Downward and Upward Communication Essay Example

Descending and Upward Communication Essay It is passing data to the subordinate. It expands proficiency by synchronizing hierarchical methodology and can guarantee that all are progressing in the direction of a similar objective. Descending Communication is the methods by which objectives are transmitted and impact is applied on representatives. It flow’s from the top level administration to low level administration through the center level administration. Model: When your manager imparts data to you, it’s descending correspondence, in light of the fact that your chief, as your director, is speaking with you as a worker. Kinds of Downward correspondence incorporates: * Job depiction * Appraisals or assessment * Organizational approach * Organizational framework Downward Communication helps in performing following administrative capacities: * Directing * Motivating * Controlling * Co-ordinating Barriers in Downward Communication: * Often single direction * Mistrust * Status nervousness * Difference in qualities and discernment. Upward correspondence : It starts from lower level of business chain of importance. This correspondence center that everyone is fit for creating musings and thoughts coming about into authoritative advancement. It builds inspiration and cause representatives to feel esteemed. It is the progression of correspondence which starts from low level administration to significant level administration through center level administration. Upward Communication is the methods by which supervisors gather from representatives the information expected to settle on trustworthy choices. Model: when you give your manager criticism about another framework or supporter, it’s upward correspondence. Kinds of upward correspondence incorporates : * Suggestion plans * Feedback gathering or overview * Grievance Procedure * Employees Manager conversation. Upward Communication helps in performing following administrative capacities: * Reporting * Decision making Barriers in Upward Communication: * Delay in dynamic * Loss of data * Fear of distortion * Communicating just the positive out comes. We will compose a custom exposition test on Downward and Upward Communication explicitly for you for just $16.38 $13.9/page Request now We will compose a custom paper test on Downward and Upward Communication explicitly for you FOR ONLY $16.38 $13.9/page Recruit Writer We will compose a custom paper test on Downward and Upward Communication explicitly for you FOR ONLY $16.38 $13.9/page Recruit Writer

The Fall of Rome

The Fall of Rome From its initial days as a government, through the Republic and the Roman Empire, Rome kept going a thousand years ... or on the other hand two. The individuals who settle on two centuries date the Fall of Rome to 1453 when the Ottoman Turks took Byzantium (Constantinople). The individuals who choose one thousand years, concur with Roman history specialist Edward Gibbon. Edward Gibbon dated the Fall to September 4, A.D. 476 when a supposed brute named Odoacer (a Germanic chief in the Roman armed force), removed the last western Roman ruler, Romulus Augustulus, who was presumably incompletely of Germanic family line. Odoacer considered Romulus so irrelevant a danger he didnt even trouble to kill him, yet sent him into retirement.* The Roman Empire Lasted Beyond the Fall The Byzantine Emperor versus the Western Emperor:Â At the hour of the overthrow and for the two going before hundreds of years, there had been two heads of Rome. One lived in the east, ordinarily in Constantinople (Byzantium). The other lived in the west, generally some place in Italy, despite the fact that not really the city of Rome. The head whom Odoacer removed had lived in Ravenna, Italy. A short time later, there was as yet one Roman sovereign, Zeno, who lived in Constantinople. Odoacer turned into the primary savage ruler of the western empire.The Roman People Lived On:Â While this bloodless overthrow in 476 is an often acknowledged date for the Fall of Rome and the start of the Middle Ages, it was not, at that point, a significant defining moment. Numerous occasions and propensities hinted at it and there were numerous individuals who kept on considering themselves and who keep on being thought of as Romans.Europes Kingdoms (From the Ashes of the Roman Empire): The accompan ying assets identify with the finish of the Roman Empire and the Fall of Rome. This incorporates speculations about the Fall of Rome (counting lead) and a few of the Roman sovereigns whose activities rushed the finish of the Roman Empire in the West. There is a segment with data on significant men whose birthplaces were a long way from the city of Rome. Reasons for the Fall of Rome Speculations on the Fall of Rome Non-Romans Who Impacted the Fall of Rome GothsGoths Origins?Michael Kulikowsky clarifies why Jordanes, our fundamental source on the Goths, who is himself viewed as a Goth, ought not be trusted.AttilaProfile of Attila, who is known as the Scourge of God.The HunsIn the overhauled release of The Huns, E. A. Thompson brings up issues about the military virtuoso of Attila the Hun.IllyriaDescendants of the early pioneers of the Balkans collided with the Roman Empire.JordanesJordanes, himself a Goth, condensed a lost history of the Goths by Cassiodorus.OdoacerThe brute who dismissed the sovereign of Rome.Sons of NubelSons of Nubel and the Gildonic WarIf the children of Nubel hadnt been so anxious to get rid of each other, Africa may have gotten autonomous of Rome.StilichoBecause of individual desire, Praetorian Prefect Rufinus forestalled Stilicho from crushing Alaric and the Goths when they had a chance.AlaricAlaric TimelineAlaric didnt need to sack Rome, however he wanted a spot for his Goths to remain and an appropriate title inside the Roman Empire. In spite of the fact that he didnt live to see it, the Goths got the main self-ruling realm inside the Roman Empire. Rome and Romans Fall of Rome Books:Â Recommended perusing for a cutting edge viewpoint on the explanations behind the fall of Rome.End of the Republic:Â Content identified with the men and occasions from the Gracchi and Marius through the fierce years between Julius Caesars death and the beginning of the principate under Augustus.Why Rome Fell:â 476 CE, the date Gibbon utilized for the fall of Rome dependent on the way that it was then that Odoacer ousted the sovereign of Rome, is dubious just like the purposes behind the fall.Roman Emperors Leading to the Fall:Â You could state Rome was nearly tumbling from the hour of its first ruler or you could state Rome fell in 476 CE or 1453, or even that it hasnt yet fallen.â End of the Republic *I think its pertinent to bring up that the last ruler of Rome was additionally not killed, yet only removed. In spite of the fact that ex-lord Tarquinius Superbus (Tarquin the Proud) and his Etruscan partners attempted to recover the seat by warlike methods, Tarquins real testimony was bloodless, as per the legends the Romans told about themselves.

