New Emission Standards Essay Example for Free

New Emission Standards Essay One of the growing Threats to Harley Davidson’s reign of dominance in US markets has been a growing awareness of the environment. In response to this, the bar for emission standards is continuously raised. In 2010 model year or newer motorcycles emission standards for both HC + NOx were reduced from 1.4 prior to 2010 to just 0.8 after 2010 (US Government Printing Office. 2013). For at least 5 years now, there have been rumors circulating that Harley Davidson would abandon the traditional air cooled designs in favor of a new liquid cooled lineup in response to ever growing emission standards (Huze. 2011). If emission control standards continue to tighten, Harley Davidson may be forced to do exactly that. Procurement of Raw Materials Historically, Harley Davidson has relied on limited number of suppliers for raw materials to provide the components used in its manufacturing plants. In some cases, the entire company’s business is dependent on just one supplier to deliver certain raw materials in time. The lack of versatility in this area means that rising input costs could lead to capacity issues in the long run. Additionally, increasing costs for commodities could lead to capacity constraints, ultimately leading to lower production (Harley Davidson, Inc. 2012). Competitive Landscape Currently, Harley Davidson holds 56% of the market share for heavyweight motorcycles, defined as those motorcycles that displace more than 650cc. To some this may be an impenetrable advantage in the market, to other, a source of vulnerability (Taylor III, 2012). For years, Harley has been the undisputed king of the â€Å"bad boys†, but in recent years up and coming companies such as Polaris are trying to muscle in to the motorcycle arena. Polaris, a company known for its snow mobiles has only recently begun selling motorcycles 14 years ago (Taylor III, 2012). Polaris has shown that it understands what its buyers want and has demonstrated success in achieving higher sales through its colorful names such as Victory, 8-Ball, and Jackpot lineup. With aggressive pricing strategies and comparable quality, Polaris has quickly passed foreign competitors such as Honda, Yamaha, and Kawasaki and is now setting its eyes on Harley Davidson after its recent acquisition of Indian. Harley Davidson – Opportunities (O) Global Expansion In response to declining domestic sales, Harley Davidson has decided to expand its global presence in China and India. In 1995 Harley Davidson entered the Hong Kong market, opening its first mainland China office just 10 years later in 2005 (Miller. 2012). Currently, Harley has 8 full service dealerships in china and have plans to open an additional 5 more within the next 5 years. Harley’s biggest challenges have been the understanding of foreign markets. In China for example, the motorcycle market was riddled with low cost economic alternatives for daily transportation. The average engine size ranges anywhere from just 50cc to 600c. With some of Harley’s heavyweights tipping the scales in the range of 800c to 1600cc beasts, the concept of heavyweight leisure riding was simply not understood yet in foreign markets. Harley would have to start from the ground up if it wants to succeed. Today, Harley has succeeded in growing its international presence in over 70 countries across the globe. Although growth into international markets has been substantial, there is still significant room for further expansion (Miller. 2012). New Product Launches Another area of opportunity for Harley Davidson lies in the arena of launching new products. In 2014 Harley expects to launch eight new models, a record number of new models within the same year. Harley realized that with the baby boomer generation coming to a close, younger audiences will demand new and innovative products. The company, now in its 110th year of operation, has launched a customer driven product development program dubbed â€Å"Project Rushmore† in hopes of succeeding in understanding what design elements are appreciated more by younger buyers (UPI.com. 2013). If they succeed, Harley could feasibly further expand its market share in both US and international markets significantly. Restructuring Plans The Milwaukee based motorcycle maker has succeeded in keeping its full-year shipment forecast intact despite signs of weakness in dealer sales in several markets (Reuters. 2013). In response to this, Harley Davidson has acknowledged the need to restructure the company into a leaner and more cost efficient beast if it is to maintain its market share and keep pace with the competition. Harley hopes to reduce the number of defaults on their loans, improve the company’s cash flows, and improve its liquidation strategy by having at least 12 months of projected liquidity in cash reserves. The company has set an ambitious goal to continue to widen its gross margin figures by nearly a full two percent in 2013 (Reuters. 