Africa. But he would not see that salvation. It lay afar off, because men were afraid of it. Because, to tell the truth, they were afraid of him, and his wife, and Msimangu, and the young demonstrator. And what was there evil in their desires, in their hunger? That man should walk upright in the nation where they were born, and be free to use the fruits of the earth, what was there evil in it?Yet men were afraid, with a fear that was deep, deep in the heart, a fear so deep that they hid their kindness, or brought it our with fierceness and anger, and hit it behind fierce and frowning eyes They were afraid because they were so few. And such fear could not be enumerate out, but by love. (310-311) Christianity plays a pivotal role in Patons Cry, The Beloved Country.Kumalo struggles throughout the story with his beliefs, having his religion shaken by what he sees in Johannesburg. More importantly , the entire plot revolves around the injustices Christianity has brought to South Africa and how it has turned wrong in the hands of white people. However, Kumalo notes that at the end of the book, Christianity is bringing fear to people, and this fear becomes the bases for the prejudice against blacks.This story is furthermore about a relationship between father and son. Because of Kumalos knowledgeable idea, the realization that it will be awhile earlier people are going to be able to love instead of fear, and the relationship told throughout the story, I believe Paton is aiming to create a Messiah same figure with Absalom. The father understands why Absaloms expiration is so important to society however, society does not and continues to be ruthless.However, Absaloms death is significant in the fact that it stands for the injustice between the white and blacks in South Africa. Much like Jesus died for our sins, Absalom dies for the injustice. Furthermore, his father is the on ly one that seems to see why his death is in important for the growth of a nation. Even though Absalom is far from the epitome of morality, he dies for a nation.

Occupational, safety, and security at work place

This boils to the fact that a rocker will perform his duties to the fullest only when he Is real that even when an shot occurs he will be taken good c be of. One of the most important things that an employer should provide to his employees Is safe even at a low risk site says Pushup Vela the human resource assistant at Prime aluminium Ltd.Brief History of natural rubber and security measure is that matchless of the effects of the Europeans coming course of time it was discovered that it was absolutely required to safeguard the interest of both the employer and the employee, hence the introduction of the arioso means of regulating employer-employee relationship.Initi every(prenominal)y, it was impression that the employees were being made objects of servitude scarce in the long run it became apparent that this cast of relationship requires the incorporation of rules to avoid either air divisiony being cheated out(a) reclaimly. Labor law which was fashioned to ameliorate the prevalent crisis engulfing the industry in England at a time was adopted in Nigeria as a film consequence of colonialism by the United Kingdom. 1. Purpose of the Study The purpose of this study is to seek and to find out whether there are challenges inherent in this land that a human resource manager faces in his day to day routine duties within the organization thereby suggesting possible solutions that sought to overcome virtually of the challenges and well as providing many recommendations for the organization and to discuss and understand how Prime Aluminum Ltd deals with their occupational sanctuary, wellness and security policy and what process it uses to achieve the encompassing objective of getting the relationship between employee and the employer and based on Prime aluminum Ltd. . 3 Scope of the Study By on the job(p) on the border paper, Vive been able to understand how the policy has been incorporated in the institution, the challenges faced in the human re source department in implementation of such a policy, the presentment of some of the solutions as well as some of the recommendations Vive been able to come up in the course of my research. When I carried out this research, I experienced some scopes in the area of this report.Bearing in mind the acquaintance and time of the paper has been wide- ranging specifically within the revision challenges inherent in the area of occupational health, safety and security suggesting the possible solutions as this rear basically deals with the top charge level and low prudence. Being a student of Human Resource instruction class, I got a chance to work with the Assistant Head of Human Resource, MrsPushup Villain, where I got a chance and experience to engage on a one on one interview where she took me through the occupational health, safety and security policy and processes and how it makes sure they achieve their goals in the organization. 2. 0 Focus on Organization 2. 1 About Prime Aluminu m Ltd Prime Aluminum Ltd has over 40 years of experience in their team of specialistsPrime Aluminum Casements Limited (PACE) has built a reputation of providing high quality solutions for aluminum windows, doors, morphologic glazing, curtain walling, ceilings, partitioning, claddings, office fit-outs and external fade related requirements across all building sectors. It is located on Mambas road as well as Lemur road. Prime prides itself in accomplishments and finding high quality solutions for all aluminum windows, doors, structural glazing, curtain walling, requirements for clients in an equitable, environmentally friendly, socially responsible and economical manner.Improve working conditions that are necessary to ensure higher grasp productivity, better quality work, better labor relations and compliance with quality standards. Health and safety in the piece of work is no much a new thing or stretch out in organizations but it is still handled as a new topic in, Prime Alumin um of today all because of the way it is being practiced. However, there appears to be some gradual improvement in awareness and practice. This part of this research work is meant to examine and describe the law as it relates to health and safety at work 3. FindingsResearch Questions Analysis This part of the study presents the response of the HER of Prime aluminum on safety health and security policies distributed questionnaires and as obtained from the interview How do gage policy, health and safety help with strategic decision making? Pushup Strategic decision making is all about risk management. Getting the right information on Security policy, health and safety improves the chances of success, and helps to prevent accidents which every business needs to do because accidents are so costly, in all sorts of shipway that many organizations do not realize before one occurs including Prime . He goal of the company is to focus on health of all the employees so as to get employers w ho are fit to do the Job. For example we had an employee who was injured where he was working in the site. He was a very potential employer but due to less safety we lost him. Thus we are now struggling to get one who did such a great Job which is cost us a lot. What is the perception of the employees on health and safety issues and policies? Pushup This part of this research work is meant to examine and describe the law as it relates to health and safety at work.By law, employers have to comfort employees health, safety and welfare at work. They have to make sure the workplace is safe and without risk to health. As part of this duty, employers must 1. Keep dust, fumes and noise under control 2. Make sure that plant and machinery are safe and regularly maintained, and that the systems used in the workplace are safe 3. Provide protective clothing where necessary 4. Report certain diseases and injuries to the relevant authority 5.Provide adequate first aid equipment and facilities At sites where soggy machinery are being used it is certain that the level is higher cause of the mechanical movement of parts of such machinery and therefore for the employee that will be observe or operating such machinery will be exposed to accidents. In a case like this, it should be known that the level of safety that will be provided will be much more than that of a site where ordinary hand tools are been used.Based on the above, we now understand that the level of safety device and Health protection will be higher nowadays because of the rapid mechanization of the manufacturing industry and the accidents that may occur will definitely be more fatal elicits. Do health and safety career problems actually reach the top and directors? Pushup Health and safety is one crucial element of the pleat of skills, along with finance, marketing and human resources, for example. It is becoming increasingly important because this could cost the firm a lot. UT in most cases the top director s are unuttered to reach because they are usually out of own doing business and it gets difficult for the HER to focus both on the workers at the sites and at the office. But these problems are normally taken care of at the end of the day. Could you give some examples of the biggest safety risks being faced by leaders? Pushup employees undoubtedly face their biggest risks term campaign for work. There is a very personal effect on individual directors a director cease set an appropriate example or an inappropriate one.For example, if they speed to reach yet an otherwise meeting, or if they work long hours and drive a long way home after draining meetings. Directors have the juristic responsibility if things go wrong and police will prosecute if, for example, a driver falls asleep at the wheel because of an over-long working day. What can they do about it other than set an example? Pushup One of Prime aluminums key issues is the management of occupational road risk driving fo r work which is why we offer a full range of solutions which can be targeted at those deemed to be the highest risk.These range from consultancy to driver training, and also computer-based driver risk assessments, which can be the cheapest, but most effective, way to start. What comes after that? Pushup Consider again the director who is a keen motorcyclist if he or she suffers an accident it could bring the business to its knees From Prime aluminums work with its key major award winners we know that one in five accidents occurs in the home or enchantment at leisure, so if we can reduce those, then the negative cost that accidents cause the workplace can be greatly reduced as well. What sort of training is available to directors?Pushup Because directors have limited time available they will often not be able to go on pre-scheduled safety training courses. Prime aluminum addresses this through two routes Conferences, where directors can network and key new developments can be high lighted by top speakers such as leading personal injury arresters and FETES 100 leading company case studies given by their chief executives Tailored consultancy and in-company safety training means that companies can have a day or two of intensifier focus on health and safety which is exactly right for their business.All of these options make directors aware of their legal duties and responsibilities, as well as the nest egg they could make if safety is properly managed, and the positive benefits of doing it properly. 4. 0 Challenges and solutions at Prime Aluminum Ltd security of his employees. Health and safety form an integral part of work environment. A work environment should enhance the well -being of employees and thus should be accident free.The terms health, safety and security are closely related to each other. Health is the general state of well-being. It not only accommodates physical well-being, but also emotional and mental well-being. Safety refers to the act of p rotect the physical well-being of an employee. It will include the risk of accidents caused due to machinery, fire or diseases. Security refers to protecting facilities and equipments from unauthorized access and protecting employees while they are on work.According to Folia, & et al (1993), asserted that a proper understanding of the various work situations in manufacturing, textile, mining, construction and other labor intensive industries reveal that workers are exposed to hazards. These include physical contact with poisons, dust inhalation, exposure to organic and inorganic chemicals, extreme temperatures of hot or cold, accidents, injuries, falls, burns and scalds, other dangers and sudden death. However, the health and safety of the workers have been recognized as a fundamental human right.The need to foster a safe work environment, protect co-workers, family members, employers, customers, suppliers, nearby communities and other members f the public impacted by the workplace environment is the primary goal of all Occupational Health and Safety(OH)practice An integrated occupational health and safety policy is essential for developing a stable and productive work environment. The government has in recent time enacted laws regulating the labor market and it has also been revised to promote healthier labor relations, appropriate working conditions, equity in the workplace and improved skills.Improved working conditions are necessary to ensure higher labor productivity, better quality work, healthier Barbour relations and compliance with quality standards. The economic gains associated with occupational health and safety policy improvements include a. Increased productivity and worker morale b. Reduction of working time lost due to injury and disease c. Reduced equipment down-time, reduced damages to materials and machinery, and savings in the costs of recruiting and training replacement employees d. Reduction in transaction costs such as insurance costs an d legal fees.Adequate occupational health and safety policy and standards are required for a nations continued integration into the world economy. International investors who subscribe to world- class occupational health and safety standards are reluctant to invest in markets in which local firms are able to compete unfairly through reduced occupational health and safety standards. Increasingly, African exporters particularly those who export to developed economies are being required to comply with international quality management standards.These standards require world-class performance in areas such as occupational health and safety policy, environmental protection and product safety A health and safety management system involves the introduction of processes designed to decrease the incidence of injury and illness in the employers operation (Alberta, 2006). The successful implementation of this resources, and a high level of employee participation. The components of effective hea lth and safety management system are briefly explained below a.Management leadership and organizational commitment. For this system to be effective, management must show leadership and commitment to the program. To achieve this, management should put the organizations expectation around health and safety into writing by developing a health and safety policy. Employees who forms part of the health and safety committee, should be involved in writing the policy, and to be signed by senior operating officer, to indicate the commitment of management. B.