Choosing a College When You Have ADHD

Choosing a College When You Have ADHD ADHD School Print Choosing a College When You Have ADHD By Keath Low Keath Low, MA, is a therapist and clinical scientist with the Carolina Institute for Developmental Disabilities at the University of North Carolina. She specializes in treatment of ADD/ADHD. Learn about our editorial policy Keath Low Updated on September 14, 2018 ADHD Overview Symptoms Causes Diagnosis Treatment Living With In Children Hero Images/Getty Images Many students with ADHD go to typical colleges without supports and are able to function well. But there are also students with ADHD who benefit tremendously from a school or supports that are specifically geared to help them achieve to their highest potential. Schools Designed to Support Students With ADHD Several schools are specifically designed for students with ADHD. If you or your child have severe symptoms or prefer to be in a smaller, more supportive setting with staff who truly understand ADHD, one of these schools may be for you. These are just three of the accredited schools that are specifically intended to support students with ADHD: Beacon College is an accredited college that offers degrees exclusively for students with ADHD.Landmark College is a fully accredited two- and four-year degree granting institution designed for students with learning disabilities and ADHD.The Metamorphosis Coaching Program at Lynn University addresses specific executive functioning issues experienced by college students with ADHD. You don’t have to narrow your search to include just these colleges, however, as there are other options available to make college a successful time for you. It is important to be thoughtful in your approach to college. Spend time researching the schools you are considering to determine how they will best meet your needs. Here are two resource guides that may be helpful during the search phase. KW Guide to Colleges: For Students with Learning Disabilities or Attention Deficit DisorderColleges with Programs for Students with Learning Disabilities or Attention Deficit Disorders ADHD Support Services There are support services available for students with learning disabilities at many colleges. The quality and extent of these services, however, varies from school to school. Be sure to contact the disability support office at each of the schools you are considering to find out about the level of support they provide. The support office may be referred to by a number of names depending on the school. Search for the Office of Student Disability Services, Disability Support, Office of Disabled Student Services, Learning Support Services, etc. Here are a few questions to ask when you contact the disability support office at each school: Is the director of disability support services a specialist in ADHD and/or learning disabilities (LD)?How many students with ADHD does the support program serve?How many ADHD specialists work with the program full time?Are ADHD specialists also available for ongoing counseling, guidance, and support?Is there a formal ADHD support group available for students on campus?Does the school offer specialized academic advising for students with ADHD and LD?What kinds of specific accommodations does the school offer?Do they offer students with ADHD early registration in order to choose preferred courses and professors?Does disability support help to communicate each student’s needs to the appropriate professors?Is specialized tutoring available for students with ADHD and LD?Are there study skills, time management, and organizing classes available specifically for students with ADHD?Is there a physician at student health who has experience treating and prescribing medication for ADHD?Does th e office have a listing of professionals in the area who are experienced in treating ADHD? You may also want to ask to meet with one or two ADHD students enrolled in the school who currently receive disability support services. They are often the best resource for practical information about the strengths and weaknesses of the program.

Mexican Cartels And Its Effects On America - 2267 Words

The Mexican cartels have impacted the society by violating many rights that human have and that are protected. The Cartels are causing too much trouble to the world especially in Mexico and its neighboring countries. They have trafficking tons and tons of drugs to world and spread its violence to nation that live in peace. The cartels have been operating for while and keep growing becoming more powerful each day and nothing has stopped them yet. The Mexican cartels have constructed their empires through the violation of human rights in the business of drug trafficking, especially through torture and deaths, resulting in worldwide corruption. The Mexican cartels main purpose at the beginning was to construct their business from the†¦show more content†¦The United States agency of drugs have discovered that for all of the illegal drugs consumed in the nation has been transported by the Mexican cartels. â€Å"U.S. federal officials say that the mexican cartels operate in doze ns of U.S. cities and analysts say they are moving to consolidate their control of the entire supply chain of illegals drugs†(Schwartz, 1). The government of the United States have been conducting the investigation to now if the cartels are north. Many big cities today have a piece of the cartel organization working there and also producing drugs to distribute within the city. According to the investigation done by the DEA and U.N officials â€Å"Mexican cartels have established command - and - control centers to orchestrate cocaine shipments by sea and air along the still wild caribbean coast with the help of local authorities†(Booth, 2). All though that the cartel are moving to the United States they are still leaving command center in mexico to conduct shipments by ground or any other shipments overseas. The cartel is able to transport such quantities of drugs with help of the local authorities and other corrupt government officials. Most of the common drugs that the cartels produce is Cocaine, Marijuana, Meth, and other chemical formulated drugs produced with stronger chemicals everyday. â€Å" About 65% of the drugs, known as Crystal, Ice, Glass, Tweak, Zip, or just Meth is now either being made inShow MoreRelatedMexican Cartels And Its Effects On America2374 Words   |  10 Pagesby several drug cartels, and leaders. These drug cartels have been increasing their territory, murder tolls, and power. The cartels have started to take over the government and Mexico has been becoming known as a lost state. Drug cartels are rising to power because of the lack of government, and structure which also brings fear to all the regular citizens throughout Mexico. Many citizens regularly get murdered for lying to the cartel, informing the authorities about the cartels, or antagonizingRead MoreMexican Drug Cartels And Its Effects On America1445 Words   |  6 Pages At first, Mexican drug cartels were structured just like family owned businesses. There would be a family member who was the main drug lord and cousins and uncles of the family would be the body guards and drug dealers. After the Institutional Revolutionary Party (PRI) lost their power within the Mexican government the structure of drug cartels became much more complex. A hierarchical diffusion is seen within the drug cartel system because the drugs spread from one important drug lord to anotherRead MoreMexican Drug War Outline816 Words   |  4 PagesWorks Cited Carpenter, Ted. Undermining Mexico’s Dangerous Drug Cartels. Cato Institute. N.p.. Web. 23 Jul 2013. lt;http://www.cato.org/publications/policy-analysis/undermining-mexicos-dangerous-drug-cartelsgt;. Al-Eryani, Ausan. Mexico’s drug war effects US as well. Collegiate Times. Virginia Tech. Web. 23 Jul 2013. lt;http://www.collegiatetimes.com/stories/18328/mexicos-drug-war-effects-us-as-well/p2gt;. Grillo, Ioan. US troops aid Mexico in drug war. Global Post. NRead MoreThe Mexican Cartel Is A Ruthless Crime Origination1182 Words   |  5 PagesThe Mexican cartel is a ruthless crime origination. It controls Mexico through bribes and fear. There are many factions of the cartel, and they are at constant war with one another. They war over many reasons from drugs to human trafficking. These battles often spill out into the streets causing the loose of many innocent lives. The cartel has a strangle hold on the Mexican people. The cartels operate much like other organized crime groups. Like the Mafia there are many factions. Each factionRead MoreThe Mexican Drug