2013). Harley Recognized that declining retail sales figures could no longer be simply shrugged off as weather related anomalies but were rather a generational decline of the number of loyal riders in the market. This suggests that younger rider’s between the ages of 18-34 are quickly becoming a significant portion of the market. Harley would need to restructure its operations in order to meet the demands of these new riders and develop both a strategy and a product that would appeal to them. Harley Davidson – Weakness (W) Product Recall Issues In late 2011, Harley Davidson recalled more than 300,000 motorcycles to fix a switch problem that presented a safety issue. The switch prevented the brake lights from coming on and could potentially cause the brakes themselves to fail as well (Associated Press. 2011). The defect has already caused at least one crash. As a result, the US Securities and Exchange Commission indicated that they expect that this recall will cost Harley anywhere from 10 to 12 Million Dollars. This coupled with another recall in 2012 for a faulty voltage regulator that affected an estimated 100,000 owners suggests that recalls are costing Harley a sizable chunk of their annual budget. Defects reasons have been tracked back to supplier quality issues arising in the manufacturing production chain. Dependence of Domestic Market Harley Davidson has recently celebrated its 110th year of business within the US. From its onset, it was clear that Harley’s target market was first and foremost the US markets. Much of Harley’s success has stemmed from targeting the baby boomer generation and appealing to big open spaces, the idea of freedom, and the feeling of exclusivity and belonging. Although business has steadily increased for Harley over the past century, it is clear that change is on the horizon. Harley’s competition is ever more aware of foreign markets and consistently devising entry strategies to further expand global market share (Burkey, 2009). Simply put, Harley has been lagging in this area. Harley’s flagship motorcycles carry a significantly higher price tag than foreign competing models making them appealing only to small elite group of riders that can afford the expense. Harley Davidson – Strengths (S) Brand Image Few people cannot instantly recognize the Harley Davidson brand. For over a century now, Harley Davidson has built a positive brand image by targeting a wide range of individuals. Harley Davidson has been continuously ranked among the top global brands in the world, holds over half of the heavyweight motorcycle market share in the US, and is ranked either first or second in the heavyweight motorcycle segment in at least nine countries across Europe (Harley Davidson, Inc. 2012). Historically, Harley Davidson has historically appealed to wide predominantly male audience ranging in ages from early 30s to late 50s. Recently, Harley Davidson has decided to further expand the brand by beginning successful marketing campaigns targeted at an audience of women. With the majority of its ads targeting a relatively specific group of individuals, Harley-Davidson has been able to build a community of enthusiasts around its brand that includes members from very diverse groups, and with almost no advertising. How does the king of heavyweight motorcycling keep its fans so loyal? It gives them a reason to belong (Rifkin. 1997). The symbolism of â€Å"belonging† is a powerful one reinforced by images of riding as part of a pack on the open road. This is further reinforced by a strong positive brand image that individuals frequently associate with superior quality and prestige of ownership. Broad Product and Service Portfolio One of Harley Davidson’s greatest strengths has been a long history of maintaining a broad product and service portfolio. Harley Davidson leverages its premium pricing model supported by a superior quality of its product line up ranging from an extensive line up of heavyweight, touring, custom, and performance motorcycles. Harley Davidson has also been able to successfully keep its owners engaged in personalizing and modifying their motorcycles by offering an extensive catalogue of parts and customization options. When you add the reliability of a two year warranty, and consider the superior level of service afforded to its owners, it is no wonder that Harley Davidson has been able to maintain the upper hand on its competition for so many years. The financial unit of the company has been successful in reducing the percentage of defaults and losses on its in house loans. While the annualized loss experienced on its managed retail motorcycle loans has come down from 1.58% to 1% in the last quarter, the retail 30+ day delinquencies on managed loans has come down from 3.68% to 2.56% (Trefis Team. 2012). Focused Research Harley Davidson continues to dominate market share in the United States commanding over 60% market share for the domestic market. Although Harley Davidson’s targeting of the youth market remains a subject of contention, their strategy remains crystal clear; to keep baby boomers in the saddle for as long as possible (Madson, 2013). To achieve this, Harley Davidson is exploring some concepts that may appeal to aging baby boomers such as a three wheeled Penster concept. The Trike concept has gained remarkable traction over the past few years and Harley Davidson has certainly taken notice. Harley Davidson Strengths and Opportunities (SO) Harley Davidson’s offensive strategy should focus on leveraging the company’s strengths such as its strong brand image and focused research and development to capitalize on opportunities such as further penetration into foreign markets. Harley Davidson is uniquely positioned with a globally recognized brand. This is a monumental advantage when comparing it to penetration strategies from relatively new motorcycle companies such as Polaris. Harley can utilize its strong focused research and development to study foreign market demands and develop a line of products to specifically appeal to that market. This would further help Harley in overcoming another one of its areas of opportunity around new product launches. Recently, Harley Davidson has shown significant progress in their expansion to the Chinese marketplace. Harley projects that within the next five years growing Chinese demand will support the opening of an additional 5 dealerships overseas (Miller. 