Teaching Literacy in the Primary School

All elements of literacy be inter-related. This essay will examine the carrying process and how the instruct of utter, auditory sense, write and examineing in all influence pupils sproutment in many ways. One pupils linguistic process and literacy conditionment will be explored in this context, with a particular emphasis on his reading progression. Literacy is the ability to use language to go bingles ideas expressively, through speaking and authorship and receptively, through listening and reading. (Palmer, S 2003). The Department for Education (2012) explains that pupils acquisition of language allows them to access acquirement across the curriculum.Notably, reading aids pupils development culturally, emotionally, spiritually and kindlyly. Since 1988 and the penetration of the National Curriculum, the government have overseen the instruct of English and literacy in informs. It was not until the publication of The Rose look into in 2006 however, that a standard system for teaching reading was devised. In his report, Rose reviewed the way early reading was taught and advised that all children should have a secure foundation of phonics knowledge so that they atomic number 18 equal to link graphemes to phonemes and blend these into words.As a solving, it became statutory for schools to use a everyday, systematic, synthetic style of teaching phonics. To help schools instigate this new teaching style, the Communication, spoken language and Literacy Development Plan (CLLD) was introduced in 2006. Local authorities were given trained consultants, often teachers, who could model high quality phonics teaching and ensure the findings of the Rose Review were implemented effectively.Ofsted (2010) reported, that several schools, from a sample demonstrating outstanding practice in their teaching of early literacy, used a scheme such as earn and Sounds, published by the Department for Education and Skills (DfES) in 2007. These schemes initially te ach phonemes alongside their create verbally representation (graphemes), followed by the accomplishment of blending and segmenting graphemes to write and rewrite individual words. It is widely recognised that the teaching of phonics modifys children to decode words, but does not teach an understanding of vocabulary. The skill of decoding is not enough to enable children to read effectively.Rose (2006) also observed this in his review, Different kinds of teaching are needed to develop word recognition skills from those that are needed to foster the comprehension of written and spoken language. Wyse and Parker (2010), cited by the Institute of Education (2012), argue in favour of contextualised teaching, which begins by looking at whole texts that pupils can relate to, therefrom motivating them to read independently. They claim that although important, the teaching of phonics, in a way where it is exaggerated above all other elements, comes with serious risk and that childrens l anguage skills develop best through classroom talk.Until deep, the importance of Speaking and Listening was overlooked by many schools. Ofsted (2005) reported that the teaching of speaking and listening had been drop and the range of contexts in which children are given the opportunity to converse with their peers was constrained. It is crucial to understand that as each strand of literacy is equally important, a child who struggles to communicate verbally will have difficulty in communicating or understanding concepts in written form. Douglas (2009) observes, Speaking and listening skills underpin all education and are the start of all other literacy skills.Rose (2006) observed, Schools provide massive opportunities and unique advantages for developing speaking and listening skills. Activities such as talking partners develop childrens vocabulary by getting them to share their ideas about set questions in short bursts, throughout the lesson. This technique can be integrated int o the teaching of any concept across the curriculum, meaning the opportunities to acquire new language are infinite. Drama is part of the Speaking and Listening strand of the National Literacy Strategy.McMaster (1998) explains that it is an priceless tool as it supports every aspect of literacy development. Drama can extend vocabulary develop decoding and conversational skills and improve understanding of syntax, as strong as metacognitive knowledge. Drama also aids personal, social and emotional development (PSED). By engaging in situations as if they were real, children build the confidence to express themselves and develop creativity and empathy. These attributes are closely associated with reading development, as they facilitate comprehension and response (Wagner, B.1988 Vygotsky, S. 1976 cited by McMaster, J. 1998)Poetry is also a useful tool to improve pupils personal, social and emotional development (PSED). Children should be encouraged to believe that poetry is a normal hu man activity, a very intense one and an activity that deal often resort to at crucial times in their lives which shows its central importance. (Longley, M. 2008 cited by McLeish, J. 2008) In Early Years Foundation symbolise (EYFS) and Key Stage 1, learning nursery rhymes and other simple poems and songs assists the development of phonological awareness.The repeated rhythm and rhyme patterns develop an understanding of how words can be gloomy into syllables. The next stage is to understand that each syllable is made up of a structure of sounds, onsets (the initial phoneme) and rimes (the remaining sound in the syllable. ) Wilson (2005) believes this is a fundamental skill to develop if a child is to blend and segment efficiently. Sharing poetry and re-telling stories provide the basis for the Talk for Writing initiative, developed by The National Strategies (2010), in conjunction with Pie Corbett.The National Strategies explain good readers learn about the skills of writing from t heir reading and draw (consciously or unconsciously) upon its models in their own work. Reciting poetry and rhymes, and re-telling stories enable children to internalise language (referred to as imitation) so that it can later be reused in their own writing automatically. Once this skill is mastered, children can continue to transpose parts of the story (innovation) using aids such as story maps and shared writing. The final stage is invention, at this point pupils use the language and writing styles they acquired to create their own pieces of writing.In his early workshops, Corbett (2008) stated that these approaches to learning also work extremely well when teaching children to write in a non-fictional context. The use of speaking and listening is also an invaluable tool when teaching children with special educational needs (SEN). Corbett (2004) states Many children with special needs have succeeded using this multi-sensory, oral strategy to developing composition. These childr en need as many opportunities as possible to internalise new vocabulary and writing styles that may be unfamiliar.The same is applicable to pupils who are learning English as an additional language (EAL). These children have the extra hurdle of comprehending vocabulary and writing styles that may differ greatly to that found in their first language. Cummins (1999) explains, There are clear differences in acquisition and developmental patterns between conversational language and schoolman language, or BICS (basic interpersonal communicative skills) and CALP (cognitive academic language proficiency). Children generally develop BICS within two years of immersion in the target language, providing they spend much of their time inschool interacting with primaeval Australian speakers.However, it typically takes children between five and seven years to develop CALP and therefore be working at a linguistic level similar to their native speaking peers. It is therefore vital to provide a w ide, varied range of opportunities for students to converse. The DFES (2004) explains, Bilingualism is an asset, and the first language has a continuing and significant role in identity, learning and the acquisition of additional languages. Children who are truly bilingual will often work at a higher academic level than those who speak one language.Child C is a six year old boy. He is a native English speaker, although he does have developmental problems with his speech and is currently see a speech therapist. He lives with both his parents and his two brothers, aged seven and two. The following information has been sourced from interviews with his teachers and bewilder, his speech and language reports and his records of attainment. Permission from Cs parents and school were sought in order to include him in this study, and for reasons of confidentiality his anonymity will be respected throughout. C was born nine days late with no issues at birth.C refused solid food until about 10 months of age, and his mother explained that he has always expressed a dislike for food that requires a lot of chewing. It was suggested to Cs parents that this may be a contributing factor to his speech difficulties. C started babbling at about 22 months, experimenting with sounds and a few words. His mother was able to understand his attempts to communicate by around 3 years of age, although other members of his family and the practitioners at his nursery school struggled to understand him, this caused C to become greatly frustrated and stressed when trying to express his wants and needs.It was at this point that C was referred for speech therapy. He was also referred for hearing tests which did not uncover any auditory problems. C and his older brother have shared books with their parents before bedtime since C was 2 years old. His mother explained that neither of the boys demonstrated a strong touch for books and requests to be read to, apart from before bed, were infrequent . Cs lack of interest was also observed by his EYFS teacher, as a result C was initially only given one book a week to share at home, as more than this tended to overwhelm him and generate a refusal read at all.It was also noted in the early months of EYFS, C disliked contributing to group discussion or conversing extensively with his peers as a result of insecurity about his speech problems. He felt much more confident talking to adults on a one to one basis. C left field EYFS with a reading level higher than the national average for his age group, although lower than that of most of his classmates. His ability to blend and segment graphemes was good, meaning he was able to read and write a range of simple words.His ability to form particular cluster sounds orally remains an issue, but his confidence to communicate with his peers and yield to group work has improved significantly. As a result, his range of vocabulary and comprehension has also improved. C has continued with his sp eech therapy in KS1 and he receives daily interventions with a teaching assistant to help with his sound formation. Cs current class teacher has observed that his reading has significantly improved recently C will now read quite complex sentences with some expression, using a range of decoding techniques such as segmenting and looking at accompanying pictures.C recently read a short passage to the rest of his class, demonstrating his improved reading skills and confidence levels. In conclusion, speaking, listening, reading and writing are all of equal importance. The strategies, tools and initiatives explored in this essay help children develop their reading skills. Each strand can be built upon each other to develop a pupils literacy development as a whole. Being literate is essential if a child is to access all areas across the curriculum.

What’s Stifling Creativity at Coolburst

Everyday the world is changing around us. It is an essential part of fruit, maturity and effectiveness. Everything from the change in weather, an age, government, or religion affect the demeanor people view, think and perform in certain situations and as a whole in society. History has turn up that creative minds can ultimately change an outcome for the better or even the worst. From telegraphs to cellular phones, McIntosh computers to IPODs and IPADs, it was the presidencys of these products that took a major risk in investing into soulfulnesss creative idea that affect our society as a whole.These companies have seen their fair share of profits lows and highs through the toughest economy eras, but they keep to persevere and prosper through such times by thinking out the realm of possibility and making their possibilities into reality. However, what happens when an organization suppresses the thought of reinventing themselves to adapt to changes in society and even the world? Coolburst is experiencing a major organizational struggle between what worked for them to get them where they are today and what it takes to continue to strive in the future.Coolburst is fit(p) in Miami, Florida. The drink products that they serve are sold in schools and restaurants. The traditional views of during business have forwarded them much success from their beginnings. They have experience great growth through the years, but recently, they profit margin has remained steadily with no boosting sales. Director of Marketing Sam Jenkins has challenged Coolbursts view and management on changing their way of thinking and opening their mind to new ideas under former CEO Garth LaRoue.Jenkinss new ideas of productivity and innovation were considered unorthodox within the organization. Ultimately, Jenkins left the Coolburst to go with a company that was very to a greater extent innovated and creativity. Witnessing these differences of opinions between Coolburst and Jenkins is new CE O Luisa Roberado. Now, Roberado is facing one of biggest challenges yet for Coolburst, what changes can be made to make Coolburst more profitable and more creativity to keep up or even surpassed the demand of an ever changing society.Was Jenkinss new idea that far fetched with the organization or was he on to something that can change the way Coolburst does business to compete with potential competitor? This case study will explore why Coolburst had a hard time accepting Jenkinss idea on making Coolburst better in the future, what the organization can do to keep with kindly changes, and how Roberado can implement and even changed the current ideology of Coolburst and help the company tapped into their resources and flourish the company to the top in innovation, creativity, and in profit margins.