War and Its Consequences1185 Words   |  5 PagesThe Mexican Drug War and Its Consequences Despite President Felipe Calderon’s good intentions and determination to eliminate the drug cartels in Mexico, the Mexican government has not been able to stop the drug cartel’s operations (smuggling drugs, people and weapons to and from the USA) for many reasons. The drug cartels have proven to be powerful, well connected politically, well trained, well- armed, and manipulation of the masses. Therefore, it has been impossible for the MexicanRead MoreThe Major Economic Problems That Have Plagued Latin America1497 Words   |  6 Pages1. What are the major economic problems that have plagued Latin America in the 20th century? During the 20th century Latin America went through a change after the U.S made the clam to directly defend Latin America. This caused a sudden trade switch from the Europe nation to the U.S. With this trade switch we start to see a big gap between the lower and upper class. With this gap the poor gets poorer and the rich become Carlos Slim. Carlos Slim was the world’s richest person form 2010-2013 and isRead MoreAnalyzing Political Corruption And Its Effects On The Mexican Economy1623 Words   |  7 Pages Analyzing Political Corruption and its Effects on the Mexican Economy Gyadisha Sulvaran Palm Beach State College Macroeconomics Warren Smith 4th of April 2014 â€Æ' Abstract For macroeconomics, we were given an assignment to examine and analyze the causes of the business cycle in foreign economies. We were divided into groups of 6 and were given the instructions to randomly select a continent. Our continent was North America. We had to choose between Canada and Mexico. The reason why we decidedRead MoreDrug Trafficking1134 Words   |  5 Pagesbeing fought daily. There are many aspects of the drug war from Mexico and other Latin American states which have effects on United States policy as well as policies from other countries that participate in the global suppression of illegal drugs. It can be hard to differentiate between conflict and issue in regards to Latin America’s drug war and International Relations. The Mexican drug war is a global issue because it interferes with the states ability to collaborate and form policy in orderRead MoreEssay on Mexican/Latin American â€Å"War† on Drugs and Trafficking1122 Words   |  5 Pagesbeing fought daily. There are many aspects of the drug war from Mexico and other Latin American states which have effects on United States policy as well as policies from other countries that participate in the global suppression of illegal drugs. It can be hard to differentiate between conflict and issue in regards to Latin America’s drug war and International Relations. The Mexican drug war is a global issue because it interferes with the states ability to collaborate and form policy in order toRead More Mexican Cartel Essay1524 Words   |  7 PagesMexican Cartel Drug War Mexico has a long history of cartels the deaths, drugs and weapon trafficking is in all time high increasing year by year. â€Å"Mexicos gangs have flourished since the late 19th century, mostly in the north due to their proximity to towns along the U.S.-Mexico border. But it was the American appetite for cocaine in the 1970s that gave Mexican drug cartels immense power to manufacture and transport drugs across the border. Early Mexican gangs were primarily situated in border

Motivation Theory Literature Review - Free Essay Example

Sample details Pages: 13 Words: 3954 Downloads: 8 Date added: 2017/06/26 Category Psychology Essay Type Analytical essay Did you like this example? INTRODUCTION: Motivation is naturally conceptualized either as an desire arising from within the human being or as an impulse arising from within the organism or as an attraction arising from an object external to the individual. According to Baron (1991) defines motivation as the internal processes that activate, guide, and maintain behaviour especially goal-directed behaviour. Also (Kanfer, 1998) defines as free will element of behaviour and the psychological mechanism governing the direction, intensity, and persistence of action not due solely to individual differences in ability or overwhelming environmental demands that force action. Don’t waste time! Our writers will create an original "Motivation Theory Literature Review" essay for you Create order Motivation has been defined as essential to adaptive functioning and quality of life (Marin Chakravorty, 2005), and as the content of the positive thinking one wishes to maintain towards the attainment of an objective (Schweingruber, 2006). According to (kanfer, 1990) states that challenges facing researchers into motivation is that it cannot be seen and most individual see is a multidimensional stream of behaviour and the products of those behaviours. Also states motivational processes can be inferred only from analysis of this continuing stream of behaviour that is determined both by environment and heredity and is observed through their effects on personality, belief, knowledge, abilities and skills. Besides that,(Herzberg,1987)describes movement as a function of fear of punishment or failure to get extrinsic rewards and motivation as a function of growth from getting intrinsic rewards out of interesting and challenging work. Motivation is to be intrinsic and extrinsic. Intrinsic motivation is internal. Intrinsic motivation is obvious as the performance of an activity for its inborn satisfaction rather than for some detachable outcome. As soon as, intrinsically motivated, a person is stimulated to take action for the pleasurable or challenge entailed relatively than because of external products, pressures (Ryan et al 2000). It occurs when people are compelled to do something out of pleasure, importance, or desire. Extrinsic motivation occurs when external factors compel the person to do something. Extrinsic motivation is a pull together that pertains whenever an action is done in order to reach some detachable result. (Rayan et al. 2000).Motivation is the driving force within individuals that compels them physiologically and psychologically to pursue one or more goals to fulfil their needs or expectations. (Lam Tang, 2003, p. 61) From my understanding whatever the behaviour, the drive pushing or polling person to act in particular way is motivation and most researchers believe motivation as the drive behind human behaviour. Furthermore, people cannot be motivated to do something if there is nothing in it for them. Obviously, reward may be the avoidance of a negative outcome as much as it may be the achievement of a positive outcome. Motivation theory has to do with why do individual make the choices they make? or what makes someone persist at one activity and yet quickly give up another. Motivation is to give reason, incentive, enthusiasm, or interest that causes a specific action or certain behaviour. Motivation is present in every life function. Simple acts such as eating are motivated by hunger. Education is motivated by desire for knowledge. Motivation is a vital quality that pervades all aspects of teaching and learning. Motivated students display interest in activities, motivated teachers feel that they can help students learn and motivated administrators make possible teaching and learning in their buildings. Self-motivatio n is what most people use to define motivation at its highest mountain of accomplishments. If you are able to motivate yourself and reach your goals, you would have conquered the ability to be disciplined and will accomplish anything you put your mind to. Motivation is very important to all decisions that you have to make. In the workplace, motivation is defined by the actions that employees take to improve the company goals as well as their aspirations for career advancement. The effective performance of employees is what makes or breaks a company. So to keep employees motivated, employers must invest in the overall morale by implementing workshops and seminars to help them. Managers are the key personnel in what makes an employee follow directions. Using rewards like promotions and incentives as well as compliments can greatly improve an employees view of the company and its managers. Employees like to be recognized for their efforts regardless of the salary bracket. Such recognit ion keeps them motivated in getting their job done. Recognition in this case is the motivation technique that managers can tap into and gain the structure over the workplace environment of how they want things done. What organisation and managers can do is provide the environment, support and resources