Accounting Information System and Special Journals

The principle of internal control violation is separation of duties. The person that records the incoming cash receipts should not be the same person that is posting the customer payments to their accounts. My recommendation to the business would be that they have one person recording the Incoming cash receipts, and they should assign another employee to take over posting the customer payments to the right accounts.This will ensure that payments will get posted to the right accounts, and reduce errors. 2. The principle of Internal control violation Is establishment of responsibility. Jeff is the custodian of the petty cash fund, and he should be the only one responsible for that petty cash fund. My recommendation for the company would be for another petty cash fund to be established for Jose when he Is needed to fill In as custodian. This will ensure that If something should happen with that petty cash fund, Jeff will not be held responsible for Joke's mistakes.Therefore, If Jose mak es a mistake as custodian of his own petty cash fund, he will be the one held responsible. 3. The principle of internal control violation is applying technological not a good for a company to store everything in the same place. My recommendation would be for the company to have all the records backed up at night as usual, and request Nadine to take the tape home with her every night. That way is something disastrous happened overnight she is able to recover all accounts with the backup tape she has taken home. The principle of internal control violation is performing regular and independent reviews. A regular and independent review should always be done whether an employee is or not meeting all Job requirements to satisfactory. My recommendation is for the manager's to immediately start regular and independent reviews. This will ensure that all employees will continue to do their Job effective and efficiently at all times. 5. The principle of internal control violation is to insure assets and bond key employees.