Zizek on Ideology and the Relationship Between Ideology and “The Real”

Zizek on political orientation and the Relationship Between Ideology and The existent CMNS 410 Professor Rick Gruneau December 13, 2011 Zizek on Ideology and the Relationship Between Ideology and The Real Slavoj Zizek is peerless of the leading theorists on political supposition since the 1990s and his fantasyions of the legitimate versus the emblematic versus the imagined ar of cross importance when dissecting the question what is political orientation? Zizeks critique of ideology and attempt to unpack its inner workings is fascinating, he is a powerful intellectual who aims to expose the fake workings of society. In this paper I pull up stakes outline Zizeks definition and approach to the accept of ideology, paying particular attention to the proportionships he draws surrounded by ideology and the real, as opposed to the imagined and the symbolic. Zizek opens the book Mapping Ideology (1994) with the introduction The Spectre of Ideology, where he defines and openly criticizes the idea of ideology and its illusory personality.First he presents us with the idea that ideology is a sort of ground substance, a generative matrix that regulates the relationship between visible and non-visible, imaginable and non-imaginable, as well as changes in that relationship (italics mine, p. 1). He further explains not everything that seems to be ideological, necessarily is, claiming that unless thither is a link to power relations in the social realm he does not exact something to be ideological.He points out that sometimes what we consider to be ideological in fact is not only withal how at other times, things which we may not perceive to be ideological, actually continue a very strong ideological orientation. He states that the starting point of the critique of ideology has to be the full acknowledgment of the fact that it is easily practical to populate in the guise of justness ideology that is and this is an important realization for it ispels a common misconception we have of ideology, especially here in the west that, ideology is about lying or misleading others and society. Instead Zizek posits the idea that the electrical capacity of a message is not what makes it ideological, but instead it is the the way this substance is connect to the subjective position implied by its own process of enunciation that makes it so (Zizek 1994, p. 8).In other words, regardless of whether the content (of a message or prey or interaction) is true or false, it becomes ideological the moment that content functions to achieve some relation of social domination and even more importantly, he adds in an inherently non-transparent way, reiterating that often times ideology is in fact of a misleading nature but not necessarily in content (italics mine, p. 8) it is from this standpoint that we can begin to understand and critique the concept of ideology.It is important to note here, although Zizek stresses the importance of recognizing dyna mics of power relations (rather than content) which constitutes ideology, he warns this can also be disadvantageous if it reduces the cognitive value of the term ideology and makes it into a mere expression of social circumstances (p. 9). Considering this, as Gerofsky (2010) explains, Zizek puddles on Hegels theory of the triad as a heuristic for further developing the theory of ideology, which is something I will address later in this paper, after we go a little bit deeper in defining ideology.According to Zezik then, a necessary condition for something to be ideological is that there must(prenominal) be a relation or motivation to power in some way, and it must be done so in a way which is not apparent to the addressees (Zizek, 1994). even so this is a rather general and overarching consideration when defining the term ideology and it is important to deconstruct the term even further before we proceed in anal retentiveyzing its inner workings and effect on society. Zizek states ideology is a systematically distorted communication a text in which under the influence of sneaking(prenominal) social interests (of domination, etc. a gap separates its official, public meaning from its actual intention that is to say, we are dealing with an unreflected tension between the explicit enunciated content of the text and its pragmatic presuppositions (Zizek, 1994, p. 10). Ideology is a system, he argues, of principles, views, theories destined to convince us of its truth, yet actually serving some unavowed particular power interest (p. 10). An example Zizek presents to illustrate this point is the way media portrayed the conflict and cause of the Bosnian war.News coverage consisted of innumerable accounts of the histories of not solitary(prenominal) Yugoslavia but the entire history of the Balkans from medieval times (p. 5). This incredible amount of selective in brass, of the struggles and relations between Bosnia and other countries over decades, if not centurie s, gives audiences the impression that they must know and understand all the background information of this issue if they are to have an opinion on it or take sides, again presenting countless hours of information and debate on the issue.Zizek explains that although this is a sort of inversion of what we normally constitute as ideological messaging, and it is unlike the misrepresentation and incessant demonization of Saddam ibn Talal Hussein which was circulated to give justice to entering into the Iraq war, the Bosnian war ideological messaging that took place is in fact more cunning, the over exaggerated and false demonization of Saddam Hussein. ecause to put it somewhat crudely, the evocation of the complexity of circumstances serves to defer us from the responsibility to act (p. 5). He explains that instead of withholding information (as the media almost often does), or misrepresenting information (Saddam Hussein), in the case of the Bosnian war the media over saturates audien ces with information to the point of immobilizing them to make a decision or take action against the fact that this war is spurred by political, economic and monetary power interests.Zizek explains the purpose of going into war was portrayed as a need to emend unacceptable human rights conditions in the country, and although human rights conditions may very well be unacceptable in that country, and then improve as a precede of the invasion, the true motivations for that war (power, domination, money) were kept hidden. This also illustrates the point made earlier about ideology not necessarily needing to be false in its information, but rather hidden in motive, for the information they presented was by no means false or limited, it was excessive, which proves to be just as debilitating a system on the general public.Zizeks examples and definitions of ideology discussed above demonstrate the division of ideology from Marxs false consciousness theory (Gerofsky, 2010), but perhaps on e of the most important classifications Zizek makes in the realm of ideology, is its connection to dislocation (dislocating truth from falsity) and how this relates to the idea of the Real (Stavrakakis,1997). Coming from the Lacanian theoretical background, the concept of Real versus Symbolic versus speculative is an integral part of Zizeks theory, one which sets him apart from traditional conceptions of ideology.The question of the Real also cannot be separated from the dislocation and presentation of the truth, so these deuce must be considered together in asserting the concept of ideology. Zizeks Real draws attention to a fascinating idea, that there is a difference between what is actually real in our universe and what is simply a created real by our social structure and by society (Stavrakakis, 1997). The Real, the true real, is the part of our world as revealed in our experience, which escapes our attempts to symbolize and represent it in a final way (1997, p. ). The real is the raw and unstructured experience of what is not yet symbolized or imaged by our social structure, by language, by symbols, and it in fact cannot be symbolized in such a way. Unlike the social reality, the true Real is unattainable to represent, explains Stavrakaki of Zizeks theory, impossible to master or symbolize, whereas the social reality is nothing but symbolism and our desire to categorize any part of our experience into a definition or material conception of some sort.The real is not only opposed to what is socially constructed as real, the symbolic, but also it is even farther removed from the imaginary, which falls farthest away on the spectrum, from true reality. The symbolic comes closer to the Real but there is settle down a gap and something will always be missing from the symbolic real for language can never be a full representation of the real, the true Real in time is always in its place. The symbolic real, however is still of importance to Zizek, for it play s the largest role in our society and is perhaps the integral component to ideology in the most general sense.The symbolic, although generally in the dimension of lauguage, Lacan (whos theories Zizek has based his own theories of ideology on) does not describe the symbolic as solely equal to language, because linguistiscs are also present in the realm of the imaginary sphere (Lucaites & Biesecker, 1998). The symbolic rather, is about the relationship to the Other, it is about difference and the signifiers which create a symbolic order. For Lacan the symbolic is characterized by the absence of any fixed relations between signifier and signified (Lucaites & Biesecker, 1998).Lastly there is the realm of the imaginary, when Lacan discusses this stage he refers to the formation of the ego. Identification is an important part of the imaginary, for the ego is formed by identifying with the counterpart or specular image (Lucaites & Biesecker, 1998). The ego, fundamentally narcissistic, is c entered on identification with insanity and this alienation is another feature of the imaginary. The imaginary is most fundamentally, however, a constitution of surface appearances, ones which are formed in deception as part of the social order.Going back to Zizeks theory on ideology, he suggests that one of the most problematic areas of the concept, is that we as theorists, try to escape from the grip of ideology in order to observe the world from an objective lensive position, however the moment we feel we have managed to take up a position of truth, from which we can condemn the lie of an ideology, we now fall back into the grip of ideology again because our understanding of the concept is structured on a binary arrangement, which is constantly playing on this relationship between reality and ideology.It is such the issue of ideology, that the moment we feel we are in the realm of truth, at last, we are in fact instantly back into the ideological exchange, without recognizing i t (Stavrakakis, 1997). Zezik does not offer a solution to this, however he offers a way to counter the problem, and this is where the concept of the Real (vs Symbolic vs Imaginary) comes into play, to attention us recognize and step external the atmosphere of ideology that surrounds us.Instead of the binary relationship between reality and ideology, now there is a trey way relationship. Zizek favours the Real over the other two constructs because he argues, the symbolic, although it is representing reality it is in fact where fiction assumes the guise of truth (Stavrakakis, p. 3), and the imaginary construct, is of course even farther away from that reality, therefore the Real should be the focus of our understanding.The Real is the only non-ideological position available, and although Zizek does not claim to offer access to the objective truth of things, he explains we must begin with assuming the existence of ideology in every aspect of our society, and to take up an actively c ritical attitude towards it. This Stavrakakis argues is the chief(prenominal) goal of Zizeks theory, to expose the need for constant critique of the ideological realm, especially in a time where our society has proclaimed that ideology is a thing of the past and no longer relevant in todays world.Zizeks theory of ideology is a contemporary one which moves beyond traditional definitions of this concept and is not concerned with the way ideological practices worked in the past and in history, instead he is intrigued with the here and now and argues strongly that the concept of ideology is far from extinct in todays society contrary to what many would like to believe. And he explains that rather than discarding the notion completely, what we need to do to understand todays politics in a completely new way of looking at it and defining what it means to be in ideological space and time.Those who believe we are past the concept of ideology, he argues, are in an archeological fantasy and this is only a sign of the great ability of ideology to ingrain itself without our recognition. In some of his famous presentations Zizek talks about the ideological meaning ingrained even in the simplest of human object and appliances, ones we dont even recognize contain an ideological message. His famous example, and one he self critically acknowledges to be some sort of anal fixation which he needs to address, is the example of toilets and how they are constructed in different ideological environments.In France he explains, toilets are constructed with the jumble at the back, so that when used, the excretory product falls directly in the hole and disappears he equates this with Frances positively liberal ideology out of sight out of mind. In Germany, the toilets are constructed with the hole at the front, in a way that holds the excrement on a shelf (not in water or instantly disappearing) but rather in a way for the individual to see and observe the specimen for worms and any other diseases he explains this is indicative of the strongly onservative ideology of Germany, where everything is business and completed as necessary. In the Anglo-Saxon world, specifically in America, he explains toilets are somewhere in between, when used the excrement falls in the water but still remains, it is not completely hidden but also not completely displayed this shows the median position the Anglo-Saxon society usually takes on, not too extreme in either respect (Zizek presentation, Youtube. com). This rather disgusting but nonetheless interesting observation does an excellent job of portraying his theory on ideology.First, ideology is very much still at play in our society and should be actively observed and considered (in order to minimize any negative and tempestuous effects it may pertain), and secondly, in order to even be able to recognize the workings of ideology in our everyday lives, we have step outside of our customary reality to which we are so well acc ustomed to, for this symbolic reality is not the Real, and in taking ourselves out of the imaginary and symbolic which appears to be truth and reality, we can then perhaps attempt to get a true glimpse of what he calls the Real.References Gerofsky, S. (2010). The impossibility of real-life word problems (according to Bakhtin, Lacan, Zizek and Baudrillard). Discourse Studies In The pagan Politics Of Education, 31(1), 61-73. doi10. 1080/01596300903465427 Lucaites, J. , & Biesecker, B. A. (1998). Rhetorical Studies and the New Psychoanalysis Whats the Real Problem?Or Framing the Problem of the Real. Quarterly daybook Of Speech, 84(2), 222. Stavrakakis, Y. (1997). Ambiguous ideology and the Lacanian twist. Journal of the Centre for Freudian Analysis and Research, 8, 117-30. Zizek, S. (1994a). The spectre of ideology. In S. Zizek (Ed. ), Mapping ideology (pp. 1-33). London & New York Verso.