2012). Growing global demand is proof that not only would this be a good business decision, but it is the necessary next step if Harley expects to keep up with its competition. Harley Davidson Strengths and Threats (ST) Harley Davidson faces some significant threats to its business in the form of changing emission standards for motorcycles, and an ever growing competitive landscape. Emission controls continue to tighten on a global scale making Harleys century long approach to air cooled engines all but obsolete (Huze. 2011). Furthermore, Harley faces some competitive threats from new entrants to the market like Polaris who are actively seeking to tap into Harley’s heavyweight motorcycle market share. Harley Davidson needs to develop a defensive strategy focusing on areas of strength such as focused research and development and broad product and service portfolio to ensure they overcome these threats. One solution would be a proactive approach to changing emission standards. Harley has been toying with the idea of introducing a lineup of liquid cooled motorcycles that would dramatically reduce their ecological footprint (Huze. 2011). Harley Davidson’s superior focused RD would have little issue with finding a way to adapt this new engine to existing models. Furthermore, the infrastructure for servicing these new engines, making adjustments, and maintaining them is already in place with is significant network of service centers around the globe. In doing so, they would not only be ready for any emission control changes that they may face in the future, but also be more competitive across product lines. Harley Davidson Weaknesses and Opportunities (WO) In recent years, Harley Davidson has literally spent millions of dollars dealing with significant recalls and associated issues (Associated Press. 2011). Most notably, two significant recalls that affected a total of more than 600,000 motorcycles over two years. Outside of the obvious cost associated with correcting the issue which is estimated to cost over 20MM over the course of 2009 and 2010, Harley Davidson is exposed to additional legal risk from possible accidents resulting from these defects (Associated Press. 2011). Harley has traced the source of the problem back to defects associated with its parts suppliers. This suggests the need for an improved quality control process prior to using the parts in production. Although this may present an additional cost initially, the reduction from costs associated with recalls off of Harley Davidson’s bottom line would be much more significant. Harley Davidson has already experienced success in its restructuring plans. The addition of inspection points could be seamlessly implemented with minimal impact to its existing business (Reuters. 2013). This would increase the chances of catching defects on parts prior to the parts being shipped to manufacturing. As a result, Harley would experience a reduction in the number of recalls in the coming years. Harley Davidson Weaknesses and Threats (WT) The combination of existing weaknesses to Harley Davidson’s business and the presence of potential threats in the market could potentially spell disaster. Harley should develop a strategy around minimizing their exposure to weakness and the avoidance of existing threats. Harley can achieve this by outsourcing the procurement of raw materials to a larger number of overseas suppliers while being supervised by in-house Harley Davidson quality control specialists to ensure a sustained level of quality of their products (Burkey. 2009). This would help to avoid one of Harley’s largest weaknesses, product recalls, as well as dramatically improve the availability of raw materials minimizing the impact that one single supplier could potentially have on their business (Taylor III. 2012). Furthermore, the fewer amount of recalls and the overall improved degree of quality would bolster the already strong Harley Davidson brand, improving its position in the competitive marketplace (Rifkin. 1997). Citations Associated Press. (Oct. 2011). Harley Recalls About 308,000 Motorcycles For Break Issue. USA Today / Money. Retrieved from: http://usatoday30.usatoday.com/money/industries/manufacturing/story/2011-10-2 4/harley-davidson-brakes-recall/50890560/1 on September 18, 2013. Burkey, Brent. (Oct 2009). Harley-Davidson Time for the tough decisions A plan to restructure core local Harley operations is in the hands of the company.York Daily Record. P4. Harley Davidson, Inc. (Aug 2012). Harley-Davidson, Inc. Financial and Strategic Analysis Review. Global Data. P1-3. Harley Davidson, Inc. (Jun 2012). SWOT Analysis. Company Report. P1-9. P9. Huze, Cyril. (Jun 2011). 