Auteur Theory : Darren Aronofsky Essay

Translated from the French, auteur simply means â€Å"author†. There have been varied perceptions regarding this theory, its importance and effectiveness. Auteur theory is essentially â€Å"a method of evaluating films based on the director’s involvement and input†. The concept of ‘Auteur’ was first introduced by Francois Truffaut in 1954 in A Certain Tendency in French Cinema. (1) In this work he claimed that film is a great medium for expressing the personal ideas of the director. He suggested that this meant that the director should therefore be regarded as an auteur. According to him, there are three forms in which a director may be regarded as an Auteur. He agreed with Andre Basin’s idea that the film must be the direct expression of the director’s vision. The second aspect was that the director must be skilled with the camera. He believed the director is to camera as the writer is to pen. Lastly he believed for a director to be considered as an Auteur, he must leave behind a distinctive signature (based on Alexander Astruc’s idea), visually or as an idea in the film. (2) Years later, this concept was reintroduced by Andrew Sarris in 1962, in a publication titled â€Å"Notes on Auteur theory† (3). Accordingly, for a director to be considered as an auteur, the director must be well versed with the technical aspects of the film. The director must have a distinct style or a signature that distinguishes his films from the others. The movies must have a theme, an inner meaning. The auteur theory has been receiving widespread criticism since the 60’s. It was argued that one person cannot control all aspects of the film. A film is a conglomeration of the efforts of lots of people. Despite this it is found to be very useful as the starting point of interpretation of some films. Auteur Theory suggests that the best films will bear their maker’s ‘signature’, which may manifest itself as the stamp of his or her individual personality or perhaps even focus on recurring themes within the body of work. (4) Keeping the concepts of the theory in mind, one can safely conclude that if the three criteria have been satisfied, the director may be considered the auteur of the film, these criteria being recurrent style, theme and visuals. Moving on to the discussion of the topic at hand, can the director of Black Swan, Darren Aronofsky be considered as an auteur? Black Swan is his fifth film as a director. All of his films share a similar theme. They all deal with an addiction in some form, the protagonist is always the one addicted. The movie shows the protagonist realizing his/her addiction, and there by degrading their personal life. ‘Black Swan’ and ‘The Wrestler’, share a single minded professionalism in the pursuit of a career, leading to the destruction of personal lives. (5) Aronofsky has had a lot of inspiration from different films, art forms and in general, all his films have an inherent trace of impending psychosis. However through the course of his five films one would see his stories delve more and more into the human psyche. With Requiem for a dream, it began as a drug addiction, a hallucination induced by drugs. In The Wrestler, he simply shows persistence, a plain disregard for anything but wrestling. Though this does not show psychosis, it clearly lights up the explosive nature of his pursuit. In some ways, this may be seen as a form of addiction too. In Black Swan, the main character is portrayed as one with OCD (Obsessive Compulsive Disorder). As the movie progresses, the psychosis of the protagonist progresses further, she begins to hallucinate and she draws parallels between her life and the ballad she is performing (the swan lake ballad). As a director, there are a lot of similarities in his films. There is always a sense of accomplishment accompanied by sense of impending doom, a tragedy as the price for the success seen earlier. One of the characters (usually the protagonist) always dies or suffers some sort of major personal tragedy. He is also greatly influenced by Stanley Kubrick, David Lynch, Roman Polanski, Satoshi Kon, Shinya Tsukamoto, Alfred Hitchcock, Spike Lee, Federico Fellini, and Jim Jarmusch. The Wrestler and Black Swan share a great resemblance to Satoshi Kon’s Perfect Blue. Though the similarities were acknowledged, Aronofsky denied it being an inspiration. (7) Perfect Blue ( Pafekuto Buru? ) is a 1997 Japanese animated psychological thriller film directed by Satoshi Kon and written by Kon and Sadayuki Murai based on the novel of the same name by Yoshikazu Takeuchi. (8) Mima Kirigoe (the protagonist) who is a member of a Japanese pop-idol group called â€Å"CHAM! † decides to pursue her career as an actress. Some of her fans are displeased with her sudden career change, particularly a stalker named Me-Mania. As her new career proceeds, Mima’s world becomes increasingly noir. Reality and fantasy spiral out of control. Shortly after leaving CHAM! , Mima receives an anonymous fax calling her a traitor. Mima finds a website called â€Å"Mima’s Room† that has public diary discussing her life in great detail. She confides in her manager Rumi Hidaka, a former pop star herself, about the site, however, she is advised to just ignore it. Othe set of Double Bind, Mima succeeds in getting a larger part. The producers have agreed to give her a leading role, however as a rape victim in a strip club. Despite Rumi’s warning that it would ruin her reputation, Mima accepts the part voluntarily. Though it is apparent