that will influence and effect motivation. Nowadays, managers are trying to find useful motivation theories to motivate their employees. While there are a wide variety of motivation theories on human motivation discussed in this literature review, which means that managers can use any or all those motivation theories to motivate their employees. Managers can use these motivation theories in ways that complement each other. MOTIVATION THEORY-MAJOR CONCEPTS AND CONTRIBUTIONS This is all about the main researchers of motivation theories and to illustrate how these theories can be apply to practical situation. Also reviewed how the major concepts and contributions can apply in work related situations and applying them in groups, teams at organisational level and individual level. Motivation is a complex concept and there are many different theories which look at the subject from different perspectives. The different theories of motivation and criticisms of the theories have been discussed as follows: 1: Hierarchy of needs theory by Maslow A.H. 2: Hygiene theory by Herzberg F. 3: X and Y theory by McGregor D.M. 4: Existence- Relatedness-Growth theory by Alderfer C. P. 5: Expectancy theory by Vroom V. 6: Goal-Setting Theory by Locke E.A. 7: Internal and External control theory by Rotter J.B. 8: Achievement theory by McClelland and Atkinson J.W. 9: Job characteristics theory by Hackman J.R. and Oldham G.R. HIERARCHY OF NEEDS THEORY: According to Maslow (1954) projected that motivation is based on a number of human need. It has to do with the hierarchy that requires the lower more basic needs to be satisfied before the increasingly less basic ones above. Maslow believed that there are five basic needs of the theory. Maslows Hierarchy of Needs is one of the most famous motivation theories, and it has been the centre of much interest by management researchers (Katz, 2004). This (Fig.1) is Maslows hierarchy of needs in pyramid with physiological needs at the base, self-actualisation at the top. 5: Self- actualisation needs: Refer to personal growth and development. 4: Esteem or Ego needs: Refer to need for status and self respect. 3: Social needs: Refer to need to part of a social group and belongingness/love 2: Safety needs: Refer to security, need for shelter or physical danger. 1: Physiological needs: Refer to basic life needs, like food, drink, shelter, etc THE SELF ACTUALISATION NEED: Maslow defined self-actualization as the complete use and utilization of talents, capacities, potentialities. (Maslow,1954). Self-actualization is not a static state. It is an ongoing procedure in which ones capacities are fully, imaginatively, and cheerfully utilized. Self-actualizing people perceive life without a doubt and they are less emotional and more objective. ESTEEM OR EGO NEEDS: According to Maslow, esteem needs were made of two types of desire. I: Desire for strength, adequacy achievement, competence and confidence in the face of organisation. 2: Desire for reputation, glory dominance, recognition, appreciation and dignity Furthermore, people need to be respected, to have self-esteem, self- respect and respect. Self-Esteem expresses the normal human desire to be acknowledged and valued by others. This has to with ones internal needs for self-respect, autonomy and external needs for status and recognition. SOCIAL NEEDS: Are the third level of human needs from the (Fig 1) above. This has to do with ones needs for affection and a sense of belonging and acceptance. Human being need to feel a sense of belonging and acceptance. They need to love and be loved. According to (Maslow, 1954) include the giving and receiving of affection. SAFETY NEEDS: As soon as peoples physical needs have satisfied, they will chase higher level needs, such as safety needs. Safety needs are determined by the need for security and protection from physical and emotion harm. Safety needs include: personal security, financial security, health and well-being, and a safety net or insurance for accidents/illness and the adverse impacts. PHYSIOLOGICAL NEEDS: Are the lowest level of needs, from the (Fig 1) above which are the most basic needs for people. Physiological needs are also measured the needs for human survival. If these needs are not met, people cannot continue to function. Physiological needs include: breathing, homeostasis, water, sleep, food, excretion, and sex. According to (Sarah Steve 2002, p175)the theory engrossed criticism regarding its principle that individuals work on satisfying needs at a higher level only once all lower- level needs have been met. Also major problem facing the theory is how to measure needs, both in terms of how powerfully they are felt and of how someone knows they have been met. Furthermore, the theory is not easy to apply directly to work because need are not met through one source. HYGIENE THEORY: According to (Herzberg et al 1959) proposed his Two-Factor Theory of Human Motivation, known as Motivation-Hygiene theory. Hygiene theory is based on the principle that things people come across satisfying in their jobs are not always the opposite of things they find dissatisfying. Herzbergs Two-Factor Theory is based on the results of research into the factors in a workplace that lead to employee satisfaction. Herzberg made use of a survey, titled, What do people want from their jobs? to achieve insights into what workplace factors people perceived as satisfying and dissatisfying. He asked people to describe in detail which workplace factors led to satisfaction and which factors led to dissatisfaction, and then he separated the results and classified them.Fig.2and Fig.3 show the factors that led to extreme satisfaction and dissatisfaction, respectively. Fig.2 Shows Factors that led to extreme satisfaction according to Herzberg Fig.3 shows Factors that led to extreme dissatis faction according to Herzberg According to(Robins 2001) The criticisms facing the theory, is methodology used by Herzberg does not take into account that often, when things are going well, people take the credit themselves, and when things are not , they blame others and their situation. There is on job satisfaction against which the people could rate their job. Also situational factors were not identified. Despite the criticisms the theory is a vital theory of motivation and used by managers to consider hygiene factors with regards to motivation. X and Y THEORY: Based on( McGregor, 1960)reviewed that the relationship between managers and employee could be immensely improved if the assumptions that had develop about how people behaved at work were changed from Theory X view to a Theory Y view. Basically one is positive and the other is negative. THEORY X: was based on scientific management school of thought and projected that enlarged productivity could result from breaking jobs down into small unit of work and giving workers a small range of obviously defined tasks to do. According to (Gray Stark, 1984) Employees do not like to work and want to evade work as much as possible. Employees do not want to take responsibility for the work; managers should take some actions, like convincing and supervising strategies, to help them finishing their mission. Theory X assumed that human actions are dictated by low-level needs, THEORY Y: Was based the predominant theory of management there could be no trust between management and employee. Also Small opportunity for relationship between management and employees in theory .According to (Gray Stark, 1984) employees treat work as entertainment and rest, it is natural. Employees have the ability to make creative decisions, not just managers. Theory Y assumed that human actions are determined by high-level needs. The theory X and Y can influence how motivating factors are regarded within an organisation. Theory X has to do with extrinsic rewards such as pay in order to motivate people to execute good in their job while theory Y has to do with working environment in which employees may discern intrinsic rewards in their job. EXISTENCE RELATEDNESS GROWTH THEORY: This has the similarities to Maslows needs hierarchy. Alderfer C.P. extended and simplified Maslows hierarchy into a shorter set of three needs: Existence, Relatedness and Growth (ERG) Theory. (Alderfer, 1969) EXISTENCE NEEDS: According to (Hollyforde Whiddett 2002) Refers Existence needs as our concern with basic material subsistence motivators and size of how satisfied this need is therefore mostly stated in conditions of what one person has in relation to others. Existence needs, is based on various forms of material and physiological needs. RELATEDNESS NEEDS: Refers to theÂÂ  motivation we have in support of maintaining interpersonal relationships GROWTH NEEDS: Refers to an intrinsic desire for personal development. Relate to personal creativity and productivity. According to (Sarah Steve 2002), the theory does not suggests how a person may meet his or needs .So managers cannot presume that everyones relationship needs will be met in the same way. Also the th eory ERG theory with its three broad categories is maybe easier to deal with than the five levels of need in hierarchy of needs theory. EXPECTANCY THEORY: Vroom suggests that this theory is Based on the principle that people expect particular actions to achieve a desired result and that the desired result is something worth striving for or avoiding. Vroom does not give attention to on needs, but rather focuses on outcomes. According to (Vroom, 1964) identifies two terms the theory. 