2012 Harley-Davidson Liquid Cooled Engines. Cyril Huze Post. Retrieved from: http://cyrilhuzeblog.com/2011/06/17/2012-harley-davidson-liquid-cooled-engines/ on September 17, 2013. Madson, Bart. (Feb. 2013). H-D RD Product Development Center. Motorcycleusa.com. Retrieved from: http://www.motorcycle-usa.com/684/15599/Motorcycle-Article/H-D-R-D-Product-Development-Center.aspx on September 18, 2013. Miller, Paula M. (Jan-Mar 2012). Harley Davidson in China. China Business Review. P41-43. PR Newswire. (Jul 2013). Harley Davidson Post Second-Quarter 2013 Earnings, Revenue and Retail Sales Growth. Regional Business News. P.16. Reuters. (2013). Harley Earnings On Target As Restructuring Pays Off. Chicago Business Tribune. Retrieved from: http://articles.chicagotribune.com/2013-04-25/business/chi-harley-earnings-20130425_1_harley-davidson-inc-winnebago-industries-sales-volumes on September 17, 2013. Rifkin, Glenn. (Oct. 1997). How Harley Davidson Revs Its Brand. Strategy + Business. Retrieved from: http://www.strategy-business.com/article/12878?gko=ffaa3 on September 17, 2013. Taylor III, Alex. (Oct. 2012). The Hurdles At Harley Davidson. Motorworld. Retrieved from: http://money.cnn.com/2012/10/03/autos/harley-davidson-polaris.fortune/index.html on September 17, 2013. Trefis Team. (May. 2012). Harley Davidson Rides To $59 As Growth Hits On All Cylinders. Forbes Magazine. Retrieved from: http://www.forbes.com/sites/greatspeculations/2012/05/03/harley-davidson-rides-to-59-as-growth-hits-on-all-cylinders/ on September 17, 2013. UPI.com.

Tourism in Malaysia Essay Example for Free

Tourism in Malaysia Essay Tourism, including Malaysian tourism, is a big industry worldwide and many countries have already cashed in on its potential. This article seeks to assess the performance of tourism in Malaysia, its development, impacts and future. Implications for students will also be discussed. Tourism success in any country begins from the priority placed on it by the government. The Malaysian government through its Ministry of Tourism Malaysia (hereafter referred to as Tourism Malaysia) plans develops and oversees tourism policies, projects and other activities to realize its vision of making Malaysia an international tourism destination. The activities of this agency are discussed in more details. Overview of Malaysian Tourism International tourist arrivals and international tourism receipts are the popular indicators used in measuringtourism performance. According to the 2010 United Nations World Tourism Organization (UNWTO) Report, Malaysia ranked 9th in international tourist arrivals, welcoming 24. 6 million visitors through its doors. This good performance represents a 3. 9% increase over 2009 performance. Malaysia achieved a third place in the Asian region after China and Turkey respectively in the world’s top ten country lists of international tourist arrivals. This performance was realized by the governments’ tourism training, incentives and promotional programs. Government incentives include tourism infrastructure fund and special tourism fund to support tourism-related development and projects. Promotional programs include the popular ‘Malaysia my second home’, student tourism programs for secondary schools and education tourism among others. Provision of tourism services according to MS ISO 9001:2000 by the Ministry is an indication of the ministry’s international quality benchmark. The impact of these efforts resulted in international tourism receipt of RM 56. 5 billion in 2010. Therefore, in order to complement this achievement, the government is stepping up efforts in its plan to boost the tourism industry. This was reflected in the 2011 budget where more tourism destinations in various parts of the country will be developed (some are already under construction as of writing), ban of import duties on certain tourism-related products, education promotions and part-funding of tourism related projects with the private sector. Highlights of Tourim sites in Malaysia The developmental efforts of the governments’ tourism agency resulting in the recognition of threeattractions as world heritage sites by UNESCO are truly commendable. Theseattractions are: 1. Gunung Mulu National Park (Sarawak) 2. Kinabalu Park (Sabah) 3. Melaka and Georgetown, historic sites of the Straits of Malacca Potential sites already submitted for inclusion by UNESCO includes: 1. Lanjak Entimau Wildlife Sanctuary (LEWS) and Ai national Park (BANP)   2.  Prehistoric Archeological Heritage of Lenggong Valley 3. The Taman Negara National Park of Peninsular Malaysia Consequently, increased tourism benefits other sectors such as increasing hotel development, training of tourism related courses in tertiary institutions, manufacturing, transportation and the aviation industry only to mention a few. Future of Malaysian tourism and implications for students As Malaysia inches closer to its vision of becoming a developed country by 2020, the race is on to ensure its tourism industry is not left behind. Continued liberalization of some government controlled sectors, increased incentives, development of more tourism destinations where possible, increased involvement of the private sector are some of the key strides that will continue to improve Malaysia tourism performance into the future. This has a huge implication for students especially students of Hospitality and Tourism management. As populations increase in Malaysia and in the world and the Malaysian government makes its tourism industry more attractive, more skilled personnel will be required to manage the increasing government functions and private businesses resulting from it. One key recommendation for the Malaysian government will be to factor in the projected skilled workforce and skill sets required to realize its future tourism ambitions while making plans for those skills and workforce today. Students interested in the tourism industry will focus on developing additional skills sets more than a passing grade that will provide the distinction to ensure a place in the ever growing Malaysian tourism industry. Conclusion This article investigated Malaysian tourism in terms of its performance, impact, development, future and implications for students.  It was found that Malaysia ranks in the world’s top ten countries in international tourist arrivals, achieved recognition for three UNESCO world heritage sites, and has a coordinated promotion, incentive and program to propel its tourism industry into the future. The implication for students especially student of the Hospitality and tourism industry werediscussed. Finally, it was recommended that a holistic tourism plan must include the projected skill set and workforce required to manage the increasing tourism projects of the future.

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

Anorexia and Bulimia Essay -- Causes of Bulimia, Eating Disorders

Each year millions of people in the United States are affected by serious and sometimes life-threatening eating disorders. The vast majorities are adolescents and young adult women. Approximately one percent of adolescent girls develops anorexia nervosa, a dangerous condition in which they can literally starve themselves to death. Another two to three percent develop bulimia nervosa, a destructive pattern of excessive overeating followed by vomiting or other " purging " behaviors to control their weight. These eating disorders also occur in men and older women, but much less frequently. The consequences of eating disorders can be severe. For example, one in ten anorexia nervosa leads to death from starvation, cardiac arrest, or suicide. Fortunately, increasing awareness of the dangers of eating disorders, sparked by medical studies and extensive media coverage, has led many people to seek help. Nevertheless, some people with eating disorders refuse to admit that they have a problem and do not get treatment. Family and friends can help recognize the problem and encourage the person to seek treatment. Anorexia nervosa is a disorder where people intentionally starve themselves. It usually starts around the time of puberty and involves extreme weight loss. Sometimes they must be hospitalized to prevent starvation because food and weight become obsessions. For some, the compulsiveness shows up in strange eating rituals, some even collect recipes and prepare gourmet feasts for family and friends. Loss of monthly menstrual periods is typical in women with this disorder and men with this disorder usually become impotent. People with bulimia nervosa consume large amounts of food and then rid their bodies of the excess calories by vomiting, abusing laxatives or exercising obsessively. Some use a combination of all these forms of purging. Many individuals with bulimia " binge and purge " in secret and maintain normal or above normal body weight, they can often successfully hide their problem from others for years. As with anorexia, bulimia typically begins during adolescence. The condition occurs most often in women but is also found in men. Many individuals with bulimia, do not seek help until they reach their thirties or forties. By then, their eating behavior is deeply ingrained and more difficult to change. Medical complications can frequentl... ... again. Family members and friends can call local hospitals or university medical centers to find out about eating disorder clinics and clinicians experienced in treating the illnesses, for the college students, treatment programs may be available in school counseling centers. Family and friends should read as mush as possible about eating disorders, so they can help the person with the illness understand his or her problem. Many local mental health organizations and the self help groups provide free literature on eating disorders. Some of these groups also provide treatment program referrals and information on local self-help groups. Once the person gets help, he or she will continue to needs lots of understanding and encouragement to stay in treatment. NIMH continues its search for new and better treatments for eating disorders. Congress has designated the 1990's as the " Decade of the Brain, " making the prevention, diagnosis, and treatment of all brain and mental disorders a national research priority. This research promises to yield even more hope for patients and their families by providing a greater understanding of the causes and complexities of eating disorders.

Hamlet: Act 2 Scene 2 :: Shakespeare Hamlet

Hamlet: Act 2 Scene 2 - Compare Hamlet's Reaction to Arrival of Rosencrantz and Guildenstern and To the Players Compare Hamlet's reaction to the arrival of Rosencrantz and Guildenstern with his reaction to the arrival of the Playyers. Account for his reactions.      Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   By comparing Hamlet's reaction to the arrival of   Rosencrantz and Guildenstern  Ã‚   with his reaction to the arrival of the Players, we can observe the different perspectives of Hamlet's character. His reaction to the arrival of his old friends is similar to his reaction to the arrival of the Players in as he is happy to see them all and he reveals his sanity to them all. When Rosencrantz and Guildenstern arrive, Hamlet is overjoyed to see his   "excellent good friends" (2.2.227) with whom he grew up. Hamlet is also delighted to meet with the Players. But this is where all the similarities end. In his conversations with Rosencrantz and Guildenstern, Hamlet becomes suspicious of the purpose of his old friends' visit and he is perceptive enough to see through the outer disguise into the interior motives. He forces them to reveal that they have been sent by the King to find out what is causing Hamlet's "transformation" ( 2.2.5). Hamlet admits his sanity by telling his good friends that his "uncle- father and aunt mother are deceived." (2.2.348)   Why does Hamlet admit his sanity to Rosencrantz and Guildenstern? Is it an another attempt to at deception, or is it to implant a sense of trust between Hamlet and his old friends? Hamlet could possibly be furthering his plans for revenge by admitting his sanity. Hamlet's friends would relay the message to the King and Claudius may think that Hamlet really is mad for admitting that he was supposedly feigning madness. On the other hand, at the conversation with the Players, his behavior is spontaneous. He welcomes his "good friends" (2.2.431) and it seems that he is "glad" (2.2.430) to see them again. He is friendly, funny and very open in his communication with the Players. Hamlet notices that one of the young players who plays female roles has grown a beard and he makes a joke about it. It also indicates that Hamlet has known them for a long time. The young Prince Hamlet also demonstrates his sanity when meeting with the Players. Hamlet shows that he is still capable of reasonable thought when he recites the lines to a play that he has heard only once. Hamlet reestablishes his friendship with the Players and shows his sanity to the Players so that he can further his plan for revenge.

Civic Virtue: The Right Thing for Our Society Essay -- society

Introduction Perhaps, the American society is the most divergent, the most accommodating and the most culturally diverse among all societies across the globe. Interestingly, most Americans reflect similar elements of behavior in many respects which are distinct to our American society. There are a number conscious and unconscious core values which are expected to guide every American character. Most of the American culture has to some extent embedded western civilization: A civilization that accommodates different cultures, merges multiple ideas, and values the freedom of choice. Still, a number of challenges in the direction of promoting our general wellbeing have been arising; thus, leading to questions on the direction that our society should direct for our common wellbeing. Here, I will be discussing approaches that can be designed in addressing the issues of wealth distribution, and the relationship between politics and community beliefs so as to have an even happier society. Inequality, Cohesiveness and Civic Virtue One among the challenges that face our society today is the widening rift between the rich and the poor. Although our politics has evaded the reality of a widening society, several philosophers have given their opinions on the challenge of wealth distribution. Our politics has become so distant from the challenge of widening social gaps that president Obama’s proposal to review tax laws in the direction of burdening the wealthy with more taxes has received heavy criticism from the republican political quarter. Apart from philosophical ideologies on the topic of wealth distribution, there is a more important challenge which requires the attention of our political leaders: Civic Virtue. Getting a picture of ... ...place in our schools and other institutions. Instead of focusing on the teaching of various religious and moral beliefs, students should be allowed to openly present their opinions on various religious matters. Such a direction must be done under an environment that respects the right of people to choose various religious and moral beliefs which are available in our society. Conclusion As I have discussed, I agree with Sandel’s opinions on the two issues (the distribution of wealth, and an open engagement between religion and politics) that I have discussed above. Wealth distribution is especially useful in guarding our democracy, promoting, cohesiveness, and improving the welfare of the poor. On the other hand, an engagement between politics and religion will be useful in promoting unity, tolerance and knowledge within our society.

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Section 8 SEKSYON 8. Hindi dapat hadlangan ang karapatan ng mga taong-bayan kabilang ang mga naglilingkod sa publiko at pribadong sektor na magtatag ng mga asosasyon, mga unyon, o mga kapisanan sa mga layuning hindi lalabag sa batas. â€Å"The right of the people, including those employed in public and private sectors, to form unions, associations, or societies for purposes not contrary to law shall not be abridged. † Freedom to form associations In large part, this section reflects the country’s bad experience during the Martial Law years, when the right to assemble and form associations was unduly abridged.Obviously, however, it is equally clear that the government can exercise its police power and abridge this right if the association in question threatens the legal order. Section 10 Section 10. No law impairing the obligation of contracts shall be passed. SEKSYON 10. Hindi dapat magpatibay ng batas na sisira sa pananagutan ng mga kontrata. Discusses the â€Å"sanct ity† of contracts and obligations Laws affecting contracts cannot be applied retroactively Aside: all contracts illegal in nature are non-bindingSection 4 â€Å"No law shall be passed abridging the freedom of speech, of expression, or of the press, or the right of the people peaceably to assemble and petition the government for redress of grievances. † 1. ) Freedom of speech is not absolute, neither is a free press (more on that on the next slide) 2. ) Freedom of assembly refers mainly to peaceful demonstrations related to public affairs – Contrast: in Singapore, for large assemblies one must secure a public entertainment license 3. Right to petition i. e. to take up one’s grievances with government without fear of persecution Freedom of Speech – means an individual is free to speak or utter whatever he wants without prior restraint. Right to a Free Press – means an individual is free to write, publish, and circulate whatever he pleases witho ut restraint. Speech and expression refer to any form of oral utterances, while press covers every sort of publication such as newspapers, magazines, books, leaflets, and the like.Radio and television are also included. Freedom of speech and expression and freedom of the press are collectively called Freedom of Expression. Freedom of Assembly – refers mainly to peaceful demonstrations related to public affairs. The Right of Petition – to take up one’s grievances with government without fear of persecution. Section 11. Free access to the courts and quasi-judicial bodies and adequate legal assistance shall not be denied to any person by reason of poverty. SEKSYON 11.Hindi dapat ipagkait sa sino mang tao ang malayang pagdulog sa mga hukuman at sa mga kalupunang mala-panghukuman at sa sat na tulong pambatas nang dahil sa karalitaan. states that paupers or person who are poor shall be given free access to courts and quasi-judicial bodies as well as free adequate lega l assistance (or free counsel to defend him in court) Section 20. No person shall be imprisoned for debt or non-payment of a poll tax. SEKSYON 20. Hindi dapat ibilanggo ang isang tao nang dahil sa pagkakautang o hindi pagbabayad ng sedula.