that Mima is indecisive, the atmosphere of the scene traumatizes her so that she increasingly becomes unable to separate reality from fantasy. She can no longer distinguish her real life from her work. She becomes paranoid. Consequently people who had been involved in tarnishing Mima’s reputation are murdered and Mima finds evidence which makes her appear as the prime suspect. Her increasing mental instability makes her doubt her own innocence. It turns out that the diarist of â€Å"Mima’s Room† is delusional and very manipulative, and that an intense folie a deux has been in play. The faux diarist (and murderer) believes that she is Mima who is forever young and graceful, has made a scapegoat of stalker Me-Mania. Mima knocks Me-Mania unconscious with a hammer when he attempts to rape her, and runs to her only support she has left alive, her manager Rumi. When Mima encounters Rumi, however, her manager is wearing a replica of Mima’s CHAM! Costume and crazily singing Mima’s pop songs. Rumi is in fact the false diarist, who believes she is the â€Å"real Mima†. Rumi is angry that Mima has been ruining the â€Å"real Mima’s† reputation, and decides to save â€Å"Mima’s† pristine pop idol image through the same means she has been using all along: murder. Mima manages to incapacitate Rumi after a chase through the city despite being wounded. Rumi remains permanently delusional and institutionalized. Mima has grown from her experiences and has moved on with her life with new found independence and confidence. One finds a striking similarity between the two storylines, if one were to imagine Mima as the White Swan – the pure innocent Nina Sayers, and Rumi as the Black Swan – the violent, sensual and dangerous psychosis of Nina. There are too many similarities between the storylines. In both cases, during the fight between the two characters (For the sake of this argument, let us assume the black swan and white swan as separate. ) The antihero of the story attacks the protagonist. In both cases the protagonist is wounded in the abdomen (almost in the same area – around the liver). In both cases, the protagonist continues their bidding after being wounded. Nina finishes the dance while Mima goes around the city in a chase. In both cases, the protagonist is affected by the job, and they receive threatening messages. Nina sees Whore while Mima receives the fax calling her a traitor. If one were to delve deeper, one even sees the similarity in the naming of characters (however this argument is based sole on conjuncture and lacks credibility. ) Despite this there is an immense similarity between the story line and the character sketches. Due to the overbearing nature of the similarity I find it hard to accept that Black Swan is not influenced by Perfect Blue. The inspiration may not have been intentional; however, the similarity is too much to be dismissed as coincidence. In this light such an undeniable similarity between the two films disqualifies Darren Aronofsky from being an auteur. Black Swan also draws heavily on what is called the Doppelganger effect and the split personality. These woven into a maze of mirror motif is the central theme throughout the film. This is a strategy which is well-known in classical Hollywood cinema such as precisely ‘The Red Shoes’. 10) This being said this film has Aronofsky’s stamp all over it. There is a great similarity between the characters ‘The Ram’ and ‘Nina Sayers’. In his own words†¦ â€Å"They are both artists who use their body Age threatens them Physical injury threatens them They only have their hands to express themselves. †(9) In all of his films, the sets have a sense of belonging. The props used are bare minimum. Only those strictly necessary are used. His set design is simple precise yet he somehow manages to bring a sense of belonging, a nativity to his sets. For example in the Black Swan recital, the set was simple, yet it somehow added to the elegance and grandeur bringing with it a certain ethereal quality with it. Another notable trait in his films is his music. Black Swan marks the fifth consecutive collaboration between Aronofsky and Clint Mansell. In all his films, music is exploited as a medium to dramatize the situation. The editing is also pertinent to the music. This relationship between the editing and BG score is most successfully exploited in Requiem for a Dream which gained acclaim for its hip-hop style editing. His attention to detail is yet another endearing stamp. For example, in the opening scene of Black Swan, the change in the tutu indicating the change in emotion is subtle almost not noticeable, yet one feels the emotion without really seeing it. The small stiff (classical pancake) tutu, for the innocent cheery bit, the long flowing (romantic) tutu for poised elegance and a graceful waltz, and, the shorter (platter) tutu, for the transition into Black Swan. Also if one were to notice keenly, one would see that almost every scene of Nina alone in the film will also include a mirror reflecting Nina in some angle. Aronofsky has made great use of this to exploit the two mindedness of Nina’s psychosis. In all respects of style, editing and mise-ene-scene Aronofsky has distinctly made his mark on all five films. In an interview, Aronofsky, says he was deeply influenced by the roller coaster Cyclone, and that he has adopted that intense structure, which keeps the audience on the edge, in his films. He aims to thrill the audience and amuse them with his films. (9) This, he certainly has achieved in his films.