1. Expectancy: means the stronger the assurance that an act will be followed by particular outcome, the stronger the expectancy is considered to be 2. Valence: Means affective orientations toward particular outcome either positive or negative. Theory expectancy and valence of Vroom concentrated on extrinsic outcomes rather than intrinsic ones. Based on research expectancy and valence are not as distinct from each other as Vroom recommended .And one the key criticisms is that the theory was tested as if it was based on behavioural and attitudinal predictions across rather than within persons. Furthermore, expectancy theory accommodates multiple wor k outcomes in forecasting motivation and predicts motivation to work hard to earn the merit pay will be low if expectancy low (Schermerhorn J. et al 1997) GOAL-SETTING THEORY: The theory is based on the principle that individuals are motivated to achieve the successful accomplishment of challenging goal. According to (Locke, 1968) based his result on three main conclusions. 1: The more hard goals result in higher level of performance than easy goal 2: The specific goals produce higher levels of performance than general goals. 3: The behavioural intentions influence the choices people make. The Goal -Setting theory is helpful to many managers because much of people management is particularly about performance against goals. Theory has great relevance for managers. According to (Robins, 2001) Implies that Goal Setting theory is restricted to those cultures that match the ones in which the research was carried out. INTERNAL AND EXTERNAL THEORY: The theory is based on locus of control. Locus of Control refers to an individuals perception about the underlying main causes of events in his/her life. According to (Rotter, 1966) invented the internal -external dimension to determine locus of control. Based on the extent to which people feel they are in control of the situations that engulfed them and also the extent to which they are convinced of directing events 1. External control: when individual feels that result of his or her actions is as the result of chance, luck as under the powerful others. 2. Internal control: When individual perceives that the occurrence is contingent upon his own behaviour. The Internal and External theory has been criticised for its centre of attention on the Internal-external dimension only as on personality variable .Besides that the theory is useful when considering motivation issues at individual level. ACHIEVEMENT THEORY: The theory is based the needs that can be classed as either Achievement(nAch), Affiliation(nAff) or Power(nPow) .Also theory is based on three factors research into behaviour in achievement-oriented activities. 1: when individual expects to receive unmistakable feedback on the marks of his or her action. 2: When individual feels liable for the outcome success or failure. 3: When individual feels a little level of uncertainty or risk in the activity Achievement theory does not explore why some people enjoyment a challenge and others fear failure. Achievement theory was not specifically about work. The theory is useful when considering why individuals react different to achieving tough goals and why they react to failure in individual ways (Sarah Steve 2002).McClelland recommends that top managers should have a high need for power coupled with a low need for affiliation. Sunil R. Cited (Kreitner kinicki 1998) JOB CHARACTERISTICS THEORY This theory is based on how to design a job so that it is motivating to individual. Also the theory deals with how individual responds to complex and challenging tasks. According to (Hackman Oldham 1976) job characteristics theory proposed that individuals intrinsic motivation is affected by psychological states: 1: Experienced meaningfulness of the work: The extent individuals perceive the job as meaningful and valuable. 2: Experienced responsibility for the outcomes of the work:-the extent individual feels personally accountable and responsible for the outcome of work done. 3: Knowledge of the result of the activities:-the extent individual knows, and understands on continuous basis how successfully he or she is performing the work Also theory has five job dimensions 1: Skill variety-this has to with number of type of skill and talents of individual. 2: Task identity-this has to with work at hand and job completion 3: task significance- this has to with job considerable impact on lives or work of other people. 4: Autonomy -All about individual in setting up the work and determining the procedures to be used in carrying it out. (Freedom and independence at work environment). 5: Feedback-All about individual getting direct and clear result about the effectiveness of his or her performance The theory is one of the few motivation theories specific to an organisational environment and has plenty of practical application. Theory centre of attention is on facilitating0 high internal work motivation. Hackman and Oldman based their theory on motivating individual through job design. CONTRIBUTIONS: According to (Whittington Evans 2005) The Hierarchy of Needs theory proposed by Maslow has been identified as a major influencing factor in the growth of other motivation and management work, such as that of McGregor (Theory X and Theory Y) and Herzberg Motivation-Hygiene theory. It has been generally applied as a basis for much research into, among other things, workforce commitment, job satisfaction and management theory (Stum, 2001). Goal-Setting Theory by Locke has been acknowledged the best goals and dispute facing management environment and Achievement theory by McClelland has identified problems facing motive to achieve success in management work. (Sarah Steve 2002).Expectancy theory has been used to identify job satisfaction, occupational choice, the likelihood of staying in a job, and the effort that one might expand at management work. Internal and External control theory by Rotter has enhanced the concept of persistence (locus of control) in management work. The job ch aracteristics theory relies on workers insight of the job dimensions for the accomplishment of the psychological states that lead to increased motivation. APPLICATION OF THE CONCEPT OF MOTIVATION TO FACILITIES MANAGEMENT Facilities management regularly deals with identifying relations that humans have with their environment. According to (Smith et al., 1997) the rates of pay had very little influence on participants job satisfaction and subsequent work motivation. Some affiliation can be made at this point to Maslows lowest levels of his hierarchy of needs physiological and safety needs. Maslow argued that individuals can only move up the hierarchy of needs to the higher levels of belongingness/love, esteem and finally self-actualisation, once the lower levels had been achieved; however, the results demonstrated in this study suggest that ancillary staff can be motivated and achieve higher levels of Maslows hierarchy without having their physiological and safety needs completely fulfilled. Individuals can only move up the hierarchy of needs to the higher levels of belongingness/love, esteem and finally self-actualisation, once the lower levels had been achieved; on the other hand, the results demo nstrated in this study suggest that facilities managers can motivate their staff and achieve higher levels of Maslows hierarchy without having their physiological and safety needs completely fulfilled. McGregors Theory X and y focused on managements assumptions about employees. Theory X and Y describe the average worker from management perspective and suggest methods by which facilities manager can get the best out of worker. Herzberg studies concentrated on satisfaction at work. Herzberg came to a conclusion that certain factors tended to lead to job satisfaction.