Persuasive research paper Essay

Everyday as we commute down the passageway we keep in line motorcyclist drive past us. What is the one thing that we can every last(predicate) agree mortals riding wheels arrest in commons? It is non a trick question. The answer is very simple they entirely share the green of riding a wheel. What is in detail is very distinct however, are the choices of attire when operating their cycle. rough individuals are brave enough to travail shorts, tank tops, and sandals. On the opposite end of the spectrum, you have both(prenominal) of the wiser ones that chose to wear a helmet, gloves, protective jacket, eye protection etc.Why the distinct difference? The fact is that a great percentage of ramrs refuse to wear the fitting protective equipment. Due to an increase in bicycle put one overrs within the recent years, a national protocol requiring certain equipment, such as a helmet, to be worn when riding a motorcycle should be instituted. there are many an(prenominal) br ing factors to motorcycle fatalities, however in that location can be a culture of change, specially with the proper acquaintance on how each piece of gum elastic equipment can encourage at preventing injury or devastation Every year that passes by, notice that more and more motorcycle share the road with ourautomobile drivers.We big businessman wonder why there has been a shift in choice of transportation. Is this a trend or fad that the population is going through? According to the Governors passage Safety Association, study data from 1976 to 2012 betoken that motorcyclist fatalities track motorcycle registrations quite closely and that registrations track inflation-adjusted gasoline prices. If the economy continues to improve and gasoline prices abide high, then motorcycle 1 Tenorio registrations, travel, and fatalities pass on continue to rise unless active measures are taken.(Hedlund). non completely do we think there are more motorcycle riders on the road, they ha ve be this to be true. Motorcycles are generally more fuel-efficient than cars, making them a very good alternative vogue of transportation when gas prices stay at a consistent high price. It is elementary mathematics if there are more motorcycles there is more individuals susceptible to accidents. Additionally, they have proven that motorcycles are more disposed(p) to be involved in a motor vehicle accident than any other vehicle. info collected in 2007 proved that per vehicle mile driven,motorcyclist were approximately 37 times more adroit to die in a motor vehicle accident and nine times more potential to be injured in an accident. They withal researched the ability of a helmet to protect against fatal injuries in motorcycle accidents. NHTSA estimates that helmets saved the lives of 1,829 motorcyclists in 2008. If all motorcyclists had worn helmets, an additional 823 lives could have been saved. (Motorcycles merchandise Safety Facts 2008 Data).As motorcycles arrive more abundant, it is imperative that we reduce the probability of death as much aspossible. As proven above, helmet wear can be a life or death-determining factor. You can force motorcycle operators to wear helmets by implementing laws, but the combination of alcohol and motorcycle outgrowth can have a withering impact despite helmet wear. When operating a motorcycle an operator needs all of their senses at adept capacity. Alcohol is central nervous system suppressant substance, causing you body to have a reduced reaction time when the situation arises.The reported helmet use evaluate for motorcycle riders with BAC levels higher thanthe legal pin down killed in traffic crashes was 46 percent, compared with 66 percent for those with no alcohol (Motorcycles Traffic Safety Facts 2008 Data). non only does alcohol reduce reaction times, it also has an impact on your ability to make perspicacious decisions. It makes individuals push the limits of their motorcycle and their riding abili ty to levels they normally 2 Tenorio would not, and the majority of the time while not wearing the equipment they should. In 2011, the NHTSA calculated 4,323 motorcyclists were killed, and 33% (1426) of the riders were underthe influence of alcohol (Watson).How can the country as a whole help reduce the amount of fatalities we currently have due to motorcycle accidents? It is not a very simple answer. It would implore involvement from both the people and the government to make this happen. star way the government can aid in the reduction of motorcycle fatalities is through the implementation of regulations, which require and enforce the wear or motorcycle protective equipment. Dating back to 1966 the government tried to confabulate the requirement of helmet wear by the states.They tried to do this by threatening with the reduction of federal-aid highway pull funds for the states that did not comply with the implementation of universal helmet use law by 1967. By 1975 all but 3 st ates had adopted and implemented such laws. Unfortunately the Supreme Court deemed this law unconstitutional. short after revoking the Act, states gradually began to weaken helmet wear laws, since it was no longer a federal requirement (Helmet Laws). a great deal like seat belt laws have been implemented across the majority of the states due to increased survivability rate wheninvolved in an accident, the wear of helmets when operating a motorcycle should be mandated.The responsibility should not only be weighted only on the federal and state governments, individuals should take responsibility also. Many non-profit organizations plump diligently to tray and raise motorcycle safety awareness with thinks like bumper stickers, fund-raising rides, and bike meets. Additionally insurance companies have aided in the increased awareness by handing out information pamphlets at locations like rack Week in Daytona Beach. Another factor thataids in the reduction of motorcycle fatalities is proper operation education.Florida is one of many states that require the operator to take a Motorcycle Basic Riders course in order to be able 3 Tenorio to receive the motorcycle endorsement on their licenses. Without this endorsement you cannot legally operate a motorcycle. With this course even people that have never been on a motorcycle can learn the basic in order to operate it on the roads. The Motorcycle Safety rump (MSF) offers motorcycle rider education and training programs and courses, and supports governmentalprograms by participating in research and general awareness campaigns and providing technical assistance to state training and licensing programs (Morris).The Department of Defense, more specifically the joined States telephone circuit Force, uses courses from the Motorcycle Safety Foundation to teach the military riders how to operate a motorcycle. In order for an individual to operate a motorcycle they have to complete the basic riders course. Within one year of the extremity of the initial course they are necessary to complete an intermediate course such as the basic riderscourse 2, advanced riders course, or the sport bikes handling course.Once these two requirements are complete, they are required to do refresher training every five years. In addition to the training, the department of defense requires all members, military and civilian, to wear protective equipment while driving on any DOD installation. That protective equipment consists of helmet, gloves, long-lasting over the ankle footwear, long sleeve shirt or jacket, long durable pants, and eye protection. If not properly equipped, individuals are not allowed to enter the installation.If the DOD is doing this to help keep the members of the military community safe, why shouldnt the rest of the country follow in those footsteps? When we think motorcycle safety, 90 percent of the time the first image that comes to mind is a helmet, as it should. The helmet is the single-handedl y the most important piece of safety equipment that a motorcycle rider shouldnt go without. However, there are many other rider protective equipment components that play a vital role in the safety of the person. Between 2001 and 2008, more than 34,000 motorcyclists were killed and an estimated 1,222,000 persons.4 Tenorio were treated in a U. S. emergency brake department for a non-fatal motorcycle-related injury (Motorcycle Crash-Related Data). This data supports the thought dish up that even though helmets are crucial at protecting against head injuries, there are many other portions of the body that are at harms way if not properly covered. 75 percent of the non-fatal emergency room visits involved parts other than the head. The other attire that might contribute to a safer ride includes, but not limited to, long durable pants, durable top, gloves, durable over-the-ankle footwear, and reflective equipment.Despite that it will probably never be deemed mandatory to wear these item s, it is important for riders everywhere to understand the devastating make an accident can have on their bodies when choosing not to wear the proper gear. There is a common mis thoughtion that the gear makes the ride more uncomfortable and, it is believed that it makes it more difficult to operate and maneuver the motorcycle. That is a falsehood Properly fitted helmets of decent quality not only will it protect your head, but also a full-faced helmet will make for a more comfortable ride. The helmet does this by preventing foreign objects and debris fromconstantly contact the riders face, and most importantly from landing in the eye.Gloves that fit snug the hand will protect it from road rash in the event that you make contact with the pavement and it also improves handgrip with the handlebars aiding with better handling. There are gloves out on the market that have additional padding in the palm of the hand, to help with comfort and provide support and a barrier in the event of a fall. The same concept can be applied to footwear. It is unbelievable that there are people out there that would ride a bike in flip-flops and think it is comfortable.Not only does it not protect the appendages, but also it makes it harder to control the bike. When choosing footwear you have to find a medium between protection and comfort. Wear something that provides the proper amount of protection but does not hinder your ability to control or maneuver the motorcycle. 5 Tenorio Choosing comfort over safety should never be an option. More specifically when you are talking close the portion of your body that controls all bodily functions. With the implementation and enforcement of a universal helmet law, the fatality rate of motorcycle accidents woulddecrease.In the past the universal helmet law failed. With that in mind, we can learn from our mistakes and see trough an stiff and legal legislation. The ultimate goal is not to interfere with he rights of individuals, but to help protect the citizens so they can continue to enjoy the freedoms we have in the United States. The amount of information revolving around motorcycle safety out for public access is almost overwhelming. Therefore, there shouldnt be an excuse why people refuse to wear gear that will only help protect them and their bodies from the dangers of riding a motorcycle.Works Cited 6 Tenorio Hedlund, James. Spotlight on Highway Safety. Motorcyclist Traffic Fatalities by State 2012 Preliminary Data. Governors Highway Safety Association, 1 Apr. 2013. Web. 09 July 2014. Helmet Laws. State Motorcycle and Bicycle. Governors Highway Safety Association, 1 July 2014. Web. 06 July 2014. Morris, C. C. , Ph. D. Motorcycle Trends in the United States Bureau of Transportation Statistics. Motorcycle Trends in the United States Bureau of Transportation Statistics. Bureau If Transportation Statistics, 14 May 2009. Web. 07 July 2014. Motorcycle Crash-Related Data. Centers for Disease Control and Preventio n. Centers for Disease Control and Prevention, 14 June 2012. Web. 06 July 2014. National Highway Traffic Safety Administration. Motorcycles Traffic Safety Facts 2008 Data (2008) 1-6. National Highway Traffic Safety Administration. NHTSAs National Center for Statistics and Analysis, 1 Dec. 2008. Web. 22 June 2014. Watson, Tim. What The Latest NHTSA Fatality Stats Reveal About Motorcycle Safety. Ride away RSS2. Ride Apart, 29 May 2013. Web. 09 July 2014. Workman, Danny. Deadly Motorcycle Accident Statistics. Examiner. com. The Examiner, 28 May 2009. Web. 09 July 2014. 7.

The Higher Arithmetic – an Introduction to the Theory of Numbers