A practical criticism of ‘For the Union Dead’ Essay

The aquarium is gone. Everywhere, giant finned cars nose forward like fish; a savage servility slides by on grease (For the Union Dead – Robert Lowell) In Lowell’s poem ‘For the Union Dead’ there is an underlying theme of the lost idealism which caused the American Civil War and its replacement with commercialism and materialism. â€Å"Giant finned cars nose forward like fish†, here Lowell reminds us of 20th century materialism by bringing about the image of the highly desired Cadillac (â€Å"finned cars†) whilst combining it with the comparison of a fish. This in my mind creates the idea of mankind following each other mindlessly to each material craze and status object. Moreover the image of men following each other (â€Å"nose forward†) alludes to the idea of slavery indicating that rather than liberating the African Americans from slavery we have in fact, all become slaves to our material needs. Also the aquatic references seen throughout the poem seem like references to Alan Tate’s poem ‘Ode to the Confederate Dead’: â€Å"Now that the salt of their blood stiffens the saltier oblivion of the sea, seals the malignant purity of the flood.† This could again be an indication that perhaps the North (and indeed all of ‘liberated’ America has in fact lost its war against slavery to materialism and are now in the â€Å"Oblivion of the sea†. Although this then creates a feeling of reliance on our objects as us fish are reliant on the oxygen in our ‘metaphoric sea of materialism’ which is also symbolized earlier in the poem by the image of Colonel Shaw’s monument in Boston Common being propped up to by planks so It can survive the building of an underground parking structure: â€Å"Propped up by a plank splint against the garage’s earthquake†. This not only shows this new found reliance on objects but represents the war Colonel Shaw was (and now again is) fighting against slavery. I pressed against the new and galvanized barbed fence on the Boston Common. Behind their cage, yellow dinosaur steam shovels were grunting as they cropped up tons of mush and grass to gouge their underworld garage (For the Union Dead – Robert Lowell) â€Å"Galvanised barbed fence [†¦] grunting [†¦] gouge† brings to mind the contrasting image that these new mechanical â€Å"Dinosaurs† are our slaves. Which is made to seem cruel not just by the use of â€Å"Grunting† but by the personification of the cranes: â€Å"They cropped up tons of mush and grass†. This suggests that the 20th century American has not just forgotten the Idealism of the North but has in fact taken up that of that South! This gives further meaning to the fact that the work these new ‘slaves’ are doing is knocking over and damaging Shaw’s monument which is close to falling into the hole in Boston Common (being dug for a new parking structure) which represents the killing of Shaw and his Idealism by Materialism. â€Å"The ditch where his sons body was thrown and lost with his Niggers† – Again.

Inclusive education Essay Example | Topics and Well Written Essays - 2500 words

Inclusive education - Essay Example It refers to an attitude, believe and value system. So in this case, once this system has been adopted by a school, then it should drive all the decisions and actions by those people who have adopted it. It is a belief that schools embrace and educate all students and not just selecting those students who can be fit in that school. It is believed that all children are unique in their different ways and they have different talents. It is through inclusive learning that children are in a position to get good education and also attain the best practices in school. This can be well illustrated by cooperative learning. In this case, you find that students are in a position to learn with each other, and offer the necessary assistance to each other where possible. It is through this cooperation that students can be in a position to do well in most of their activities. Inclusive learning also reflects multi intelligence, multi level instructions, good learning and also teaching styles where children are exposed to competent teachers who express their teaching skills fully and they are also exposed to differentiated curriculum. (Armstrong, Barton and Armstrong, 2000) With inclusive schooling, it emphasizes on various issues. First it ensures that students get equal educational opportunities. It is through this that students will be in a position to have access to the various educational programs hence leading to their success in school.... It is through inclusive schooling that the necessary resources will be provided and these resources are usually provided to ensure that students are in a position to learn well. Third there is respecting differences whereby students in this case will be in a position to respect differences which might arise. (Bender, 2002) Inclusive schooling is said to teach mutual respect where students are in a position to respect each other, responsibility in the sense that these children will be in a position to demonstrate responsibility by obeying commands and taking charge of their responsibilities, they will also learn to be generous to others and last they will learn to be independent. (Dei, Zine, and James-Wilson, 2002) Therefore the rationale behind inclusive schooling can be summarized as intending to ensure that all children regardless of the differences the may exist between them are put under the same class environment in order to take advantage of their diversity and help them to learn well. In this regard it is also expected to lower the cost of education to the parent and to the government since students will be able to use the same resources and amenities rather than providing each student with their own resources. It is an approach that is expected to make the education process smooth for the teachers, parents and the students. It aims at eliminating any discrimination and achieving equality for all in education. Values, policies and strategies of inclusive education One of the most important emphasis of the inclusive education is the way in which it advocates for changes and modification in the content, the approach structures, policies and strategic of the education system. The model is enabled by a value system which ensures that there is child

Like water for chocolate Research Paper Example | Topics and Well Written Essays - 1500 words