(motivators),while others led to frequently to dissatisfaction(Hygiene factors).these factor are related to job context they are concerned with job environment as an extrinsic to the job itself. Shane, S., 2001. Technology opportunities and new firm creation. Management Science 47 9, pp. 1173-1181. LATEST IDEAS ON MOTIVATION THEORY First of all, fields of management background study like leadership, decision making and organization plan persist to develop conceptually speculative developments focusing on work motivation have not held in reserve speed. These changes can have a philosophical manipulate on how companies challenge to attract, retain and motivate their employees in labour market. JOB DESIGN: This is about changing the content or process of job to increase job satisfaction and performance. This is usually easier for new organisational or new business unit. Furthermore, most organisations find themselves designing as the result of organisational change and to motivate their Staff. Job design is relating to job characteristics theory, suggests job should be design in such a way as to maximise intrinsic motivation. (Sarah Steve 2002) EFFECTIVE COMMUNICATION: This gives managers strategies, to promotes positive improvement in organizations particularly motivation and performance improvement o f employees. For organization to improve well, messages should be designed in such a way as to maximise motivation. Effective communication strategies have strong effect to motivation of employees. (Sarah Steve 2002) OBJECTIVCE- SETTING: This is important to managers to focus individuals work efforts. It is important to commit employee with goals to motivate them. Most organisations applied Management by Objective (MBO) to motivate their employee. This is relating to Goal-setting, suggests individuals tend to produce higher levels of performance when they have specific goal. (Sarah Steve 2002) MANAGING VIRTUAL TEAMS: Is made of group people who regularly work mutually dependent for joint purpose across time, distance and organization. So motivating people to apply for jobs in virtual teams is one of the latest ideas. CONCULSIONS Motivational techniques, therefore, are useful to teachers, leaders, parents, employers, and almost anyone. The key is in understanding that you are not motivating someone else. Instead, you are simply providing a circumstance that triggers that person to be motivated. However, The developments in motivation and management theory. (Whittington Evans 2005) conclude that the road to building high commitment organisations requires the solid foundation laid by Maslow, McGregor and Herzberg. Motivation theories can therefore offer guidance to anyone interest in motivational issues. Motivation theories discussed above can help mangers consider how and why people are motivated. For example, Hierarchy of need theory may help manager figures out where someone is in term of the satisfaction his or her short and long term needs -which may in turn influence how they seek to create the situation that may be the most motivating. Motivation theories discussed were based on work related environme nts. However the can be applied to help individuals understand human behaviour at work. This literature review also emphasized the important of motivation theories to facilities manager with regards to working environments. Also provide frameworks enable facilities manager to manage their working environment. Finally, one of the challenge facing researchers into motivation, it cannot be seen.

Morality vs. Obedience Free Essays

â€Å"If you can’t give a good reason for believing what you believe, then it’s not your belief; it’s someone else’s. † Morality vs. Obedience How would someone tease apart this blanket statement and how would they compare it to morality and obedience in less than three pages? Well, this is how I would. We will write a custom essay sample on Morality vs. Obedience or any similar topic only for you Order Now First, I’d start by making clear that belief is different from knowledge. Knowledge can be defined as â€Å"a clear perception of a truth or fact, erudition; skill from practice. A belief can be defined as â€Å"an assent to anything proposed or declared, and its acceptance as fact by reason of the authority from whence it proceeds, apart from personal knowledge; faith. † So, to be clear, knowledge is dropping a ball and knowing that it will fall to the ground because of prior experience and the perception that the existence of gravity is a fact. Belief would be dropping the ball and believing it will float because your older brother told you it would. Next, I would establish that this statement is not staking claim on any measure of truth as it relates to beliefs or knowledge. Beliefs and knowledge are both dynamic concepts on all organizational levels of the human experience, from the individual to the global. Lets take for example the statement, â€Å"The Earth is flat. † Five hundred years ago, this statement was knowledge. Today, it’s knowledge that this statement is false. Knowledge changes based on the information and evidence available to us at any point in time. This is also true of beliefs but in a different way. When most of us were young, we believed in the Tooth Fairy. Why? Because someone told us she was real. However, this belief changed as we began to mature and rely more upon our own abilities to discern truth from fiction. Thus, we have established that both knowledge and belief are subjective but for different reasons. Belief requires no evidence, while knowledge requires evidence and/or experience. I would now like to make a comparative argument regarding morality and obedience. Simply put, morality is ‘doing what’s right, regardless of what anyone says’ and obedience is ‘doing what your told, regardless of what’s right. One can think of many examples regarding these concepts as they are defined but I will leave this to the reader’s imagination. There is a simple truth to defining obedience and morality in this manner that I find directly comparable to the examined quotation. Neither belief nor obedience require an individual or group to actively engage in an activity or thought process independently. There is no critical thinking involved and one must put their faith in the ‘rightness’ or ‘truth’ of another’s belief, knowledge or otherwise. Both knowledge and morality require active engagement. One cannot possess knowledge without having evidence to back it up and one cannot act morally without understanding what it is that makes a situation right or wrong in their eyes. This is especially true when acting morally requires someone to go against social norms or established laws. The point of this paper is to shed light on the fact that we are all responsible for ourselves to the extent that we can be. No one person can know everything about everything, so belief and obedience are often useful and easy fallbacks. For example, I don’t snowboard on black diamond slopes. This is not because I have knowledge of how hurt I will get but because I belief the sign posting and obediently board within my level to avoid possible injury. However, in every decision one makes there should be a component of knowledge and morality applied. In application to the snowboarding example, I know, from experience, that my snowboarding skills are not on par with a black diamond. I choose not to risk my health and the resources of first responders by making the moral decision to stay off of it, despite my desire to challenge myself. How to cite Morality vs. Obedience, Papers

History Of Math Essay Example For Students

History Of Math Essay Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past, mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or of the generalization of these two fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to be regarded increasingly as the science of relations, or as the science that draws necessary conclusions. This latter view encompasses mathematical or symbolic logic, the science of using symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates, and rules for combining and transforming primitive elements into more complex relations and theorems. This brief survey of the history of mathematics traces the evolution of mathematical ideas and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity itself; evidence of a sense of geometry and interest in geometric pattern has been found in the designs of prehistoric pottery and textiles and in cave paintings. Primitive counting systems were almost certainly based on using the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today. Ancient Mathematics The earliest records of advanced, organized mathematics date back to the ancient Mesopotamian country of Babylonia and to Egypt of the 3rd millennium BC. There mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry and with no trace of later mathematical concepts such as axioms or proofs. The earliest Egyptian texts, composed about 1800 BC, reveal a decimal numeration system with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as in the system used by the Romans. Numbers were represented by writing down the symbol for 1, 10, 100, and so on as many times as the unit was in a given number. For example, the symbol for 1 was written five times to represent the number 5, the symbol for 10 was written six times to represent the number 60, and the symbol for 100 was written three times to represent the number 300. Together, these symbols represented the number 365. Addition was done by totaling separately the units-10s, 100s, and so forth-in the numbers to be added. Multiplication was based on successive doublings, and division was based on the inverse of this process. The Egyptians used sums of unit fractions (a), supplemented by the fraction B, to express all other fractions. For example, the fraction E was the sum of the fractions 3 and *. Using this system, the Egyptians were able to solve all problems of arithmetic that involved fractions, as well as some elementary problems in algebra. In geometry, the Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the volumes of figures such as bricks, cylinders, and pyramids. To find the area of a circle, the Egyptians used the square on U of the diameter of the circle, a value of about 3.16-close to the value of the ratio known as pi, which is about 3.14. The Babylonian system of numeration was quite different from the Egyptian system. In the Babylonian system-which, when using clay tablets, consisted of various wedge-shaped marks-a single wedge indicated 1 and an arrowlike wedge stood for 10 (see table). Numbers up through 59 were formed from these symbols through an additive process, as in Egyptian mathematics. The number 60, however, was represented by the same symbol as 1, and from this point on a positional symbol was used. That is, the value of one of the first 59 numerals depended henceforth on its position in the total numeral. For example, a numeral consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 ? 602 + 27 ? 60 + 10. This principle was extended to the representation of fractions as well, so that the above sequence of numbers could equally well represent 2 ? 60 + 27 + 10 ? († ), or 2 + 27 ? († ) + 10 ? († -2). With this sexagesimal system (base 60), as it is called, the Babylonians had as convenient a numerical system as the 10-based system. The Babylonians in time developed a sophisticated mathematics by which they could find the positive roots of any quadratic equation (Equation). They could even find the roots of certain cubic equations. The Babylonians had a variety of tables, including tables for multiplication and division, tables of squares, and tables of compound interest. They could solve complicated problems using the Pythagorean theorem; one of their tables contains integer solutions to the Pythagorean equation, a2 + b2 = c2, arranged so that c2/a2 decreases steadily from 2 to about J. The Babylonians were able to sum arithmetic and some geometric progressions, as well as sequences of squares. They also arrived at a good approximation for ?. In geometry, they calculated the areas of rectangles, triangles, and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders. However, the Babylonians did not arrive at the correct formula for the volume of a pyramid. Greek Mathematics The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians. The new element in Greek mathematics, however, was the invention of an abstract mathematics founded on a logical structure of definitions, axioms, and proofs. According to later Greek accounts, this development began in the 6th century BC with Thales of Miletus and Pythagoras of Samos, the latter a religious leader who taught the importance of studying numbers in order to understand the world. Some of his disciples made important discoveries about the theory of numbers and geometry, all of which were attributed to Pythagoras. In the 5th century BC, some of the great geometers were the atomist philosopher Democritus of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of circles are equal to areas of certain triangles. This discovery is related to the famous problem of squaring the circle-that is, constructing a square equal in area to a given circle. Two other famous mathematical problems that originated during the century were those of trisecting an angle and doubling a cube-that is, constructing a cube the volume of which is double that of a given cube. All of these problems were solved, and in a variety of ways, all involving the use of instruments more complicated than a straightedge and a geometrical compass. Not until the 19th century, however, was it shown that the three problems mentioned above could never have been solved using those instruments alone. In the latter part of the 5th century BC, an unknown mathematician discovered that no unit of length would measure both the side and diagonal of a square. That is, the two lengths are incommensurable. This means that no counting numbers n and m exist whose ratio expresses the relationship of the side to the diagonal. Since the Greeks considered only the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express this ratio of diagonal to side. (This ratio, ?, would today be called irrational.) As a consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also discovered a method for rigorously proving statements about areas and volumes by successive approximations. Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and who also wrote on optics, astronomy, and music. The 13 books that make up his Elements contain much of the basic mathematical knowledge discovered up to the end of the 4th century BC on the geometry of polygons and the circle, the theory of numbers, the theory of incommensurables, solid geometry, and the elementary theory of areas and volumes. The century that followed Euclid was marked by mathematical brilliance, as displayed in the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga. Archimedes used a method of discovery, based on theoretically weighing infinitely thin slices of figures, to find the areas and volumes of figures arising from the conic sections. These conic sections had been discovered by a pupil of Eudoxus named Menaechmus, and they were the subject of a treatise by Euclid, but Archimedes writings on them are the earliest to survive. Archimedes also investigated centers of gravity and the stability of various solids floating in water. Much of his work is part of the tradition that led, in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman soldier during the sack of Syracuse. His younger contemporary, Apollonius, produced an eight-book treatise on the conic sections that established the names of the sections: ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry until the time of the French philosopher and scientist Ren? Descartes in the 17th century. After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable stature. The writings of Hero of Alexandria in the 1st century AD show how elements of both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside the logical edifices of the great geometers. Very much in the same tradition, but concerned with much more difficult problems, are the books of Diophantus of Alexandria in the 3rd century AD. They deal with finding rational solutions to kinds of problems that lead immediately to equations in several unknowns. Such equations are now called Diophantine equations (see Diophantine Analysis). Career Development through International Mobility EssayThe greatest mathematician of the 18th century was Leonhard Euler, a Swiss, who made basic contributions to calculus and to all other branches of mathematics, as well as to the applications of mathematics. He wrote textbooks on calculus, mechanics, and algebra that became models of style for writing in these areas. The success of Euler and other mathematicians in using calculus to solve mathematical and physical problems, however, only accentuated their failure to develop a satisfactory justification of its basic ideas. That is, Newtons own accounts were based on kinematics and velocities, Leibnizs explanation was based on infinitesimals, and Lagranges treatment was purely algebraic and founded on the idea of infinite series. All these systems were unsatisfactory when measured against the logical standards of Greek geometry, and the problem was not resolved until the following century. 19th Century In 1821 a French mathematician, Augustin Louis Cauchy, succeeded in giving a logically satisfactory approach to calculus. He based his approach only on finite quantities and the idea of a limit. This solution posed another problem, however; that of a logical definition of ?real number.? Although Cauchys explanation of calculus rested on this idea, it was not Cauchy but the German mathematician Julius W. R. Dedekind who found a satisfactory definition of real numbers in terms of the rational numbers. This definition is still taught, but other definitions were given at the same time by the German mathematicians Georg Cantor and Karl T. W. Weierstrass. A further important problem, which arose out of the problem-first stated in the 18th century-of describing the motion of a vibrating string, was that of defining what is meant by function. Euler, Lagrange, and the French mathematician Jean Baptiste Fourier all contributed to the solution, but it was the German mathematician P eter G. L. Dirichlet who proposed the definition in terms of a correspondence between elements of the domain and the range. This is the definition that is found in texts today. In addition to firming the foundations of analysis, as the techniques of the calculus were by then called, mathematicians of the 19th century made great advances in the subject. Early in the century, Carl Friedrich Gauss gave a satisfactory explanation of complex numbers, and these numbers then formed a whole new field for analysis, one that was developed in the work of Cauchy, Weierstrass, and the German mathematician Georg F. B. Riemann. Another important advance in analysis was Fouriers study of infinite sums in which the terms are trigonometric functions. Known today as Fourier series, they are still powerful tools in pure and applied mathematics. In addition, the investigation of which functions could be equal to Fourier series led Cantor to the study of infinite sets and to an arithmetic of infinite numbers. Cantors theory, which was considered quite abstract and even attacked as a ?disease from which mathematics will soon recover,? now forms part of the foundations of mathemat ics and has more recently found applications in the study of turbulent flow in fluids. A further 19th-century discovery that was considered apparently abstract and useless at the time was non-Euclidean geometry. In non-Eculidean geometry, more than one parallel can be drawn to a given line through a given point not on the line. Evidently this was discovered first by Gauss, but Gauss was fearful of the controversy that might result from publication. The same results were rediscovered independently and published by the Russian mathematician Nikolay Ivanovich Lobachevsky and the Hungarian J?nos Bolyai. Non-Euclidean geometries were studied in a very general setting by Riemann with his invention of manifolds and, since the work of Einstein in the 20th century, they have also found applications in physics. Gauss was one of the greatest mathematicians who ever lived. Diaries from his youth show that this infant prodigy had already made important discoveries in number theory, an area in which his book Disquisitiones Arithmeticae (1801) marks the beginning of the modern era. While only 18, Gauss discovered that a regular polygon with m sides can be constructed by straightedge and compass when m is a power of 2 times distinct primes of the form 2n + 1. In his doctoral dissertation he gave the first satisfactory proof of the fundamental theorem of algebra. Often he combined scientific and mathematical investigations. Examples include his development of statistical methods along with his investigations of the orbit of a newly discovered planetoid; his founding work in the field of potential theory, along with the study of magnetism; and his study of the geometry of curved surfaces in tandem with his investigations of surveying. Of more importance for algebra itself than Gausss proof of its fundamental theorem was the transformation of the subject during the 19th century from a study of polynomials to a study of the structure of algebraic systems. A major step in this direction was the invention of symbolic algebra in England by George Peacock. Another was the discovery of algebraic systems that have many, but not all, of the properties of the real numbers. Such systems include the quaternions of the Irish mathematician William Rowan Hamilton, the vector analysis of the American mathematician and physicist J. Willard Gibbs, and the ordered n-dimensional spaces of the German mathematician Hermann G?nther Grassmann. A third major step was the development of group theory from its beginnings in the work of Lagrange. Galois applied this work deeply to provide a theory of when polynomials may be solved by an algebraic formula. Just as Descartes had applied the algebra of his time to the study of geometry, so the German mathematician Felix Klein and the Norwegian mathematician Marius Sophus Lie applied the algebra of the 19th century. Klein applied it to the classification of geometries in terms of their groups of transformations (the so-called Erlanger Programm), and Lie applied it to a geometric theory of differential equations by means of continuous groups of transformations known as Lie groups. In the 20th century, algebra has also been applied to a general form of geometry known as topology. Another subject that was transformed in the 19th century, notably by Laws of Thought (1854), by the English mathematician George Boole and by Cantors theory of sets, was the foundations of mathematics (Logic). Toward the end of the century, however, a series of paradoxes were discovered in Cantors theory. One such paradox, found by English mathematician Bertrand Russell, aimed at the very concept of a set ( Set Theory). Mathematicians responded by constructing set theories sufficiently restrictive to keep the paradoxes from arising. They left open the question, however, of whether other paradoxes might arise in these restricted theories-that is, whether the theories were consistent. As of the present time, only relative consistency proofs have been given. (That is, theory A is consistent if theory B is consistent.) Particularly disturbing is the result, proved in 1931 by the American logician Kurt G?del, that in any axiom system complicated enough to be interesting to most mathematic ians, it is possible to frame propositions whose truth cannot be decided within the system. Current Mathematics At the International Conference of Mathematicians held in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at G?ttingen, the former academic home of Gauss and Riemann. He had contributed to most areas of mathematics, from his classic Foundations of Geometry (1899) to the jointly authored Methods of Mathematical Physics. Hilberts address at G?ttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century. These problems have indeed stimulated a great deal of the mathematical research of the century. When news breaks that another of the ?Hilbert problems? has been solved, mathematicians all over the world await the details of the story with impatience. Important as these problems have been, an event that Hilbert could not have foreseen seems destined to play an even greater role in the future development of mathematics-namely, the invention of the programmable digital computer (Computer). Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th century, it was Charles Babbage in 19th-century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. Babbages imagination outran the technology of his day, however, and it was not until the invention of the relay, then of the vacuum tube, and then of the transistor, that large-scale, programmed computation became feasible. This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics. It has suggested new areas for mathematical investigation, such as the study of algorithms. It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-color problem first proposed in the mid-19th century. The theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors. The theorem was finally proved in 1976 by means of a large-scale computer at the University of Illinois. Mathematical knowledge in the modern world is advancing at a faster rate than ever before. Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications. Mathematics