This page intentionally left blank Now into its eighth con stateeing and with additional material on primality examination, written by J. H. Davenport, The high Arithmetic introduces concepts and theorems in a track t wear does non require the evidencereader to suck an in-depth spangledge of the theory of yields entirely alike touches upon matters of duncish mathematical signi? butt jointce. A companion website (www. cambridge. org/davenport) provides more(prenominal) details of the latest advances and sample code for important algorithms. Reviews of earlier editions . . . the well-known and charming introduction to morsel theory . . bum be recommended both for free lance field of honor and as a reference text for a normal mathematical audience. European Maths alliance Journal Although this book is non written as a textbook but rather as a work for the bothday reader, it could sure enough be officed as a textbook for an undergrad course in flake theory and, in the reviewers opinion, is far superior for this nominate to whatever other book in English. Bulletin of the American Mathematical Society THE HIGHER arithmetical AN INTRODUCTION TO THE THEORY OF NUMBERS Eighth edition H. Davenport M. A. , SC. D. F. R. S. late Rouse Ball professor of mathematics in the University of Cambridge and Fellow of Trinity College Editing and additional material by James H. Davenport CAMBRIDGE UNIVERSITY PRESS Cambridge, b be-assed York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. cambridge. org In chassisation on this title www. cambridge. org/9780521722360 The estate of H. Davenport 2008 This publication is in copyright.Subject to statutory draw offion and to the provision of relevant collective licensing agreements, no reproduction of each part whitethorn take place withou t the written permission of Cambridge University Press. First published in instill produceat 2008 ISBN-13 ISBN-13 978-0-511-45555-1 978-0-521-72236-0 eBook (EBL) paperback Cambridge University Press has no responsibility for the persistence or the aline of urls for external or third-party internet websites referred to in this publication, and does non guarantee that both content on oft(prenominal) websites is, or pull up stakes remain, accurate or appropriate. CONTENTS Introduction I calculateisation and the run agrounds 1. 2. 3. 4. . 6. 7. 8. 9. 10. The laws of arithmetic deduction by induction superlative poesy The fundamental theorem of arithmetic Consequences of the fundamental theorem Euclids algorithm around other reasonedation of the fundamental theorem A property of the H. C. F Factorizing a follow The series of prep atomic procedure 18s page octet 1 1 6 8 9 12 16 18 19 22 25 31 31 33 35 37 40 41 42 45 46 II Congruences 1. 2. 3. 4. 5. 6. 7. 8. 9. The congruousness nonation Linear congruitys Fermats theorem Eulers function ? (m) Wilsons theorem algebraic congruousnesss Congruences to a top modulus Congruences in several unknowns Congruences covering all add up v vi troika Quadratic Residues 1. 2. 3. 4. . 6. Primitive roots Indices Quadratic residues Gausss lemma The law of reciprocity The distribution of the quadratic equivalence residues Contents 49 49 53 55 58 59 63 68 68 70 72 74 77 78 82 83 86 92 94 99 103 103 104 108 111 114 116 116 117 120 122 124 126 128 131 133 IV Continued Fractions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Introduction The ordinary continued fraction Eulers ob dish The convergents to a continued fraction The equating ax ? by = 1 In? nite continued fractions Diophantine judgment Quadratic ir symmetrynals Purely half-yearly continued fractions Lagranges theorem Pells equivalence A geometrical version of continued fractionsV Sums of Squ atomic consequence 18s 1. 2. 3. 4. 5. takingss re hold able by cardinal squ atomic build 18s Primes of the counterfeit 4k + 1 Constructions for x and y office by cardinal squargons Representation by three squ ares VI Quadratic Forms 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Equivalent forms The discriminant The design of a subroutine by a form Three recitations The reduction of positive de? nite forms The reduced forms The centre up of representations The class-number Contents VII Some Diophantine Equations 1. Introduction 2. The compare x 2 + y 2 = z 2 3. The equation ax 2 + by 2 = z 2 4. Elliptic equations and curves 5.Elliptic equations modulo qualitys 6. Fermats Last Theorem 7. The equation x 3 + y 3 = z 3 + w 3 8. Further learnings vii 137 137 138 140 145 151 154 157 159 165 165 168 173 179 185 188 194 199 200 209 222 225 235 237 VIII Computers and Number possibility 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Testing for primality Random number generators Pollards operatoring manners comp geniusnt partisation and pri mality via elliptic curves Factoring large add up The Dif? eHellman cryptographic method The RSA cryptographic method Primality testing revisited Exercises Hints Answers Bibliography IndexINTRODUCTION The higher(prenominal) arithmetic, or the theory of amount, is confound-to doe with with the properties of the immanent song 1, 2, 3, . . . . These bring forth must tolerate exercised human curiosity from a very early period and in all the records of ancient civilizations there is evidence of some preoccupation with arithmetic over and supra the needs of everyday life. But as a systematic and in subordinate science, the higher arithmetic is entirely a creation of modern seasons, and can be said to fight from the discoveries of Fermat (16011665).A peculiarity of the higher arithmetic is the great dif? culty which has often been experienced in proving unsophisticated popular theorems which had been suggested quite a born(p)ly by quantitative evidence. It is just this, sa id Gauss, which craps the higher arithmetic that magical charm which has make it the favourite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other move of mathematics. The theory of number is planetaryly considered to be the purest branch of pure mathematics.It certainly has very few occupy applications to other sciences, but it has ace feature in uncouth with them, namely the inspiration which it derives from experiment, which takes the form of testing possible oecumenical theorems by mathematical examples. Such experiment, though necessary in some form to progress in every part of mathematics, has played a greater part in the development of the theory of be than elsewhere for in other branches of mathematics the evidence found in this demeanor is too often fragmentary and misleading.As regards the present book, the compose is well aware that it entrust not be read without effort by those who are not, in some sense at to the lowest degree, mathematicians. But the dif? culty is partly that of the subject itself. It cannot be evaded by using fallible analogies, or by presenting the validations in a dash viii Introduction ix which may convey the main idea of the descent, but is inaccurate in detail. The theory of numbers is by its nature the most involve of all the sciences, and demands exactness of thought and expo from its devotees. The theorems and their proofs are often illustrated by numerical examples.These are globally of a very simplex kind, and may be despised by those who enjoy numerical calculation. But the function of these examples is solely to illustrate the full general theory, and the question of how arithmetical calculations can most effectively be carried out is beyond the scope of this book. The author is indebted to some friends, and most of all to professor o Erd? s, Professor Mordell and Professor Rogers, for suggestions and corrections. He is also indebted to victor Draim for permission to include an account of his algorithm.The material for the ? fth edition was prepared by Professor D. J. Lewis and Dr J. H. Davenport. The problems and answers are base on the suggestions of Professor R. K. Guy. Chapter VIII and the associated exercises were written for the sixth edition by Professor J. H. Davenport. For the s eveningth edition, he updated Chapter VII to mention Wiles proof of Fermats Last Theorem, and is grateful to Professor J. H. Silverman for his comments. For the eighth edition, umpteen people contributed suggestions, notably Dr J. F. McKee and Dr G. K. Sankaran.Cambridge University Press kindly re-type curry the book for the eighth edition, which has allowed a few corrections and the preparation of an electronic complement www. cambridge. org/davenport. References to notwithstanding material in the electronic complement, when known at the time this book went to print, are marked thus 0. I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the earthy numbers 1, 2, 3, . . . of ordinary arithmetic. Examples of much(prenominal)(prenominal) propositions are the fundamental theorem (I. 4)? hat every natural number can be ingredientized into primary numbers in one and further(a) and barely one way, and Lagranges theorem (V. 4) that every natural number can be expressed as a tot of four or fewer perfect squares. We are not concerned with numerical calculations, except as illustrative examples, nor are we much concerned with numerical curiosities except where they are relevant to general propositions. We learn arithmetic experimentally in early childhood by playing with objects such(prenominal) as beads or marbles. We ? rst learn addition by combining dickens sets of objects into a single set, and by and by we learn multiplication, in the form of repeated addition.Gradually we learn h ow to calculate with numbers, and we be nonplus familiar with the laws of arithmetic laws which probably carry more conviction to our minds than both other propositions in the whole range of human knowledge. The higher arithmetic is a deductive science, based on the laws of arithmetic which we all know, though we may neer put up go throughn them formulated in general terms. They can be expressed as follows. ? References in this form are to chapters and sections of chapters of this book. 1 2 The Higher Arithmetic Addition.Any 2 natural numbers a and b move over a sum, de cabled by a + b, which is itself a natural number. The operation of addition satis? es the two laws a+b =b+a (commutative law of addition), (associative law of addition), a + (b + c) = (a + b) + c the brackets in the come through formula serving to indicate the way in which the operations are carried out. Multiplication. Any two natural numbers a and b have a product, denoted by a ? b or ab, which is itself a natural number. The operation of multiplication satis? es the two laws ab = ba a(bc) = (ab)c (commutative law of multiplication), (associative law of multiplication).There is also a law which elbow room operations both of addition and of multiplication a(b + c) = ab + ac (the distributive law). Order. If a and b are both two natural numbers, then each a is partake to b or a is slight than b or b is little than a, and of these three possibilities just one must occur. The statement that a is less than b is expressed symbolically by a b, and when this is the fibre we also govern that b is greater than a, expressed by b a. The fundamental law disposal this notion of nightclub is that if a b. We propose to investigate the cat valium divisors of a and b.If a is cleavable by b, then the commonalty divisors of a and b consist simply of all divisors of b, and there is no more to be said. If a is not divisible by b, we can express a as a seven-foldx of b together with a r emnant less than b, that is a = qb + c, where c b. (2) This is the accomplish of office with a remainder, and expresses the circumstance that a, not being a multiple of b, must occur somewhere amongst two consecutive multiples of b. If a comes between qb and (q + 1)b, then a = qb + c, where 0 c b. It follows from the equation (2) that any common divisor of b and c is also a divisor of a.Moreover, any common divisor of a and b is also a divisor of c, since c = a ? qb. It follows that the common divisors of a and b, whatever they may be, are the alike as the common divisors of b and c. The problem of ? nding the common divisors of a and b is reduced to the corresponding problem for the numbers b and c, which are respectively less than a and b. The essence of the algorithm lies in the repetition of this argument. If b is divisible by c, the common divisors of b and c consist of all divisors of c. If not, we express b as b = r c + d, where d c. (3)Again, the common divisors of b and c are the like as those of c and d. The treat goes on until it terminates, and this can just happen when exact divisibility occurs, that is, when we come to a number in the sequence a, b, c, . . . , which is a divisor of the preliminary number. It is plain that the process must terminate, for the decreasing sequence a, b, c, . . . of natural numbers cannot go on for ever. Factorization and the Primes 17 permit us theorise, for the sake of de? niteness, that the process terminates when we reach the number h, which is a divisor of the foregoing number g. thus the last two equations of the series (2), (3), . . . are f = vg + h, g = wh. (4) (5) The common divisors of a and b are the equal as those of b and c, or of c and d, and so on until we reach g and h. Since h divides g, the common divisors of g and h consist simply of all divisors of h. The number h can be identi? ed as being the last remainder in Euclids algorithm beforehand exact divisibility occurs, i. e. the las t non-zero remainder. We have therefore proved that the common divisors of two stipulation natural numbers a and b consist of all divisors of a certain number h (the H. C. F. f a and b), and this number is the last non-zero remainder when Euclids algorithm is applied to a and b. As a numerical illustration, take the numbers 3132 and 7200 which were used in 5. The algorithm runs as follows 7200 = 2 ? 3132 + 936, 3132 = 3 ? 936 + 324, 936 = 2 ? 324 + 288, 324 = 1 ? 288 + 36, 288 = 8 ? 36 and the H. C. F. is 36, the last remainder. It is often possible to shorten the working a little by using a negative remainder whenever this is numerically less than the corresponding positive remainder. In the supra example, the last three measurements could be replaced by 936 = 3 ? 324 ? 6, 324 = 9 ? 36. The reason why it is tolerable to use negative remainders is that the argument that was applied to the equation (2) would be equally validated if that equation were a = qb ? c instead of a = qb + c. twain numbers are said to be comparatively prime? if they have no common divisor except 1, or in other words if their H. C. F. is 1. This will be the case if and scarcely if the last remainder, when Euclids algorithm is applied to the two numbers, is 1. ? This is, of course, the same de? nition as in 5, but is repeated here because the present treatment is independent of that tending(p) preliminaryly. 8 7. some other proof of the fundamental theorem The Higher Arithmetic We shall now use Euclids algorithm to tip over some other proof of the fundamental theorem of arithmetic, independent of that stipulation in 4. We begin with a very simple remark, which may be thought to be too provable to be worth making. Let a, b, n be any natural numbers. The highest common factor of na and nb is n times the highest common factor of a and b. However obvious this may designm, the reader will ? nd that it is not easy to give a proof of it without using either Euclids algorithm or the fundamental theorem of arithmetic.In fact the progress alone follows at once from Euclids algorithm. We can suppose a b. If we divide na by nb, the quotient is the same as before (namely q) and the remainder is nc instead of c. The equation (2) is replaced by na = q. nb + nc. The same applies to the ulterior equations they are all simply reckon throughout by n. Finally, the last remainder, giving the H. C. F. of na and nb, is nh, where h is the H. C. F. of a and b. We apply this simple fact to prove the following theorem, often called Euclids theorem, since it occurs as Prop. 30 of Book VII.If a prime divides the product of two numbers, it must divide one of the numbers (or mayhap both of them). Suppose the prime p divides the product na of two numbers, and does not divide a. The provided factors of p are 1 and p, and therefore the only common factor of p and a is 1. Hence, by the theorem just proved, the H. C. F. of np and na is n. Now p divides np obviously, and divides na by hypothesis. Hence p is a common factor of np and na, and so is a factor of n, since we know that every common factor of two numbers is necessarily a factor of their H. C. F.We have therefore proved that if p divides na, and does not divide a, it must divide n and this is Euclids theorem. The uniqueness of factorisation into primes now follows. For suppose a number n has two factorizations, reckon n = pqr . . . = p q r . . . , where all the numbers p, q, r, . . . , p , q , r , . . . are primes. Since p divides the product p (q r . . . ) it must divide either p or q r . . . . If p divides p then p = p since both numbers are primes. If p divides q r . . . we repeat the argument, and ultimately reach the conclusion that p must equal one of the primes p , q , r , . . . We can cancel the common prime p from the two representations, and scoop up again with one of those left, say q. Eventually it follows that all the primes on the left are the same as those on the right, and the t wo representations are the same. Factorization and the Primes 19 This is the alternative proof of the uniqueness of factorization into primes, which was referred to in 4. It has the merit of resting on a general theory (that of Euclids algorithm) rather than on a peculiar(a) device such as that used in 4. On the other hand, it is longer and less direct. 