Like water for chocolate - Research Paper Example e releasing of a newer version by replacing various charts (Gilly 23).The author sets this story at a time of Mexican Revolution that took place between 1910 and 1920 in Northern Mexico. It concerns a family where the youngest called Tita is an excellent cook and originator of various food recipes. The Mexican Revolution concerned redistribution of land and government reform to the various women and men who played a pivotal role in the revolution. The process involved restructuring the society to include women in the public sphere fully. The main objective of this revolutionary was to create a new constitution. The congress that took place in1916 and the number of Mexican women who attended the congress was seven hundred (Esquivel 29). This group had the intention of reforming the 1884 Civil Code. This Civil Code denied women a right to act in an independent manner from the male who are the leaders of household in all aspects which include child guardianship and inheritance. The feminist congress had concentrations in education, voting, and issues concerning the holding of various public offices. In 1917, the government drafted the revolutionary constitution and women acquired various rights (McLynn 19). They acquired the right to vote and started becoming active in politics. Despite the fact that Mexico was independent of Spain, the governments had internal and external conflicts. The revolution tore the country in the twentieth century. Madero who was a liberal leader led a revolt in 1910 after he lost a rigged election. This culminated in Diaz resigning and Madero replacing him as president in 1911. The Mexican revolution concerned different beliefs, different political parties, and how the country should be governed. All these disagreed with each other and fought for power and struggled to emerge as the ruler of the country. One finds the same observation in Tita’s family. Tita is not comfortable with Mama Elena’s traditions and disagrees with them. She

Law 1183 for Washington Essay Example | Topics and Well Written Essays - 1000 words

Law 1183 for Washington - Essay Example â€Å"The main issue however concerns the taxation and distribution of liquor in Washington† (Smith, 2010). There are proponents who argue that privatizing the liquor sales and distribution and availing the liquor on groceries would greatly increase the availability of the liquor sales, which will be another important part of the income to the city authority or the state government. The opponents of this initiative on the other hand advocate for state control in obtaining revenue as well as streamlining the drinking patterns within the community. Initiative 1183 is a measure that is aimed at directing the liquor board to close down all the state owned liquor stores, terminate the contracts offered by the state to private stores selling liquor, and allow the state to issue private licenses that would allow the liquor to be sold, imported and distributed by private parties in groceries and other easy access points (Mercier, 2010). This initiative is aimed at repealing the unifor m pricing as decided by the state liquor board, and other requirements that are required in sales and distribution of hard liquor. In fact, stores that previously had contracts from the state controlled liquor distribution under this initiative might be converted to retail licenses. About 1,428 retail outlets in initiative 1183 would be licensed compared to3, 357 in initiative 1100 (Mercier, 2010). A study carried out by the Office of Financial Management (OFM) has been the impetus to the proponents of this initiative. There are estimations that upon implementing this initiative, the State General Fund revenue would increase from $216 to $253 million, while the local revenues are expected to improve from $186 to $227 million (Mercier, 2010). This report has been used to consolidate proponents to vote for the initiative. In addition the initiative would retain the tax structure and license issuance would be about 17% of all liquor sales and an annual fee of $166 (Corte, 2010). One as pect in states protectionist monopoly is to reduce the overuse of hard liquor among its citizens. This makes the state to control not only the sales but also the consumption of the liquor. There are however, certain contradictions: â€Å"does state monopoly control the patterns and trends in alcohol intake? State protectionism on alcohol consumption might not directly affect the alcohol consumption patterns in the state† (Smith, 2010). For example, California has an average consumption of 2.34 gallons of alcohol per an average person, while Washington has an average of 2.35 gallons of alcohol consumed per an average person, as per the National Institute of Health figures 2007 (Smith (2010). California has a full privatized hard liquor licenses and sales and distribution of other alcoholic drinks. This makes alcohol readily available at groceries and other near points, where consumers may access the alcohol with ease (Smith, 2010). Proponents have a view that there is no diffe rence between alcohol consumed in the two states, meaning that the protectionism in alcohol distribution would not be effective in controlling consumption. â€Å"Washington would be the fifth state in protectionism from the second position after initiative 1183† (Smith, 2010).This argument has been used to challenge the assertion that the use of alcohol would be a menace to good governance (Timberlake,

Voter Turmont vs Ballot Initiatives Research Paper

Voter Turmont vs Ballot Initiatives - Research Paper Example Whether or not ballot initiatives have an effect on voter turnout in the United States is a contested question. Voter turnout referrers to the number of people who take part in a voting forum like election, referendum or other gatherings. Voter turnout exhibits some hearty pattern that explains why the number of voters varies from one place to another. According to most researches done on voter turnout, the main factor that affects voter turnouts is institutional variables (Jackman, 1987). Ballot initiative on the other side is referred to a process of whereby the people are authorized to enact or refute legislations at the polls hence superseding the legislative body. An initiative is a type of election facilitated by the people with the aim of resolving issues that elected leaders fail to raise or attend contrary to public desires. In 1962, Powell’s book, ‘Contemporary Democracies’ was the first book to be published on the study of voter turnout. His 1986 articl e, ‘American Political Science Review Articles’ established that countries with nationally competitive districts whose parties and members usually have enticements to persuade voters to turn up at the polls, or those that had strong party-group association such as churches and unions were likely to have high voter turnout (Powell, 1986, p 21-22). In his conclusion, Powell said that the turnout in America is inhibited by its institutional context, and the main emphasis, which is also the most powerful variable, is on party-group associations. Voter turnout in the past years has been on a declining trend in the united State, with only a few exceptions. Although some sources from defenders of participatory, normative theorists and to some extent journalist have indicated that ballot measures that are initiated by citizens are likely to increase voter turnout, other researches refute the assertions, despite use of direct democracy having been embraced in the United States f or the last 25 years. Whereas those who prefer direct democracy dispute that citizen participation, efficacy and confidence in the government can only be increased by permitting citizens to vote directly on policy issues, those who oppose say the process will only have minimal change, and threatens to deteriorate state legislatures and replace representative democracy (Broder, 2000). Most of the conclusions based on the comparative cross-national research are vigorous and as a result, there lacks a compelling foundation over the connection between voter turnout and ballot initiatives. Institutional variables in the end get to be overstated. Use of the initiative process for over 26 years in 50 states has been linked to higher turnout rates. The initiative process is evidently assisting in increasing the number of turnout in electoral participation. For example, in the 1990’s the discrepancy in turnout rates between initiative and non initiative states has been on the rise ove r time, estimated at 3% to 4.5% higher in presidential elections and between 7% to 9% higher in midterm elections (Tolbert Grummel & Smith, 2001). The rate of ballot initiative measures is increasing in the United States, with an increase on the use of initiatives to decide policy matters. In states such as California, Mississippi, Colorado,

MAE Assingment Assignment Example | Topics and Well Written Essays - 2000 words

MAE Assingment - Assignment Example Dev.   0.025827   11240942   65635008   0.019383   Observations   45   45   45   45 Table 1 above presents the descriptive statistics for our variables of interest. The only point of concern that may arise in this situation is that all the variables reflect some degree of skewness which violates the normality assumption. Additionally, the fact that the number of observations is only 45 may also be a point of concern since this can lead to small sample bias. 2. Time plots Figure 1: Time plot of P There are no seasonal patterns evident in the time plot of P. Figure 2: Time plot of Q The time plot of Q exhibits strong seasonal variations. Figure 3: Time plot of G As is evident from figure 3 above, similar to the time plot of P, the time plot of G also does not exhibit seasonal fluctuations. Figure 4: Time plot of X Figure 4 shows that X also follows a seasonally fluctuating pattern 3. Thus, there is strong evidence of seasonal fluctuations among the Q and X series. Thi s is visible in the oscillatory patterns that these series seem to follow. The series P and G exhibit no seasonal patterns. Additionally, all the series reflect a steady upward trend. Therefore inclusion of seasonal dummies is important since our dependent variable Q does exhibit seasonal fluctuations. ... 18.31869 0.0000 P -7530.197 6092.988 -1.235879 0.2235 G -84559.50 9770.479 -8.654591 0.0000 X 1.865016 0.111494 16.72746 0.0000 R-squared 0.915605   Ã‚  Ã‚  Ã‚  Mean dependent var 4442.111 Adjusted R-squared 0.909430   Ã‚  Ã‚  Ã‚  S.D. dependent var 505.4463 S.E. of regression 152.1132   Ã‚  Ã‚  Ã‚  Akaike info criterion 12.97181 Sum squared resid 948675.2   Ã‚  Ã‚  Ã‚  Schwarz criterion 13.13241 Log likelihood -287.8658   Ã‚  Ã‚  Ã‚  Hannan-Quinn criter. 13.03168 F-statistic 148.2710   Ã‚  Ã‚  Ã‚  Durbin-Watson stat 1.390217 Prob(F-statistic) 0.000000 From table 2 above we find that the estimated coefficients for both P and X are significantly different from zero (evident from the t-statistic). G however is not a significant determinant of Q. The coefficients reflect that the demand for drink and tobacco is negatively influenced by the price of the items and positively influenced by the total consumer expenditure. The coefficient on G is also negative but since it is not significantly different from 0 at the 5% level, we conclude that it does not have an influence on drink and tobacco demand. Thus, our results imply that an increase in the prices of drinks and tobacco will lead to a reduction in its demand while an increase in overall consumer expenditure leads to an increase in the demand. 5. Attempting to include all four dummies leads to perfect multicollinearity. Thus we modify the equation and include dummies for the 1st 3 quarters only. Table 3 presents the results. Table 3: OLS estimation with quarterly dummies Dependent Variable: Q Method: Least Squares Date: 09/01/11 Time: 01:01 Sample: 1980Q1 1991Q1 Included observations: 45 Coefficient Std. Error t-Statistic Prob.  Ã‚   C 5127.935 356.9563 14.36572 0.0000 P -8713.964 2700.994 -3.226206 0.0026 X 0.805451 0.096091 8.382187 0.0000 G -23150.70

INFORMING AND INVOLVING EMPLOYEES HRM Essay Example | Topics and Well Written Essays - 3000 words

INFORMING AND INVOLVING EMPLOYEES HRM - Essay Example Employee participation and involvement are two different terms with two different outcomes. Cox, Marchington and Suter (2009) consider employee involvement and participation (EIP) a loose term as it can give rise to different perceptions of the terms. Different definitions and interpretations have been given of the term EIP which ranges from encouraging commitment to achieve organizational success or exercising influence over their work or personal involvement of organization al workers. According to the Chartered Institute of Personnel and Development (CIPD) employee involvement (EI) is defined as ‘a process of employee involvement designed to provide employees with the opportunity to influence and where appropriate, take part in decision making on matters which affect them’ (LTSN, n.d.). Employee participation (EP), on the other hand, is ‘a process of employee involvement designed to provide employees with the opportunity to influence and where appropriate, take part in decision making on matters which affect them’. Thus employee participation has a direct impact on decision-making whereas employee involvement amounts to support, understanding, commitment and contribution (AC219, n.d.). EI has been identified as a means to secure commitment and high performance which has resulted in an increase in the interest in employee involvement and participation. EI has been a major area of growth in the UK since early 1980s. It includes team-working (including self-managed teams), team briefing, downward communications, two-way communications, suggestion schemes, problem-solving groups, and financial participation (including profit-sharing schemes). Performance is a function of ability, motivation and Oppurtunity. This suggests that the selection process should be rigorous and the training systems should be improved to increase the ability levels.