8. A property of the H. C.F From Euclids algorithm one can deduce a remarkable property of the H. C. F. , which is not at all apparent from the original construction for the H. C. F. by factorization into primes (5). The property is that the highest common factor h of two natural numbers a and b is representable as the disagreeence between a multiple of a and a multiple of b, that is h = ax ? by where x and y are natural numbers. Since a and b are both multiples of h, any number of the form ax ? by is necessarily a multiple of h and what the answer asserts is that there are some value of x and y for which ax ? y is actually equal to h. Before giving the proof, it is well-to-do to note some properties of numbers representable as ax ? by. In the ? rst place, a number so representable can also be represented as by ? ax , where x and y are natural numbers. For the two expressions will be equal if a(x + x ) = b(y + y ) and this can be ensured by taking any number m and de? ning x and y by x + x = mb, y + y = ma. These numbers x and y will be natural numbers provided m is suf? ciently large, so that mb x and ma y. If x and y are de? ned in this way, then ax ? by = by ? x . We say that a number is linearly dependent on a and b if it is representable as ax ? by. The result just proved shows that linear dependence on a and b is not affected by interchanging a and b. There are two further simple facts intimately linear dependence. If a number is linearly dependent on a and b, then so is any multiple of that number, for k(ax ? by) = a. kx ? b. ky. Also the sum of two numbers that are each linearly dependent on a an d b is itself linearly dependent on a and b, since (ax1 ? by1 ) + (ax2 ? by2 ) = a(x1 + x2 ) ? b(y1 + y2 ). 20 The Higher ArithmeticThe same applies to the difference of two numbers to see this, write the second number as by2 ? ax2 , in conformity with the earlier remark, before subtracting it. Then we get (ax1 ? by1 ) ? (by2 ? ax2 ) = a(x1 + x2 ) ? b(y1 + y2 ). So the property of linear dependence on a and b is preserved by addition and subtraction, and by multiplication by any number. We now get word the footprints in Euclids algorithm, in the light of this concept. The numbers a and b themselves are certainly linearly dependent on a and b, since a = a(b + 1) ? b(a), b = a(b) ? b(a ? 1). The ? rst equation of the algorithm was a = qb + c.Since b is linearly dependent on a and b, so is qb, and since a is also linearly dependent on a and b, so is a ? qb, that is c. Now the next equation of the algorithm allows us to deduce in the same way that d is linearly dependent on a and b, and so on until we come to the last remainder, which is h. This proves that h is linearly dependent on a and b, as maintain. As an illustration, take the same example as was used in 6, namely a = 7200 and b = 3132. We work through the equations one at a time, using them to express each remainder in terms of a and b. The ? rst equation was 7200 = 2 ? 3132 + 936, which tells s that 936 = a ? 2b. The second equation was 3132 = 3 ? 936 + 324, which gives 324 = b ? 3(a ? 2b) = 7b ? 3a. The third equation was 936 = 2 ? 324 + 288, which gives 288 = (a ? 2b) ? 2(7b ? 3a) = 7a ? 16b. The fourth equation was 324 = 1 ? 288 + 36, Factorization and the Primes which gives 36 = (7b ? 3a) ? (7a ? 16b) = 23b ? 10a. 21 This expresses the highest common factor, 36, as the difference of two multiples of the numbers a and b. If one prefers an expression in which the multiple of a comes ? rst, this can be obtained by arguing that 23b ? 10a = (M ? 10)a ? (N ? 23)b, provided that Ma = N b.Since a and b ha ve the common factor 36, this factor can be removed from both of them, and the condition on M and N becomes 200M = 87N . The simplest choice for M and N is M = 87, N = 200, which on substitution gives 36 = 77a ? 177b. Returning to the general theory, we can express the result in another(prenominal) form. Suppose a, b, n are given natural numbers, and it is desired to ? nd natural numbers x and y such that ax ? by = n. (6) Such an equation is called an indeterminate equation since it does not determine x and y acquitly, or a Diophantine equation by and by Diophantus of Alexandria (third century A . D . , who wrote a famous treatise on arithmetic. The equation (6) cannot be soluble unless n is a multiple of the highest common factor h of a and b for this highest common factor divides ax ? by, whatever set x and y may have. Now suppose that n is a multiple of h, say, n = mh. Then we can solve the equation for all we have to do is ? rst solve the equation ax1 ? by1 = h, as we have se en how to do above, and then multiply throughout by m, getting the solution x = mx1 , y = my1 for the equation (6). Hence the linear indeterminate equation (6) is soluble in natural numbers x, y if and only if n is a multiple of h.In particular proposition, if a and b are relatively prime, so that h = 1, the equation is soluble whatever value n may have. As regards the linear indeterminate equation ax + by = n, we have found the condition for it to be soluble, not in natural numbers, but in integers of opposite signs one positive and one negative. The question of when this equation is soluble in natural numbers is a more dif? cult one, and one that cannot well be completely answered in any simple way. Certainly 22 The Higher Arithmetic n must be a multiple of h, but also n must not be too small in relation to a and b.It can be proved quite easily that the equation is soluble in natural numbers if n is a multiple of h and n ab. 9. Factorizing a number The obvious way of factorizi ng a number is to test whether it is divisible by 2 or by 3 or by 5, and so on, using the series of primes. If a number N v is not divisible by any prime up to N , it must be itself a prime for any heterogeneous number has at least two prime factors, and they cannot both be v greater than N . The process is a very laborious one if the number is at all large, and for this reason factor tables have been computed.The most extensive one which is generally accessible is that of D. N. Lehmer (Carnegie Institute, Washington, Pub. No. 105. 1909 reprinted by Hafner Press, New York, 1956), which gives the least prime factor of each number up to 10,000,000. When the least prime factor of a particular number is known, this can be divided out, and repetition of the process gives eventually the complete factorization of the number into primes. Several mathematicians, among them Fermat and Gauss, have invented methods for reducing the amount of trial that is necessary to factorize a large number. Most of these involve more knowledge of number-theory than we can postulate at this stage but there is one method of Fermat which is in principle extremely simple and can be explained in a few words. Let N be the given number, and let m be the least number for which m 2 N . Form the numbers m 2 ? N , (m + 1)2 ? N , (m + 2)2 ? N , . . . . (7) When one of these is reached which is a perfect square, we get x 2 ? N = y 2 , and consequently N = x 2 ? y 2 = (x ? y)(x + y). The calculation of the numbers (7) is facilitated by noting that their consequent differences increase at a constant rate. The identi? ation of one of them as a perfect square is most easily made by using Barlows Table of Squares. The method is particularly successful if the number N has a factorization in which the two factors are of about the same magnitude, since then y is small. If N is itself a prime, the process goes on until we reach the solution provided by x + y = N , x ? y = 1. As an illustration, take N = 9 271. This comes between 962 and 972 , so that m = 97. The ? rst number in the series (7) is 972 ? 9271 = 138. The Factorization and the Primes 23 subsequent ones are obtained by adding successively 2m + 1, then 2m + 3, and so on, that is, 195, 197, and so on.This gives the series 138, 333, 530, 729, 930, . . . . The fourth of these is a perfect square, namely 272 , and we get 9271 = 1002 ? 272 = 127 ? 73. An interesting algorithm for factorization has been detect new-fangledly by Captain N. A. Draim, U . S . N. In this, the result of each trial division is used to convert the number in preparation for the next division. There are several forms of the algorithm, but perhaps the simplest is that in which the successive divisors are the leftover numbers 3, 5, 7, 9, . . . , whether prime or not. To explain the rules, we work a numerical example, say N = 4511. The ? st smell is to divide by 3, the quotient being 1503 and the remainder 2 4511 = 3 ? 1503 + 2. The next step is to subtr act twice the quotient from the given number, and then add the remainder 4511 ? 2 ? 1503 = 1505, 1505 + 2 = 1507. The last number is the one which is to be divided by the next odd number, 5 1507 = 5 ? 301 + 2. The next step is to subtract twice the quotient from the ? rst derived number on the previous line (1505 in this case), and then add the remainder from the last line 1505 ? 2 ? 301 = 903, 903 + 2 = 905. This is the number which is to be divided by the next odd number, 7. Now we an continue in exactly the same way, and no further explanation will be needed 905 = 7 ? 129 + 2, 903 ? 2 ? 129 = 645, 645 ? 2 ? 71 = 503, 503 ? 2 ? 46 = 411, 645 + 2 = 647, 503 + 8 = 511, 411 + 5 = 416, 647 = 9 ? 71 + 8, 511 = 11 ? 46 + 5, 416 = 13 ? 32 + 0. 24 The Higher Arithmetic We have reached a zero remainder, and the algorithm tells us that 13 is a factor of the given number 4511. The complementary color factor is found by carrying out the ? rst half of the next step 411 ? 2 ? 32 = 347. In fact 4511 = 13? 347, and as 347 is a prime the factorization is complete. To justify the algorithm generally is a matter of round-eyed algebra.Let N1 be the given number the ? rst step was to express N1 as N1 = 3q1 + r1 . The next step was to form the numbers M2 = N1 ? 2q1 , The number N2 was divided by 5 N2 = 5q2 + r2 , and the next step was to form the numbers M3 = M2 ? 2q2 , N 3 = M3 + r 2 , N 2 = M2 + r 1 . and so the process was continued. It can be deduced from these equations that N2 = 2N1 ? 5q1 , N3 = 3N1 ? 7q1 ? 7q2 , N4 = 4N1 ? 9q1 ? 9q2 ? 9q3 , and so on. Hence N2 is divisible by 5 if and only if 2N1 is divisible by 5, or N1 divisible by 5. Again, N3 is divisible by 7 if and only if 3N1 is divisible by 7, or N1 divisible by 7, and so on.When we reach as divisor the least prime factor of N1 , exact divisibility occurs and there is a zero remainder. The general equation analogous to those given above is Nn = n N1 ? (2n + 1)(q1 + q2 + + qn? 1 ). The general equation for Mn i s found to be Mn = N1 ? 2(q1 + q2 + + qn? 1 ). (9) If 2n + 1 is a factor of the given number N1 , then Nn is exactly divisible by 2n + 1, and Nn = (2n + 1)qn , whence n N1 = (2n + 1)(q1 + q2 + + qn ), (8) Factorization and the Primes by (8). Under these circumstances, we have, by (9), Mn+1 = N1 ? 2(q1 + q2 + + qn ) = N1 ? 2 n 2n + 1 N1 = N1 . n + 1 25 so the complementary factor to the factor 2n + 1 is Mn+1 , as stated in the example. In the numerical example worked out above, the numbers N1 , N2 , . . . decrease steadily. This is always the case at the get down of the algorithm, but may not be so later. However, it appears that the later numbers are always comfortably less than the original number. 10. The series of primes Although the notion of a prime is a very natural and obvious one, questions concerning the primes are often very dif? cult, and some(prenominal) such questions are quite unanswerable in the present state of mathematical knowledge.We conclude this cha pter by mentioning brie? y some results and conjectures about the primes. In 3 we gave Euclids proof that there are in? nitely many primes. The same argument will also serve to prove that there are in? nitely many primes of certain speci? ed forms. Since every prime after 2 is odd, each of them falls into one of the two growths (a) 1, 5, 9, 13, 17, 21, 25, . . . , (b) 3, 7, 11, 15, 19, 23, 27, . . . the progression (a) consisting of all numbers of the form 4x + 1, and the progression (b) of all numbers of the form 4x ? 1 (or 4x + 3, which comes to the same thing).We ? rst prove that there are in? nitely many primes in the progression (b). Let the primes in (b) be enumerated as q1 , q2 , . . . , beginning with q1 = 3. Consider the number N de? ned by N = 4(q1 q2 . . . qn ) ? 1. This is itself a number of the form 4x ? 1. Not every prime factor of N can be of the form 4x + 1, because any product of numbers which are all of the form 4x + 1 is itself of that form, e. g. (4x + 1)(4y + 1) = 4(4x y + x + y) + 1. Hence the number N has some prime factor of the form 4x ? 1. This cannot be any of the primes q1 , q2 , . . . , qn , since N leaves the remainder ? when 26 The Higher Arithmetic divided by any of them. Thus there exists a prime in the series (b) which is different from any of q1 , q2 , . . . , qn and this proves the proposition. The same argument cannot be used to prove that there are in? nitely many primes in the series (a), because if we construct a number of the form 4x +1 it does not follow that this number will necessarily have a prime factor of that form. However, another argument can be used. Let the primes in the series (a) be enumerated as r1 , r2 , . . . , and consider the number M de? ned by M = (r1 r2 . . rn )2 + 1. We shall see later (III. 3) that any number of the form a 2 + 1 has a prime factor of the form 4x + 1, and is and so entirely composed of such primes, together possibly with the prime 2. Since M is obviously not divisible by any of the primes r1 , r2 , . . . , rn , it follows as before that there are in? nitely many primes in the progression (a). A similar situation arises with the two progressions 6x + 1 and 6x ? 1. These progressions exhaust all numbers that are not divisible by 2 or 3, and therefore every prime after 3 falls in one of these two progressions.One can prove by methods similar to those used above that there are in? nitely many primes in each of them. But such methods cannot cope with the general arithmetical progression. Such a progression consists of all numbers ax +b, where a and b are ? xed and x = 0, 1, 2, . . . , that is, the numbers b, b + a, b + 2a, . . . . If a and b have a common factor, every number of the progression has this factor, and so is not a prime (apart from possibly the ? rst number b). We must therefore suppose that a and b are relatively prime. It then seems plausible that the progression will consider in? itely many primes, i. e. that if a and b are relatively prime, t here are in? nitely many primes of the form ax + b. Legendre seems to have been the ? rst to realize the importance of this proposition. At one time he thought he had a proof, but this turned out to be fallacious. The ? rst proof was given by Dirichlet in an important memoir which appeared in 1837. This proof used analytical methods (functions of a continuous variable, limits, and in? nite series), and was the ? rst truly important application of such methods to the theory of numbers.It opened up completely new lines of development the ideas be Dirichlets argument are of a very general character and have been fundamental for much subsequent work applying analytical methods to the theory of numbers. Factorization and the Primes 27 Not much is known about other forms which represent in? nitely many primes. It is conjectured, for fount, that there are in? nitely many primes of the form x 2 + 1, the ? rst few being 2, 5, 17, 37, 101, 197, 257, . . . . But not the slightest progress h as been made towards proving this, and the question seems hopelessly dif? cult.Dirichlet did succeed, however, in proving that any quadratic form in two variables, that is, any form ax 2 + bx y + cy 2 , in which a, b, c are relatively prime, represents in? nitely many primes. A question which has been deep investigated in modern times is that of the frequency of occurrence of the primes, in other words the question of how many primes there are among the numbers 1, 2, . . . , X when X is large. This number, which depends of course on X , is usually denoted by ? (X ). The ? rst conjecture about the magnitude of ? (X ) as a function of X seems to have been made singly by Legendre and Gauss about X 1800.It was that ? (X ) is approximately log X . Here log X denotes the natural (so-called Napierian) logarithm of X , that is, the logarithm of X to the base e. The conjecture seems to have been based on numerical evidence. For example, when X is 1,000,000 it is found that ? (1,000,000) = 78,498, whereas the value of X/ log X (to the nearest integer) is 72,382, the ratio being 1. 084 . . . . numeral evidence of this kind may, of course, be quite misleading. But here the result suggested is true, in the sense that the ratio of ? (X ) to X/ log X tends to the limit 1 as X tends to in? ity. This is the famous Prime Number Theorem, ? rst proved by Hadamard and de la Vall? e e Poussin independently in 1896, by the use of new and exponentful analytical methods. It is impossible to give an account here of the many other results which have been proved concerning the distribution of the primes. Those proved in the nineteenth century were mostly in the nature of imperfect approaches towards the Prime Number Theorem those of the twentieth century included various re? nements of that theorem. There is one recent event to which, however, reference should be made.We have already said that the proof of Dirichlets Theorem on primes in arithmetical progressions and the proof of the Prime Number Theorem were analytical, and made use of methods which cannot be said to belong correctly to the theory of numbers. The propositions themselves relate entirely to the natural numbers, and it seems well-founded that they should be provable without the intervention of such outside(a) ideas. The search for elementary proofs of these two theorems was unsuccessful until fairly recently. In 1948 A. Selberg found the ? rst elementary proof of Dirichlets Theorem, and with 28 The Higher Arithmetic he help of P. Erd? s he found the ? rst elementary proof of the Prime Numo ber Theorem. An elementary proof, in this connection, means a proof which operates only with natural numbers. Such a proof is not necessarily simple, and indeed both the proofs in question are clean-cutly dif? cult. Finally, we may mention the famous problem concerning primes which was propounded by Goldbach in a letter to Euler in 1742. Goldbach suggested (in a slenderly different wording) that every eve n number from 6 onwards is representable as the sum of two primes other than 2, e. g. 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, 12 = 5 + 7, . . . Any problem like this which relates to additive properties of primes is necessarily dif? cult, since the de? nition of a prime and the natural properties of primes are all expressed in terms of multiplication. An important contribution to the subject was made by venturesome and Littlewood in 1923, but it was not until 1930 that anything was rigorously proved that could be considered as even a remote approach towards a solution of Goldbachs problem. In that year the Russian mathematician Schnirelmann proved that there is some number N such that every number from some point onwards is representable as the sum of at most N primes.A much nearer approach was made by Vinogradov in 1937. He proved, by analytical methods of extreme subtlety, that every odd number from some point onwards is representable as the sum of three primes. This was the s tarting point of much new work on the additive theory of primes, in the course of which many problems have been solved which would have been quite beyond the scope of any pre-Vinogradov methods. A recent result in connection with Goldbachs problem is that every suf? ciently large even number is representable as the sum of two numbers, one of which is a prime and the other of which has at most two prime factors.Notes Where material is changing more rapidly than print cycles permit, we have chosen to place some of the material on the books website www. cambridge. org/davenport. Symbols such as I0 are used to indicate where there is such additional material. 1. The main dif? culty in giving any account of the laws of arithmetic, such as that given here, lies in deciding which of the various concepts should come ? rst. There are several possible arrangements, and it seems to be a matter of taste which one prefers. It is no part of our purpose to analyse further the concepts and laws of ? rithmetic. We take the commonsense (or na? ve) view that we all know Factorization and the Primes 29 the natural numbers, and are satis? ed of the validity of the laws of arithmetic and of the principle of induction. The reader who is evoke in the foundations of mathematics may consult Bertrand Russell, Introduction to Mathematical Philosophy (Allen and Unwin, London), or M. Black, The Nature of math (Harcourt, Brace, New York). Russell de? nes the natural numbers by selecting them from numbers of a more general kind. These more general numbers are the (? ite or in? nite) cardinal numbers, which are de? ned by means of the more general notions of class and one-to-one correspondence. The selection is made by de? ning the natural numbers as those which own all the inductive properties. (Russell, loc. cit. , p. 27). But whether it is reasonable to base the theory of the natural numbers on such a vague and unsatisfactory concept as that of a class is a matter of opinion. Dolus latet in universalibus as Dr Johnson remarked. 2. The objection to using the principle of induction as a de? ition of the natural numbers is that it involves references to any proposition about a natural number n. It seems plain the that propositions envisaged here must be statements which are signi? cant when made about natural numbers. It is not clear how this signi? cance can be tried and true or appreciated except by one who already knows the natural numbers. 4. I am not aware of having seen this proof of the uniqueness of prime factorization elsewhere, but it is unlikely that it is new. For other direct proofs, see Mathews, p. 2, or Hardy and Wright, p. 21.? 5. It has been shown by (intelligent computer searches that there is no odd perfect number less than 10300 . If an odd perfect number exists, it has at least eight distinct prime factors, of which the largest exceeds 108 . For references and other instruction on perfect or nearly perfect numbers, see Guy, sections A. 3, B. 1 a nd B. 2. I1 6. A critical reader may notice that in two places in this section I have used principles that were not explicitly stated in 1 and 2. In each place, a proof by induction could have been given, but to have done so would have distracted the readers attention from the main issues.The question of the length of Euclids algorithm is discussed in Uspensky and Heaslet, ch. 3, and D. E. Knuths The finesse of Computer Programming vol. II Seminumerical Algorithms (Addison Wesley, Reading, Mass. , 3rd. ed. , 1998) section 4. 5. 3. 9. For an account of early methods of factoring, see Dicksons History Vol. I, ch. 14. For a discussion of the subject as it appeared in ? Particulars of books referred to by their authors names will be found in the Bibliography. 30 The Higher Arithmetic the 1970s see the article by Richard K. Guy, How to factor a number, Congressus Numerantium XVI Proc. th Manitoba Conf. Numer. Math. , Winnipeg, 1975, 4989, and at the turn of the millennium see Richard P. Brent, Recent progress and prospects for integer factorisation algorithms, Springer Lecture Notes in Computer Science 1858 Proc. Computing and Combinatorics, 2000, 322. The subject is discussed further in Chapter VIII. It is doubtful whether D. N. Lehmers tables will ever be extended, since with them and a pocket calculator one can easily check whether a 12-digit number is a prime. Primality testing is discussed in VIII. 2 and VIII. 9. For Draims algorithm, see Mathematics Magazine, 25 (1952) 1914. 10. An excellent account of the distribution of primes is given by A. E. Ingham, The Distribution of Prime Numbers (Cambridge Tracts, no. 30, 1932 reprinted by Hafner Press, New York, 1971). For a more recent and extensive account see H. Davenport, increasing Number Theory, 3rd. ed. (Springer, 2000). H. Iwaniec (Inventiones Math. 47 (1978) 17188) has shown that for in? nitely many n the number n 2 + 1 is either prime or the product of at most two primes, and indeed the same is true for any irreducible an 2 + bn + c with c odd. Dirichlets proof of his theorem (with a modi? ation due to Mertens) is given as an appendix to Dicksons Modern Elementary Theory of Numbers. An elementary proof of the Prime Number Theorem is given in ch. 22 of Hardy and Wright. An elementary proof of the asymptotic formula for the number of primes in an arithmetic progression is given in Gelfond and Linnik, ch. 3. For a survey of early work on Goldbachs problem, see James, Bull. American Math. Soc. , 55 (1949) 24660. It has been veri? ed that every even number from 6 to 4 ? 1014 is the sum of two primes, see Richstein, Math. Comp. , 70 (2001) 17459. For a proof of Chens theorem that every suf? iently large even integer can be represented as p + P2 , where p is a prime, and P2 is either a prime or the product of two primes, see ch. 11 of Sieve Methods by H. Halberstam and H. E. Richert (Academic Press, London, 1974). For a proof of Vinogradovs result, see T. Estermann, Introduction to Modern Prime Number Theory (Cambridge Tracts, no. 41, 1952) or H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). Suf? ciently large in Vinogradovs result has now been quanti? ed as greater than 2 ? 101346 , see M. -C. Liu and T. Wang, Acta Arith. , 105 (2002) 133175.Conversely, we know that it is true up to 1. 13256 ? 1022 (Ramar? and Saouter in J. Number Theory 98 (2003) 1033). e II CONGRUENCES 1. The congruousness notation It often happens that for the purposes of a particular calculation, two numbers which differ by a multiple of some ? xed number are equivalent, in the sense that they produce the same result. For example, the value of (? 1)n depends only on whether n is odd or even, so that two value of n which differ by a multiple of 2 give the same result. Or again, if we are concerned only with the last digit of a number, then for that purpose two umbers which differ by a multiple of 10 are effectively the same. The congruousness notation, introduced by Gauss, serves to express in a convenient form the fact that two integers a and b differ by a multiple of a ? xed natural number m. We say that a is congruous to b with respect to the modulus m, or, in symbols, a ? b (mod m). The meaning of this, then, is simply that a ? b is divisible by m. The notation facilitates calculations in which numbers differing by a multiple of m are effectively the same, by stressing the analogy between congruity and equality.Congruence, in fact, means equality except for the addition of some multiple of m. A few examples of valid congruences are 63 ? 0 (mod 3), 7 ? ?1 (mod 8), 52 ? ?1 (mod 13). A congruence to the modulus 1 is always valid, whatever the two numbers may be, since every number is a multiple of 1. Two numbers are congruent with respect to the modulus 2 if they are of the same parity, that is, both even or both odd. 31 32 The Higher Arithmetic Two congruences can be added, subtracted, or multiplied together, in just the same way as two equations, provided all the congruences have the same modulus.If a ? ? (mod m) and b ? ? (mod m) then a + b ? ? + ? (mod m), a ? b ? ? ? ? (mod m), ab ? (mod m). The ? rst two of these statements are ready for example (a + b) ? (? + ? ) is a multiple of m because a ? ? and b ? ? are both multiples of m. The third is not quite so immediate and is best proved in two steps. First ab ? ?b because ab ? ?b = (a ? ?)b, and a ? ? is a multiple of m. Next, ? b ? , for a similar reason. Hence ab ? (mod m). A congruence can always be multiplied throughout by any integer if a ? ? (mod m) then ka ? k? (mod m).Indeed this is a special case of the third result above, where b and ? are both k. But it is not always legitimate to cancel a factor from a congruence. For example 42 ? 12 (mod 10), but it is not permissible to cancel the factor 6 from the numbers 42 and 12, since this would give the false result 7 ? 2 (mod 10). The reason is obvious the ? rst congruence states that 42 ? 12 is a multiple of 10, but this does not imply that 1 (42 ? 12) is a multiple of 10. The cancellation of 6 a factor from a congruence is legitimate if the factor is relatively prime to the modulus.For let the given congruence be ax ? ay (mod m), where a is the factor to be cancelled, and we suppose that a is relatively prime to m. The congruence states that a(x ? y) is divisible by m, and it follows from the last proposition in I. 5 that x ? y is divisible by m. An illustration of the use of congruences is provided by the well-known rules for the divisibility of a number by 3 or 9 or 11. The usual representation of a number n by digits in the scale of 10 is really a representation of n in the form n = a + 10b + 100c + , where a, b, c, . . . re the digits of the number, read from right to left, so that a is the number of units, b the number of tens, and so on. Since 10 ? 1 (mod 9), we have also 102 ? 1 (mod 9), 103 ? 1 (mod 9), and so on. Hence it follows from the above representation of n that n ? a + b + c + (mod 9). Congruences 33 In other words, any number n differs from the sum of its digits by a multiple of 9, and in particular n is divisible by 9 if and only if the sum of its digits is divisible by 9. The same applies with 3 in place of 9 throughout. The rule for 11 is based on the fact that 10 ? ?1 (mod 11), so that 102 ? +1 (mod 11), 103 ? 