STI and HIV Essay Example | Topics and Well Written Essays - 2500 words

STI and HIV - Essay Example Nearly 88% of these people are in the sexually active and economically productive age group of 15 to 49 years (National AIDS Control Organization 2006, 11). Therefore, most people living with HIV are in the prime of their working lives with many of them supporting families. The remaining 8% of the infected population are above 50 years and another 4% are children (National AIDS Control Organization 2006, 11). The spread of HIV in India has been very uneven. Although the overall rate of infection in India has been very low, certain regions and certain population groups within the country have extremely high rate of infection. The infection rates are extremely high in the southern states of Andhra Pradesh, Maharashtra, Tamil Nadu and Karnataka and the far north-east states of Manipur and Nagaland. Together these states account for 64% of the HIV burden in India (National AIDS Control Organization 2006, 14). Prevalence Rates Overall the prevalence of HIV for adult males and females has shown a declining trend in the past five years. In 2006, it was 0.36% while in 2002 it was 0.45% in 2002 (National AIDS Control Organization 2006, 11). ... 2007). One of the major concerns regarding the epidemic of HIV in India is the increase seen in the proportion of infections among children and adults above 50 years. Among children, the prevalence of HIV was 3% in 2002 which increased to 4% in 2006 (National AIDS Control Organization 2006, 13). Similarly, the prevalence of the disease was 6% among adults above the age of 50 which has increased to 8% in 2006 (National AIDS Control Organization 2006, 13). Transmission route In India, nearly 88% of the transmission of STI/HIV happens through heterosexual contact (The World Bank 2008). Other routes of transmission include perinatal (4.7%), unsafe blood and blood products (1.7%), infected needles and syringes (1.8%) and other unspecified routes of transmission (4.1%) (The World Bank 2008). It is interesting to note that in the high prevalence southern states of India, STI/HIV has been found to spread primarily through heterosexual contact while in the high prevalence north-eastern states , the disease has been found to spread mainly among injecting drug users and sex workers (National AIDS Control Organization 2006). Researchers believe that the HIV epidemic in India has followed the ‘type 4’ pattern (The World Bank 2008). This is a pattern where new infections occur among the most vulnerable populations like the female sex workers, men who have sex with men or the injecting drug users. The infection then spreads to ‘bridge’ populations like the clients of sex workers or sexual partners of drug users and finally it enters the general population. Studies have revealed that long distance truck drivers and male migrant workers make up significant proportions of clients of sex workers (UNGASS 2008). In

Original writing Essay Example for Free

Original writing Essay I desperately waited for the answer. Laura has been murdered I was speechless once again come on honey, think, youve been in this situation before. My brain started to hurt, I was in utter confusion. She was only 9 years old and she was a Buddhist, she had no health problems, she had no enemies well not that I could think of 20 seconds from then has just gone passed without any speech, I received a fax from the south, it was a picture, I took it, I gazed at it in horror. What I saw was something that would never leave my mind. It was a picture of my best friend, brutally murdered a piece of her body probably her arm had been cut into almost equal boxes of about 10 cm each and placed in a certain way to spell something something that still continues today. the LTT. LTT is the Liberation Tigers of Tamil, a terrorist organization in Sri Lanka that began in 1970 as a student protest over the limited university access for Tamil students; currently seeks to establish an independent Tamil state; relies on guerrilla strategy including terrorist tactics that target key government and military personnel; the Tamil Tigers perfected suicide bombing as a weapon of war. They attack the southern part of Sri Lanka where there are many Buddhists, Muslims and Catholic. I never thought the LTT would have gotten this far. but they have. The questions and the disappointment that reached my brain were agonizing. I thought she trusted me, I thought she would tell me everything, I thought she would never doubted me. I thought she told me that there was NO ENEMIES, although I did know she had her little plans for peace, but not clearly. That instance I realised all these years of knowing each other was a bogus we didnt really no each other if we did then why am I so confused? 1. Today is the 22nd of August 2002, 9:15 pm. I stared at the newspaper in my bedroom. The newspaper I remember its the last thing we were talking about before we left, you were very clever for your age, I valued your words highly, I sure didnt seem to care about them, but I did keep it in my head, thats the only thing I have to remind myself of you. Remember once you were reading this newspaper about enlightenment? You gave me the English section; I thought it was pretty stupid, I remember reading it, and putting it away. Do you also remember the next day you went home angry at me for putting curd in your shoes and tying the shoelaces together? I felt bad okay, I waited with the newspaper for you to come back and give it to you, as stupid as it may sound of giving you an old newspaper, I kept it in a way to apologize to you after annoying you and show you that I do take care of your things. but you never came back. I cant ask you anymore, the answers to the questions, the answers to this mystery, and the answers to life. I cant talk to you no more, Mum will think Im gone insane because only mad people talk to a newspapers. But then I wouldnt mind because if murdering people is how sane people are, then Id love to be insane. I can however read it to you; this page will always be in my mind. Ill hold you to my ear so you can tell me what happened, but I know newspapers cant talk. Ill draw you in the newspaper and then you can talk But only computer animations do that. Remember yesterday when you flew over to the temple through my window, and you were floating with the wind? Remember I ran after you screaming and shouting your name? Remember people staring at me, thinking I was a stupid child running after a newspaper? I dont care what they think, what matters is what I think, I think of you, Ill cuddle you, Ill die for you, you are that one person that I ALWAYS trust, that one person who is very dear to me, even if you are just a friend, you were part of the family. Sorry for all the things I told you, it was a misunderstanding, you understand right? Thats what best friends do, thats what humans do. Please come back, at least give me a clue pointing me to the direction of where this happened, maybe how the angels pointed to the illumination church in Vatican city just like Dan Brown says in his book of Angels and Demons, but yours with newspapers perhaps? Remember when we played this little game of treasure hunt in the back garden? Please tell me this is one of that, you know I hate mysteries, then why did have to leave me mystified? Give me the answer to this mystery, Ill try my best, point me towards it, whatever, Ill find my way, I know you will guide me so I wont get scared. Ill promise you Ill do whatever you wanted me to do; Ill play lots and lots of treasure hunt games with you, even if I think its stupid. Are you satisfied now? Please come back. Please tell me this is a joke. Ill email you, but I cant, I dont have your address, Theres no point in emailing you because maybe you dont have computers up there or you just dont a email account. Is it [emailprotected] com? You have to live in a place you have computers to read my mail. I cant send you a post card or a letter can I? I dont have your address; do you have a postman up there? Im sorry I spilled water in your painting, I hope thats not why you went, I promise I wont do it again, you know Im clumsy and thats how I learn. Only I know how painful it is to tell you this, but I regret shouting at you, annoying you just because I was bored, I know you were joking those times but I took it seriously. But anyhow and anyway I wish you were back. Ill staple my mouth so I wont scream at you and loose you again. I read this newspaper over and over again, to see if you had left me a clue, the only clue I found was heaven, is that it? If I keep this, it will remind me of you, Ill place it carefully and keep it organised the way you like it. Ill make sure your letters wont dissolve in my tears. I should stop now, I wouldnt want to wet you, dont worry Ill look after you. I take care of you like you took care of me and stood beside me in everything I did. Oh how I wish you were back. Please, please, please come back. I miss you awfully.