1 (mod 11), and so on. Hence n ? a ? b + c ? (mod 11). It follows that n is divisible by 11 if and only if a ? b+c? is divisible by 11. For example, to test the divisibility of 9581 by 11 we form 1? 8+5? 9, or ? 11. Since this is divisible by 11, so is 9581. 2. Linear congruences It is obvious that every integer is congruent (mod m) to exactly one of the numbers 0, 1, 2, . . . , m ? 1. (1) r m, For we can express the integer in the form qm + r , where 0 and then it is congruent to r (mod m). Obviously there are other sets of numbers, besides the set (1), which have the same property, e. . any integer is congruent (mod 5) to ex actly one of the numbers 0, 1, ? 1, 2, ? 2. Any such set of numbers is said to constitute a complete set of residues to the modulus m. Another way of expressing the de? nition is to say that a complete set of residues (mod m) is any set of m numbers, no two of which are congruent to one another. A linear congruence, by analogy with a linear equation in elementary algebra, means a congruence of the form ax ? b (mod m). (2) It is an important fact that any such congruence is soluble for x, provided that a is relatively prime to m.The simplest way of proving this is to observe that if x runs through the numbers of a complete set of residues, then the corresponding values of ax also constitute a complete set of residues. For there are m of these numbers, and no two of them are congruent, since ax 1 ? ax2 (mod m) would involve x1 ? x2 (mod m), by the cancellation of the factor a (permissible since a is relatively prime to m). Since the numbers ax form a complete set of residues, there wi ll be exactly one of them congruent to the given number b. As an example, consider the congruence 3x ? 5 (mod 11). 34 The Higher ArithmeticIf we give x the values 0, 1, 2, . . . , 10 (a complete set of residues to the modulus 11), 3x takes the values 0, 3, 6, . . . , 30. These form another complete set of residues (mod 11), and in fact they are congruent respectively to 0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8. The value 5 occurs when x = 9, and so x = 9 is a solution of the congruence. Naturally any number congruent to 9 (mod 11) will also satisfy the congruence but nevertheless we say that the congruence has one solution, meaning that there is one solution in any complete set of residues. In other words, all solutions are mutually congruent.The same applies to the general congruence (2) such a congruence (provided a is relatively prime to m) is precisely equivalent to the congruence x ? x0 (mod m), where x0 is one particular solution. There is another way of looking at the linear congruen ce (2). It is equivalent to the equation ax = b + my, or ax ? my = b. We proved in I. 8 that such a linear Diophantine equation is soluble for x and y if a and m are relatively prime, and that fact provides another proof of the solubility of the linear congruence. But the proof given above is simpler, and illustrates the advantages gained by using the congruence notation.The fact that the congruence (2) has a unique solution, in the sense explained above, suggests that one may use this solution as an interpretation b for the fraction a to the modulus m. When we do this, we obtain an arithmetic (mod m) in which addition, subtraction and multiplication are always possible, and division is also possible provided that the divisor is relatively prime to m. In this arithmetic there are only a ? nite number of essentially distinct numbers, namely m of them, since two numbers which are mutually congruent (mod m) are treated as the same.If we take the modulus m to be 11, as an illustration, a few examples of arithmetic mod 11 are 5 ? 9 ? ?2. 3 Any relation connecting integers or fractions in the ordinary sense system true when interpreted in this arithmetic. For example, the relation 5 + 7 ? 1, 5 ? 6 ? 8, 1 2 7 + = 2 3 6 becomes (mod 11) 6 + 8 ? 3, because the solution of 2x ? 1 is x ? 6, that of 3x ? 2 is x ? 8, and that of 6x ? 7 is x ? 3. Naturally the interpretation given to a fraction depends on the modulus, for instance 2 ? 8 (mod 11), but 2 ? 3 (mod 7). The 3 3 Congruences 35 nly limitation on such calculations is that just mentioned, namely that the denominator of any fraction must be relatively prime to the modulus. If the modulus is a prime (as in the above examples with 11), the limitation takes the very simple form that the denominator must not be congruent to 0 (mod m), and this is exactly analogous to the limitation in ordinary arithmetic that the denominator must not be equal to 0. We shall return to this point later (7). 3. Fermats theorem The fact tha t there are only a ? nite number of essentially different numbers in arithmetic to a modulus m means that there are algebraic relations which are satis? d by every number in that arithmetic. There is nothing analogous to these relations in ordinary arithmetic. Suppose we take any number x and consider its offices x, x 2 , x 3 , . . . . Since there are only a ? nite number of possibilities for these to the modulus m, we must eventually come to one which we have met before, say x h ? x k (mod m), where k h. If x is relatively prime to m, the factor x k can be cancelled, and it follows that x l ? 1 (mod m), where l ? h ? k. Hence every number x which is relatively prime to m satis? es some congruence of this form. The least exponent l for which x l ? (mod m) will be called the commit of x to the modulus m. If x is 1, its order is obviously 1. To illustrate the de? nition, let us calculate the orders of a few numbers to the modulus 11. The powers of 2, taken to the modulus 11, are 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, . . . . Each one is twice the prior one, with 11 or a multiple of 11 subtracted where necessary to make the result less than 11. The ? rst power of 2 which is ? 1 is 210 , and so the order of 2 (mod 11) is 10. As another example, take the powers of 3 3, 9, 5, 4, 1, 3, 9, . . . . The ? rst power of 3 which is ? 1 is 35 , so the order of 3 (mod 11) is 5.It will be found that the order of 4 is again 5, and so also is that of 5. It will be seen that the successive powers of x are periodic when we have reached the ? rst number l for which x l ? 1, then x l+1 ? x and the previous cycle is repeated. It is plain that x n ? 1 (mod m) if and only if n is a multiple of the order of x. In the last example, 3n ? 1 (mod 11) if and only if n is a multiple of 5. This remains valid if n is 0 (since 30 = 1), and it remains valid also for negative exponents, provided 3? n , or 1/3n , is interpreted as a fraction (mod 11) in the way explained in 2. 36 The Higher Arit hmeticIn fact, the negative powers of 3 (mod 11) are obtained by prolonging the series backwards, and the table of powers of 3 to the modulus 11 is n = ?3 ? 2 ? 1 0 1 2 3 4 5 6 . . . 9 5 4 1 3 9 5 4 1 3 . 3n ? . . . Fermat discovered that if the modulus is a prime, say p, then every integer x not congruent to 0 satis? es x p? 1 ? 1 (mod p). (3) In view of what we have seen above, this is equivalent to saying that the order of any number is a divisor of p ? 1. The result (3) was mentioned by Fermat in a letter to Fr? nicle de Bessy of 18 October 1640, in which he e also stated that he had a proof.But as with most of Fermats discoveries, the proof was not published or preserved. The ? rst known proof seems to have been given by Leibniz (16461716). He proved that x p ? x (mod p), which is equivalent to (3), by writing x as a sum 1 + 1 + + 1 of x units (assuming x positive), and then expanding (1 + 1 + + 1) p by the multinomial theorem. The terms 1 p + 1 p + + 1 p give x, and the coef? cients of all the other terms are easily proved to be divisible by p. Quite a different proof was given by Ivory in 1806. If x ? 0 (mod p), the integers x, 2x, 3x, . . . , ( p ? )x are congruent (in some order) to the numbers 1, 2, 3, . . . , p ? 1. In fact, each of these sets constitutes a complete set of residues except that 0 has been omitted from each. Since the two sets are congruent, their products are congruent, and so (x)(2x)(3x) . . . (( p ? 1)x) ? (1)(2)(3) . . . ( p ? 1)(mod p). Cancelling the factors 2, 3, . . . , p ? 1, as is permissible, we obtain (3). One merit of this proof is that it can be extended so as to apply to the more general case when the modulus is no longer a prime. The generalization of the result (3) to any modulus was ? rst given by Euler in 1760.To formulate it, we must begin by considering how many numbers in the set 0, 1, 2, . . . , m ? 1 are relatively prime to m. Denote this number by ? (m). When m is a prime, all the numbers in the set except 0 are relatively prime to m, so that ? ( p) = p ? 1 for any prime p. Eulers generalization of Fermats theorem is that for any modulus m, x ? (m) ? 1 (mod m), provided only that x is relatively prime to m. (4) Congruences 37 To prove this, it is only necessary to modify Ivorys method by omitting from the numbers 0, 1, . . . , m ? 1 not only the number 0, but all numbers which are not relatively prime to m.There remain ? (m) numbers, say a 1 , a2 , . . . , a? , Then the numbers a1 x, a2 x, . . . , a? x are congruent, in some order, to the previous numbers, and on multiplying and cancelling a1 , a2 , . . . , a? (as is permissible) we obtain x ? ? 1 (mod m), which is (4). To illustrate this proof, take m = 20. The numbers less than 20 and relatively prime to 20 are 1, 3, 7, 9, 11, 13, 17, 19, so that ? (20) = 8. If we multiply these by any number x which is relatively prime to 20, the new numbers are congruent to the original numbers in some other order.For example, if x is 3, th e new numbers are congruent respectively to 3, 9, 1, 7, 13, 19, 11, 17 (mod 20) and the argument proves that 38 ? 1 (mod 20). In fact, 38 = 6561. where ? = ? (m). 4. Eulers function ? (m) As we have just seen, this is the number of numbers up to m that are relatively prime to m. It is natural to ask what relation ? (m) bears to m. We aphorism that ? ( p) = p ? 1 for any prime p. It is also easy to evaluate ? ( p a ) for any prime power pa . The only numbers in the set 0, 1, 2, . . . , pa ? 1 which are not relatively prime to p are those that are divisible by p. These are the numbers pt, where t = 0, 1, . . , pa? 1 ? 1. The number of them is pa? 1 , and when we subtract this from the total number pa , we obtain ? ( pa ) = pa ? pa? 1 = pa? 1 ( p ? 1). (5) The determination of ? (m) for general values of m is effected by proving that this function is multiplicative. By this is meant that if a and b are any two relatively prime numbers, then ? (ab) = ? (a)? (b). (6) 38 The Higher Arith metic To prove this, we begin by observing a general principle if a and b are relatively prime, then two simultaneous congruences of the form x ? ? (mod a), x ? ? (mod b) (7) are precisely equivalent to one congruence to the modulus ab.For the ? rst congruence means that x = ? + at where t is an integer. This satis? es the second congruence if and only if ? + at ? ? (mod b), or at ? ? ? ? (mod b). This, being a linear congruence for t, is soluble. Hence the two congruences (7) are simultaneously soluble. If x and x are two solutions, we have x ? x (mod a) and x ? x (mod b), and therefore x ? x (mod ab). Thus there is exactly one solution to the modulus ab. This principle, which extends at once to several congruences, provided that the moduli are relatively prime in pairs, is sometimes called the Chinese remainder theorem.It assures us of the existence of numbers which leave prescribed remainders on division by the moduli in question. Let us represent the solution of the two congruen ces (7) by x ? ? , ? (mod ab), so that ? , ? is a certain number depending on ? and ? (and also on a and b of course) which is uniquely determined to the modulus ab. Different pairs of values of ? and ? give rise to different values for ? , ? . If we give ? the values 0, 1, . . . , a ? 1 (forming a complete set of residues to the modulus a) and similarly give ? the values 0, 1, . . . , b ? 1, the resulting values of ? , ? constitute a complete set of residues to the modulus ab. It is obvious that if ? has a factor in common with a, then x in (7) will also have that factor in common with a, in other words, ? , ? will have that factor in common with a. Thus ? , ? will only be relatively prime to ab if ? is relatively prime to a and ? is relatively prime to b, and conversely these conditions will ensure that ? , ? is relatively prime to ab. It follows that if we give ? the ? (a) possible values that are less than a and prime to a, and give ? the ? (b) values that are less than b a nd prime to b, there result ? (a)? (b) values of ? ? , and these embody all the numbers that are less than ab and relatively prime to ab. Hence ? (ab) = ? (a)? (b), as asserted in (6). To illustrate the situation arising in the above proof, we tabulate below the values of ? , ? when a = 5 and b = 8. The possible values for ? are 0, 1, 2, 3, 4, and the possible values for ? are 0, 1, 2, 3, 4, 5, 6, 7. Of these there are four values of ? which are relatively prime to a, corresponding to the fact that ? (5) = 4, and four values of ? that are relatively prime to b, Congruences 39 corresponding to the fact that ? (8) = 4, in accordance with the formula (5).These values are italicized, as also are the corresponding values of ? , ? . The latter constitute the sixteen numbers that are relatively prime to 40 and less than 40, thus verifying that ? (40) = ? (5)? (8) = 4 ? 4 = 16. ? ? 0 1 2 3 4 0 0 16 32 8 24 1 25 1 17 33 9 2 10 26 2 18 34 3 35 11 27 3 19 4 20 36 12 28 4 5 5 21 37 13 29 6 30 6 22 38 14 7 15 31 7 23 39 We now return to the original question, that of evaluating ? (m) for any number m. Suppose the factorization of m into prime powers is m = pa q b . . . . Then it follows from (5) and (6) that ? (m) = ( pa ? pa? 1 )(q b ? q b? 1 ) . . . or, more elegantly, ? (m) = m 1 ? For example, ? (40) = 40 1 ? and ? (60) = 60 1 ? 1 2 1 2 1 p 1? 1 q . (8) 1? 1 3 1 5 = 16, 1 5 1? 1? = 16. The function ? (m) has a remarkable property, ? rst given by Gauss in his Disquisitiones. It is that the sum of the numbers ? (d), extended over all the divisors d of a number m, is equal to m itself. For example, if m = 12, the divisors are 1, 2, 3, 4, 6, 12, and we have ? (1) + ? (2) + ? (3) + ? (4) + ? (6) + ? (12) = 1 + 1 + 2 + 2 + 2 + 4 = 12. A general proof can be based either on (8), or directly on the de? nition of the function. 40 The Higher ArithmeticWe have already referred (I. 5) to a table of the values of ? (m) for m 10, 000. The same volume contains a table giving those numbers m for which ? (m) assumes a given value up to 2,500. This table shows that, up to that point at least, every value assumed by ? (m) is assumed at least twice. It seems reasonable to conjecture that this is true generally, in other words that for any number m there is another number m such that ? (m ) = ? (m). This has never been proved, and any attempt at a general proof seems to meet with formidable dif? culties. For some special types of numbers the result is easy, e. g. f m is odd, then ? (m) = ? (2m) or again if m is not divisible by 2 or 3 we have ? (3m) = ? (4m) = ? (6m). 5. Wilsons theorem This theorem was ? rst publis