The Last Spin Essay Example for Free

The Last Spin Essay Two boys, named Tigo and Dave, who were both enemies and both belonged in two different gang’s had never met each other before and engaged themselves in the game known as the ‘Russian Roulette’. They had to settle a situation for their gang’s. Apparently, Dave and his gang member’s had set foot on the territory that belonged to Tigo and his gang member’s. When they engaged themselves in the ‘Russian Roulette’, Tigo wore a green silk jacket with an orange stripe on each sleeve and had thick black hair and a nose that was a bit too long. Meanwhile, Dave wore a blue and gold jacket and had large eyes that were moist – looking. They were both sitting in the middle of a basement room on two different chairs with a table in the middle. On the centre of the table there was a Smith and Wesson . 38 Police Special that was worth forty five dollars and three . 38 Special cartridges. Tigo had referred to the gun by saying ‘I like a good piece’. It all began when Tigo loaded the first cartridge, twirled the cylinder, placed the gun on the side of his head, squeezed the trigger and nothing happened. They kept on going while talking amongst themselves, until Tigo added an extra cartridge to change and lower the odds. Afterwards, they still kept on going, while talking amongst them, until Tigo re added an extra cartridge to make it even money and decided that they could not keep on playing this game for the rest of the night. He had said ‘To hell with the club! ’ and decided to ignore the situation. Meanwhile, Dave had wondered if they should become friends and said ‘†¦Friends? ’ Tigo agreed and decided that they shall do ‘The Last Spin’. The tragic event occurred when Dave picked up the gun, placed it on the side of his head and fired. There was an explosion and half of Dave’s head was ripped away and shattered his face. A small, sharp cry had escaped Tigo’s throat, a look of incredulous shock knifed his eyes and he placed his head on the table and wept. 2. At the beginning of the story, Dave and Tigo are separated because: They are both each other’s enemies. They both belong in different gang’s. They both have no blood for each other. They have never met each other before. They have never crossed paths before. As the story continues, Dave and Tigo are brought together because: They both have to settle a situation for each other’s gang. They both have girlfriends. They are pretty lucky. They both have to stick to their gang’s They both went through the spinning of the cylinder without one of the cartridges coming out of the barrel several times when they squeezed the trigger. They both agree to become friends when Dave says ‘†¦Friends? ’ They both want to go to the lake on Sunday with their girlfriend’s in one boat. If I were to film the story I would chose Johnny Depp and Al Pacino to cast in the roles of Tigo and Dave because they have both played in different gangster movies when they were younger or recently. For example, Al Pacino acted as an Italian mafia named Michael Corleone in the ‘Godfather Trilogy’ in 1972, 1974 and 1990 and also as a young Cuban refugee who turns into a gangster named Tony Montana in the movie ‘Scarface’ in 1983. Besides, Johnny Depp has acted as an American bank robber in the Midwest during the early 1930s named John Dillinger in the movie ‘Public Enemies’ in 2009. Furthermore, the set I would use would be in a small basement room with a bit of light that is coming from a light bulb that is hanging from the ceiling, a few windows but some of them would be broken and damaged, a door that is old and rusted, a grey floor covered with some dust and dirt, a grey ceiling that has some cracks and some spider webs, and some grey walls that are covered with graffiti. Inside the basement room I would have a table, two chairs and some boxes that are spread around and are stacked up one on top of the other. Some of the directions I would give to the actors, Al Pacino and Johnny Depp would be to mention that Tigo and Dave are enemies and at the end of the short story, they become friends. I would also tell them to express their feelings and emotions such as when they are about to squeeze the trigger and shoot the Smith and Wesson . 38 Police Special next to their heads, when they speak about their girlfriends and when Tigo starts weeping because Dave’s head has been ripped away from the explosion. Last of all, I would tell them to use gangster voices and expressions. 4. Some of the words that I think are KEY to our understanding of the story would be: Enemy. Club. Settle. Situation Friends Explosion. Weeping. 5. Some grammatically incorrect sentences that I picked out are: I seen pieces before I’ve seen pieces before. I got no bad blood for you I’ve got no bad blood for you. We going to sit and talk all night Are we going to sit and talk all night. I man, you got to admit your boys shouldn’t have come into our territory last night I mean, you’ve got to admit that your boys shouldn’t have come into our territory last night. I got to admit nothing I’ve got to admit nothing. I never seen you either I’ve never seen you either. Where you from originally? – Where are you from originally? Why you giving me a break? Why are you giving me a break? The they get to be our people’s age and they turn to fat They get to be our people’s age and then they turn to fat. You’re the one needs the courage: You’re the one who needs the courage. There should be some body you can trust There should be somebody you can trust. Well here goes Well here it goes. We keep this up all night We can keep this up all night. 6. To begin with, in the passages of description, the author, Evan Hunter, describes Tigo and Dave and the different objects. Secondly, in the lines of dialogue, he describes the different discussions amongst themselves. Thirdly, in the passages of description and narrative he uses the Third Person Point of View by using the word ‘He’. However, in the lines of dialogue he uses Second Person Point of View by using the word ‘You’. Furthermore, in the passages of description and narrative, Evan Hunter, pretends to be in the same room as Dave and Tigo so therefore he can give more information to the readers. Last of all, the purpose of the Second Person Point of View by using the words ‘You’ in the lines of dialogue is used so that the author can pretend to be part of Dave and Tigo’s conversation so he can give more information to the readers. 7. Today’s age is more or less the same as the past generations. In the past, there were many wars were many people were killed. For example, in Europe there was World War One from 1914 to 1918 in which the Allied Forces (France, England, United States, Russia and Italy) fought versus Germany who belonged in the Central Powers and later on, there was World War Two from 1939 to 1945 where again, the Allied Forces fought versus Germany, Italy and Japan who belonged in the Axis Powers. Secondly, there also were many famous criminals. For example, there was John Dillinger from the United States, Jack the Ripper from England and Jacques Mesrine from France. Last of all, there were many different Civil Wars that erupted in different countries. For example, there was the American Civil War in the United States from 1861 to 1865 and the Irish Civil War in Ireland from 1922 to 1923. However, in nowadays, there are still some Civil Wars such as the Somali Civil War in Somalia that erupted in 1991 and the Civil War in Afghanistan that erupted in 1978. In nowadays, there is still some violence because nearly every single day, there are many stories on the news about children that have been kidnapped by one or several pedophiles. For example, there is Marc Dutroux from Belgium. Secondly, there is the Death Penalty which was used in many different countries in the past but today, it is still used in countries such as the United States, Korea DPR, China and Cuba. Last of all, there were many gangs in the past but they still exist today in the United States, South America, Eastern Europe and China. For example, in China you have the gang known as the ‘Triads’. To conclude, there is violence because of Terrorism. There is always a Terrorist Group such as Al Qaeda, Hamas or the Revolutionary Forces of Colombia (FARC) or an individual person who always has to commit an act of Terrorism to induce fear in victims who are ruthless and not protected against Terrorism. Therefore, they kill, injure, maim, destroy and terrorize many citizens. The Terrorist Group or individual person commits the Act of Terrorism for various reasons such as Revenge, Communism, Separatism, Poverty and Economic Disadvantages, Globalization, Religion, Social and Political Injustice. Additionally, there is violence within the Child Soldiers in countries such as Rwanda and the Democratic Republic of the Congo because they are brainwashed by their factions, clans or leaders and are ordered to kill other people who are defenseless when they are given an AK – 47. A perfect example would be the movie ‘Blood Diamond’ with Leonardo Dicaprio and directed by Edward Zwick. Last of all, there is violence in many different sports such as football (soccer) and basketball. It may happen when the supporters of a certain team will fight against the supporters of another team if they have any rivalries. For example, the Chelsea supporters will fight against the Arsenal, Tottenham Hotspurs or Manchester United supporters. This would create violence because many of them would get killed, wounded and stabbed. A second example would be a football player that would get killed by its own supporters. An incident of this type occurred in the 1994 World Cup in America where a Colombian football player, Andres Escobar, had accidentally scored an own goal in a match against the United States, a match which Colombia lost 2:1. On his return to Colombia, Andres Escobar had been confronted outside a bar in Medellin by a gunman who shot him six times. When he shot him, he always repeated ‘Goal ‘after each shot.

Vedic Mathematics Multiplication

Vedic Mathematics Multiplication Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 Ãâ€" 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have –2 Ãâ€" 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∠´ Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana VyavakalanÄ bhyÄ m Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 – 5x – 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) ( x – 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 Ãâ€" 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol à ª5à ª, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 Ãâ€" 58, and the second 23 Ãâ€" 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 Ãâ€" 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure Vedic Mathematics Multiplication Vedic Mathematics Multiplication Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 Ãâ€" 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have –2 Ãâ€" 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∠´ Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana VyavakalanÄ bhyÄ m Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 – 5x – 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) ( x – 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 Ãâ€" 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol à ª5à ª, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 Ãâ€" 58, and the second 23 Ãâ€" 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 Ãâ€" 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure