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

A Study of Warfare in Europe Between 1300 and 1500 Essay

Since the introduction of gunpowder into Europe, it has gone on to dominate warfare into the twentieth century. With the development of the first European guns in the fourteenth century, armies were given use of a weapon which was to radically alter most of the ways of making war which had been established during the Middle Ages, and changes began to be seen within only a few years. It is, however, questionable whether the nature of these early changes constituted a revolution in the methods of war, and even more so whether guns had by 1500 made a great deal of impact on the character of war as it had existed in 1300. In assessing whether a revolution had taken place (or at least whether one was in the process of happening) by 1500, it is necessary to examine three areas: the effectiveness of guns during the period; the extent of their use in conflicts; and finally the changes which resulted from the employment of the new weapons in war. The first reliable sources which assert the existence of guns appeared in the 1320s, and from the late 1330s the number of references to them rose dramatically. The early guns were of large calibre and used almost exclusively for sieges, although as early as Crà ©cy in 1346, the English â€Å"fired off some cannons which they had brought to the battle to frighten the Genoese.† Guns were made in one of two ways. Firstly, there were cast metal guns, usually of bronze, which were made at the foundry. These were usually the better weapons because they were made of a single piece of metal and therefore were less likely to burst apart on firing. The second method was arranging wrought iron strips into tubes which were then bound together with iron hoops in much the same way as barrels were made. The advantage of these guns was that iron was a much cheaper metal than bronze (but could not be cast), but being made of many pieces faults were more likely to develop, causing either the release of explosive pressure through the sides of the barrel and therefore a reduction in the power of the shot, or even the complete bursting of the gun. This structural weakness was compounded by the inclusion of a detachable breech (such guns were far easier to build) which often detached itself on the discharging of the weapon. Despite their lack of quality, however, iron guns were the more numerous due to their comparative cheapness, although smaller guns tended to be made chiefly from bronze both because of the difficulties of constructing small guns from iron  strips, and because they required less metal than the great bombards. The sizes and types multiplied from their modest beginnings until there were guns ranging in size and type from great bombards used for reducing entire cities, to handguns used as anti-personnel weapons. The early role played by artillery was in sieges, where its effectiveness was soon widely appreciated. â€Å"Broadly speaking, the use of guns meant that sieges could be brought to a conclusion much more quickly.† Cannons, with the ability to throw stones with great force over a flatter trajectory and with more accuracy than the old siege engines, could bring an end to a siege in a few weeks where previously it might have taken several months. In 1437, the castle of Castelnau-de-Cernà ¨s was â€Å"broken down during the said siege by cannon and engines, and a great part of the walls of the same thrown to the ground, so that it was in no way defensible against the king’s enemies.† On occasion, the mere presence of bombards could be enough to induce swift surrender, cities preferring capitulation to large-scale destruction by cannon fire. As artillery was added to armies in increasing amounts during the 1400s, wars became far more fast-moving affairs. The French employed artillery on a large scale in reducing the English fortresses in Normandy and Gascony, while huge, cumbersome bombards were used to good effect by the Spanish in the Reconquista. Such was the effectiveness of artillery by 1500 that Machiavelli could declare that â€Å"No wall exists, however thick, that artillery cannot destroy in a few days.† The successes which the early guns had in siege warfare led to the bombard being a vital part of any country’s armoury. The use of the counterweight trà ©buchet, which had been in existence in Roman times, failed to decline until the 1380s and was still listed as an active weapon in some French arms inventories until the 1460s. This shows the gradual nature of the introduction of gunpowder artillery (perhaps caused by shortages of materials for the cannons or unwillingness to invest in them when a prince already owned trà ©buchets), but there was little doubt that guns were becoming by far the better siege weapons. Their importance gave rise to a kind of ‘arms race’ in France in particular, with cities in the contested areas of the Hundred Years War assigning the acquisition of guns a high  priority. Charles the Bold’s army included a substantial artillery element in his war with the Swiss (although he was perhaps not a good enough general to make effective use of it), while in 1494, Charles VIII invaded Italy with â€Å"†¦an army of 18,000 men and a horse-drawn siege-train of at least forty guns. Even contemporaries realised that this marked a new departure in warfare: in 1498 the Venetian Senate declared that ‘the wars of the present time are influenced more by the force of bombards and artillery than by men at arms’.† Despite their undoubted worth, however, artillery did have a number of disadvantages at this time. Perhaps the main one of these is the chronic difficulty of moving the heavy guns, especially over land. Philip the Good of Burgundy experienced such problems in his war with Ghent in 1452-3: â€Å"Such was the weight of a great bombard which he borrowed from the town of Mons that all the bridges between Mons and Lille had to be strengthened with iron supports for its passage. During the journey, the gun fell into a ditch, and took two days to be extricated by men using lifting equipment specially constructed for the purpose.† Transport was easier by river, but clearly this limited the movement of the artillery. It is partly because of these transport problems that artillery (with the exception of handguns) was used little during open field battles. Particularly when manoeuvring was of critical importance to an army, the last thing a commander would want would be to have to wait for the artillery, which would be slower than the rest of his force, and be unlikely to be able to move away from roads. Due to a lack of enthusiasm for such a cumbersome battlefield weapon, field artillery developed little in the early days of gunpowder, and the large cannons which were used on battlefields tended to be immobile siege guns which had been hastily adapted. Not only did the lack of mobility of cannons cause problems for armies on the march, but it also restricted their usefulness on the battlefield itself. The absence of effective gun carriages meant that artillery tended to be fixed rather than able to be aimed, the guns being mounted on wooden frames or simply positioned on mounds of earth. Their slow rate of fire (not only because of the time taken to load them, but because it took time for the  guns to cool down between shots) and their limited range at this time was another weakness, which led to them being easily overrun. Soldiers could wait at the limit of the guns’ range until the first salvo had been fired, then charge, reaching them well before the next shots could be fired, and disable the guns. The weaknesses of artillery on the battlefield were such that, â€Å"Even during the second half of the fifteenth century cannons were only used occasionally in pitched battles.† Handguns were of more use on the battlefield, having none of the transport problems of the heavy artillery and having a great deal in common with the crossbow, a weapon of proven worth. Like the crossbow, the handgun was a specialised anti-personnel weapon, and was ideal for firing at large exposed masses of soldiers where it could inflict considerable damage. The advantages which handguns had over crossbows was their superior hitting power, (of which Pope Pius II remarked, â€Å"No armour can withstand the blow of this torment, and even oaks are penetrated by it,†) and their relative cheapness due to the simplicity of their construction. As their accuracy improved and the numbers of trained marksmen increased, they came to supersede the crossbows in European armies, but by 1500 this process was by no means complete, the two weapons frequently working side-by-side in battles. While the slow rate of fire of handguns meant they could not stand independently in battle and needed the support of troops armed with close-combat weapons, they became an accepted auxiliary weapon in many armies. Despite the increased use of gunpowder weapons in battle, they were by no means always successful. Superiority in artillery was no guarantee of victory, as Charles the Bold discovered at Grandson and Morat in 1476. At Agincourt, the French gunners were pushed to one side by their own men at arms and played no significant role in the course of the battle. There are, however, examples of the successes of guns in battle, hinting at the success they were to achieve in the future. The Battle of Castillon in 1453 showed the devastating effects of crossfire: â€Å"Talbot imprudently attacked the [French] camp which led to the intervention of the French battery commanded by Giraud de Samian, a highly respected cannoneer. ‘He grievously injured  them because with each shot five or six fell dead to the ground’.† With the increased use of guns, deaths and injuries caused by them came to be recorded in greater number: â€Å"In 1442, John Payntour, an English esquire, was killed by a culverin shot at La Rà ©ole. Four years previously, Don Pedro, brother of the king of Castile, had been decapitated by a gunshot during the siege of the castle of Capuana at Naples†¦In April 1422, one Michael Bouyer, esquire, was languishing in prison at Meaux, ‘gravely ill and mutilated in one of his legs by a cannon shot, in such a way that he cannot aid himself’†¦It was becoming obvious that the gun could not only batter down fortifications, but could kill, and kill selectively from afar.† It is clear that by 1500, guns had come to be an everyday part of European armies. While the use of firearms on the battlefield tended to be limited to handguns, these were gradually replacing the older bows as the main auxiliary shot weapon. Cannons had made a huge impact on the conduct of siege warfare, bringing sieges to an end comparatively swiftly, and becoming indispensable in great armies. Although there were bound to be initial troubles with what was after all a relatively new weapon, notable successes were being recorded, especially in sieges, and the gun was definitely here to stay. To constitute a revolution, though, the growing use of such weapons would have to have changed not only the methods of making war, but also the outcome and the character of conflicts. What, then, were the consequences of the increased use of gunpowder? One of the largest series of changes happened in the area where the new guns were at their most effective, that of siege warfare. The advantages which a defending army could gain by hiding within fortifications had been understood for a long time. During the ‘High Middle Ages’, the war zones of Western Europe had become studded with castles and forts, and wars came to be characterised not by swift manoeuvre and open field battles, but more through long, drawn-out sieges. The failure of an attacking army to take a castle was likely to cause it a great deal of problems. If bypassed by an army, a defending garrison could retake control of the surrounding countryside from its secure central base and conduct raids on the enemy’s army and supplies (especially as fortifications tended to be located at  communications centres). Failure to take a large number of castles could result in their garrisons uniting to form an army capable of defeating the enemy force in the open field. In short, territory could not be conquered without gaining control of the fortifications within it. The effect of the introduction of effective siege guns with the capability of breaching the walls of castles was to bring the advantage in siege warfare away from defence and more towards the attacking force. With guns able to bring about the capitulation of fortresses within a few weeks or even a few days, there was a diminishing prospect of the defending country being able to organise an army in time to relieve the besieged. It would seem that the introduction of cannons had, for a time at least, called into question the efficacy of defence by small, dispersed garrisons defending fortifications. Had the use of siege guns not produced a defensive reaction, it seems possible that the castle could have been made redundant and defensive armies driven to do battle in the open field on equal terms with their adversaries. Attacking innovation, however, did produce defensive reaction, which in turn provoked counter-reaction from besiegers, and this greatly altered the nature of siege warfare during the fourteenth and fifteenth centuries. Guns, of course, were not exclusively weapons of attack. Defensive firearms were an early experiment which had some effect in maintaining some sort of defensability of fortifications in two ways. Firstly, by firing at the besieging army from the castle walls, defensive marksmen and cannoneers could take advantage of the short range of early guns by making it very hard to position bombards close enough to the walls to cause damage. By refusing to give the besieging army the freedom to position guns wherever they liked, the defenders could in this way keep the enemy at ‘arms length’. The second way in which guns could be used to defend fortifications was not to defend the walls from destruction by cannon fire, but to provide crossfire against enemy troops when the time came for them to attempt to storm the castle, a function which crossbows were able to perform, but were of inferior effectiveness and more expensive than handguns. This use of defensive firearms caused changes in the way in which attackers  approached sieges. Guns being fired into the besiegers’ camp necessitated the greater use of cover, in particular for the bombards which were placed nearest to the castle walls for much of the siege. To this end, trenches were dug and wooden shields or hoardings were constructed to protect the soldiers and their guns. Trenches provided some degree of effective protection against most weapons, while the hoardings gave protection to the guns, which usually had to be positioned in more exposed locations in order that they could target the walls before them, against everything but powerful firearms. â€Å"Jean de Bueil could advocate the siting of a besieger’s camp before a beleaguered fortress on the model of the fortified entrenchments dug at Maulà ©on, Guissen, Cherbourg, Dax, and Castillon fifteen years before. Trenches, he wrote, were to be dug from one part of the siege to another, covered by hoardings. Ease of contact and movement between the units of the encircling forces could thus be ensured.† Further success could be gained by the besieging forces by employing not only large bombards to break down the walls, but also smaller guns to pick off individual defending troops. This would not only prevent defensive gunners from having the luxury of completely free shots at their enemies, but would limit the effectiveness of attempts to repair breaches in the walls. The use of guns in both attack and defence produced perhaps the biggest changes in siege warfare in the form of changes to the fortifications themselves. The castles which existed before the widespread use of cannons were ill-suited to withstand the accurate and powerful impact of cannonballs. Trebuchà ©ts had been of more use in lobbing stones over the walls to cause damage inside castles than in actually causing breaches, and so the walls were built tall and flat so as to be better able to resist being scaled by soldiers. Such walls provided a large target for cannon fire, and their flatness meant that the full force of the shot was directed straight into them. Rather than rebuild entire castles, lords were often forced through reasons of cost to opt for the next-best solution of adaptation. Scarping the walls with banks of earth or masonry was tried in an attempt both to thicken the walls, and to turn the blows of cannons into more glancing shots, with the added bonus that sloping walls meant that siege ladders became ineffective. While it was a sound theory to avoid  square-on impacts from cannonballs, scarping was of limited value, and where this was an adaptation to an older castle, in many cases it weakened the walls by placing extra weight on them. Gunports were a further adaptation to fortifications which occurred as a result of the introduction of guns. These were holes put in the walls of the castles, often where arrow slits had been, to allow small guns to be fired from a position of relative safety. They were frequently positioned in the towers or gatehouses of castles to provide flanking fire along the walls where it was anticipated any attacking soldiers would have to stop before being able to push on into the castle. This modification was quite cheap and easy to put in place, and was used across Europe. One method of improving the efficacy of defensive fire both against attacking troops and forces sitting back and besieging was the greater use of defensive outworks built of earth or masonry. Not only could this ‘forward defensive’ strategy force enemy guns further back from the castle proper, but it also provided a further opportunity to enfilade soldiers as they advanced. Boulevards or artillery towers could be built in ditches forward of the walls with a clear line of fire along the trench. As the enemy soldiers advanced, they would have to spend time negotiating the ditch during which the fire from the outworks could take its toll. â€Å"At Dax, Guissen and Cadillac, in 1449 and 1451, the French encountered heavy resistance from such outer works constructed by the defenders.† The ultimate defence against besiegers armed with guns, though, was the fortification based on the angled bastion. It was this defence which was coming increasingly into use by 1500 which decisively swung the balance of power back in favour of the defenders. The bastion was essentially a gun platform for siting heavy guns which needed the freedom to be turned and fired against the enemy camp wherever it was in relation to the castle, a freedom which could not be obtained firing through gunports. These towers were thrust forward of the walls to keep the enemy back and were built the same height as the rest of the castle (unlike traditional towers), perhaps to facilitate the movement of guns around the walls, or perhaps because of the high cost of taller towers. The entire structure was squat, making it a smaller target and allowing the guns at the top of the walls to maintain fire at targets close to the foot of the fortification, and scarped to produce more glancing blows from cannonballs. This last objective was also achieved by projecting the bastions at a different angle to the rest of the wall, so that in effect only the wall would receive square-on blows. It could be said that round castles and round towers would present no flat surfaces to be hit squarely, but to build such fortifications would make flanking fire along the walls at best difficult. Round bastions were built, but left dead ground where guns could not reach, while entire castles built on a circular model would need many projecting towers to provide fire along the walls. With the angled bastion, fire could be given along the entire base of the tower from guns positioned in the walls, while fire along the walls could be provided from gunports in alcoves in the bastion. It seemed for a while that the destructive power of cannons would lead to a decisive shift towards the attackers in siege warfare which would perhaps bring an end to the dominant role of fortifications in warfare. However, defensive tactics adapted to the situation in a number of ways, ensuring the survival of the castle and the siege. It can be said, though, that although the nature of warfare overall was not changed, the nature of sieges changed significantly as a result of the use by both attackers and defenders of gunpowder weapons, and because a new type of castle had been born. If guns provided a temporary revolution in the balance of sieges, then the bastion was equally as revolutionary in restoring the old balance. â€Å"By resisting the new artillery and providing platforms for heavy guns [the bastion] revolutionised the defensive-offensive pattern of warfare.† While cannons produced many changes in the conduct of sieges, changes of similar magnitude cannot be seen in open field warfare. Cannons, lacking effective carriages to allow them to keep pace with their armies and to manoeuvre on the battlefield, were little used until the late fifteenth century. Handguns, despite coming to be as accepted a weapon as the crossbow, failed to produce any noteworthy changes. Possessing greater hitting power than the crossbow, but similar weaknesses, including slow rate  of fire, they were unable to establish themselves as anything more than an auxiliary weapon. While the Swiss were to use handguns in their successful pike square formations, their role could be (and often was) performed equally well by crossbowmen, and European armies continued to be based on knightly cavalry and close-combat infantry. The handgun of the fifteenth century was simply another auxiliary shot weapon, and, â€Å"The arquebus, or match-lock musket, did not finally oust the crossbow from French armies until 1567.† Nonetheless, the importance of guns increased during the fourteenth and fifteenth centuries until they became an essential part of major wars. One of the results of this was to make war a much more large-scale thing in terms of money, and to put serious warfare (involving conquest and therefore sieges) out of reach of the pockets of anybody but princes. Artillery was very expensive. â€Å"It was one thing, in accordance with ancient ways, to expect a man at arms to come to the host equipped with his own horses and armour, but no one, in the new conditions of war, expected a master of artillery to provide his own cannon.† On a national level, the introduction of guns further widened the gap in military potential between rich and poor countries, underlining the superiority of countries like France over countries like the Italian states. It can be asked to what extent gunpowder weapons revolutionised notions of chivalry and whether the attitudes of people altered as a result of their experiences of the new guns. There is a good deal of late mediaeval literature which shows that many people despised them. They were an indiscriminate weapon which had no respect for social status, meaning that princes could now be killed from afar by peasants and artisans. This went against the traditional chivalric notion of individual combat among equals. Guns were also seen as cowardly, because of the belief that the gunner, hiding behind the smoke from his gun, did not put himself in mortal danger by firing, yet could still take the lives of others. Many saw guns as being instruments of the devil, with the noise and fire created being seen as having come from Hell itself. A popular attitude during the early days of guns in Europe is shown by Cervantes when he wrote, blessed be those happy  ages that were strangers to the dreadful fury of these devilish instruments of artillery, whose inventor I am satisfied is now in Hell, receiving the reward of his cursed invention, which is cause that very often a cowardly base hand takes away the life of the bravest gentleman; and that in the midst of that vigour and resolution which animates and inflames the bold, a chance bullet (shot perhaps by one who fled, and was frightened by the very flash the mischievous piece gave, when it went off) coming nobody knows how, or from where, in a moment puts a period to the brave designs and the life of one that deserved to have survived many years†¦ It is unlikely, however, that this attitude was held by the majority of people at the time. Shot weapons were nothing new, and had been in existence and used on a large scale for many years. There was little difference between a knight being killed by an arquebus or by a longbow. The large-scale use of guns by most European armies demonstrates that while there might have been some degree of chauvinism against firearms, princes were still quite prepared to use them, and indeed the church positively encouraged their use at a time when the Turkish threat to Christendom was increasing. There is little evidence that captured gunners were treated any differently to any other captured commoners (and by 1500 it was by no means guaranteed that gunners would not in fact be noble), and overall, society had little difficulty in accepting the place of artillery in modern warfare. Guns were ‘domesticated’ and given names, taking on characters of their own, and, â€Å"By the end of the fifteenth century and the beginning of the early modern era, gunpowder weaponry had simply become a feature of everyday life. Guns had become so conventional that they began to be used in celebrations, in fashion, and in crime. Ultimately, guns even became virility symbols.† This growing acceptance represents in part a change in attitudes brought about by the realisation that guns were of considerable use, but mainly it is a result of the rather superficial nature of chivalry at that time. There was a tendency for people only to behave according to the rules of chivalry when either it suited them to or when they could afford to. Princes, when faced with an adversary armed with artillery could not afford to confine themselves to criticising such ‘bad sportsmanship’ but had to respond in kind, an option which they were more than willing to take. In assessing whether gunpowder’s introduction caused revolutionary changes in Europe before 1500, it is necessary not only to examine the specific changes which arose, but moreover to assess whether warfare in 1300 had significantly changed in character by 1500 as a result of the use of guns. The answer to this has to be a definite no. The armies of 1500 made extensive use of guns, but these had not revolutionised the makeup of armed forces. The dominance of cavalry had persisted throughout the two centuries, and its only serious challenge had come in the late fifteenth century with the pike square, which by no means relied on guns. While the use of cannons had transformed the methods used in conducting sieges, only temporarily had there been the prospect of altering the nature of war away from the innumerable fortress battles which characterised the period. Gunpowder weapons had failed to bring an end to the siege as an important aspect of war, and could only act as a supplem entary weapon on the battlefield. Overall, despite the numerous changes which the increasing use of guns had caused, it is possible to agree with J. R. Hale’s assertion: â€Å"Gunpowder, in short, revolutionised the conduct but not the outcome of wars.†

Modern architecture and traditional architecture

Modern architecture and traditional architecture Nowadays, as we known the architectural community has had a strong and continuing interest in traditional and modern architecture. Architecture, this word possesses an immense creativity in itself. Usually, when we hear this word, picture of creative design of physical structures flashes in our mind. Integral to the identity of any country is its architectural heritage, combining modern and traditional architectural designs or product of the blend between splendid modern and traditional architecture.Based on what have found, architecture has been Rosen down into many categories to fit the lifestyle of people in a particular place at a particular time. There are basically two types of architect which are modern architecture and traditional architecture. According to architect Eric Spry, the word â€Å"modern† provokes such strong reactions in the world of residential architecture. Some people might imagine wonderful homes of stee l and glass with open, flowing floor plans; others might imagine sterile homes that feel like museums, complete with men in red suits watching carefully that nothing is touched.Strong pinions abound about modern architecture, as they do regarding the wide variety of other architectural styles. Five hundred years ago, Native Americans was built with adobe and Europeans built with stone. Homes had thick walls, small and deep- set windows, and small interior rooms. Technologies such as steel later allowed large expanses of space and large expanses of glass. In our lifestyles today are considerably different than the lifestyles of 50 years ago, let alone the lifestyles from 100 or 200 years ago. Architecture must represent the way we live today, not the way we lived hundreds of years ago.Remember parlors? Not many would. These were sitting rooms common a hundred years ago where guests were greeted. Our lifestyle changed, and parlors were weeded out. (Discover Modern Architecture's Appea l . Eric Spry). What is a modern architecture mean? Modern architecture is known as the movement of architecture that began in the 20th century, it is also architecture that is characterized by the simplification of forms and subtraction of ornaments, modern architecture can be some of the most futuristic, colorful, innovative designs ever. Traditional vs. Modern Architecture' (Ranches . 011). Modern architecture these days there are so many materials that architects can use to create different effects on buildings. In history, Modern architecture developed during the early 20th century but gained popularity only after the Second World War. For decades, modernism became the dominant structure for institutions and corporate buildings even up to the recent period. Architectures of this type exhibit functionalism and rationalism in its structure. (What is the difference between post-modern and modern architecture?. 000). Characteristics of modern architecture include he functional requ irements of the structure, lesser ornaments used and eliminations of dispensable details, and the application of the concept of â€Å"form follows function†. ‘Comparative investigation of traditional and modern architecture' (A. S. Delia, M. A. Ensnare, T. Zachary Beverages . 2011). Generally, modern design is simple, sober and features minimal accessories. The modern design is characterized with angular frames, low profiles, geometric and abstract patterns in textiles, upholstery as well as in artwork.Natural materials like linen, leather and teak wood are mostly used. The lines are unembellished as well as straight. In modern design, the furniture is often raised from the floor with the help of legs in order to create an airy and open atmosphere. Colors used in modern design are neutral shades that are highlighted with splashes of color. Walls are generally cream and white in color. Floors are mostly made of cement or bare wood. In addition, sculptures and paintings a re used as an integral part of modern design.If you are in the process of designing or renovating your home, you may be wondering whether to include modern design in the design layout. Well, the terms – modern is closely related and people tend to use the terms interchangeably. However, in the world of design and d ©core, both the terms represent distinct and different styles. To be modern a building should be light and airy, it must push technology to its limits even effecting new invention in the process; to be architecture it must provide utility, stability, commodity and delight and all of this done in sympathy with Nature.Being novel is not to be confused with being modern. ‘Sustainable systems in Iranian traditional architecture' Avid Iraqi , Sabina Kabuki Madman . 011). As result, architecture has been going backward since the mid-20th Century because the technology available at the time still has not been fully utilized, for example, space frames, and especial ly the engineering concepts of Businessmen Fuller, such as geodesic domes large enough to cover entire cities and his lightweight temerity towers; such technology is essential to conserve scare resources in order to assure economic growth, as well as to provide for increases in population. Modern & Traditional Houses' can Weiss. 2009). Basically ,a modern home should represent how we live today. It should reflect current construction methods and materials. It should have integrity by avoiding trends. Modern architecture offers an opportunity for an original beauty, not by imitating another style from another time or place, but by considering the present and, with imagination, creating a fresh aesthetic. Secondary, we might ask what is traditional architecture?Traditional architecture is that way of building which makes serious use of the familiar symbolic forms of a particular culture of a particular people in a particular place. It is different from modern buildings because of thei r method of construction, to because of their age or their listed status. Traditional buildings have an appeal due their special character, history and location. Furthermore, when looking for a property to buy it's easy to fall in love with an old building. ‘Architecture – modernism vs. traditionalism' (Lance Baker . 2011).Traditional architecture is the term used to categorize methods of construction which use local anesthetically available resources and traditions to address local needs. Some believed that, by using local practices, such as using local materials in construction, building costs will decrease, hence being economically more advantages. By the professor Lucien Steel, traditional architecture requires a high ethical commitment to the people, their places, their beliefs and their particular traditions. This commitment is not a slavish one, nor is it a servile opportunism.Ethical attitudes are not reducible to the uncritical acceptance of dominant sets of va lues and moral conventions. They require the distinction between civic and private virtues on one hand and willful customs and obsolete practices of false morality and corrupted policies on the other. So if modernity in some way would contribute to discern the most appropriate and the cost efficient, the most human and the most ecological aspects of the contemporary potential, every traditional architect and city-builder couldn't be but a committed modern.Traditional architecture and city-building are based on a positive philosophy of life, on faith in humanity, on respect of environment and historical cultures as a common heritage of mankind, and on an inviolable legacy of genius and know-how from proceeding generations of craftsmen and committed citizen. Traditional architecture and city-building imply a sense of modesty and humility of he individual creator within the sacred creation of the universe, as well as the powerful intuition that concepts of beauty, harmony, Justice, tru th, rightness are embedded in permanence and universality.Tradition forwards a selected knowledge, a tested experience as well as an heritage of models, types, techniques and formal vocabularies. It is a dynamic process, an on-going effort and development, not a static heritage of dogmas and immutable recipes. Tradition shoulders the responsibility of carrying on an inherited culture beyond the contingencies and improvisations of the moment. In order to remain vital, alive and relevant it needs to be earned, consolidated and enriched by each single generation in the perspective of universal ideals of civilization.It implies a constant effort of appropriation of knowledge, experience and cultural values, a permanent effort of intellectual, artistic and material reconstruction. (Tradition and Modernity in Contemporary Practice. Lucien Steel). Traditional architecture are mainly classified as historical buildings that have a lot of character and culture incorporated into them and artis ts were commissioned to put some color into the building giving each one an individual stamp.Now a day's traditional architecture is the widespread form of building since many years, constructed through traditional way of building methods by local builders without using the services of a professional architect. Due to western influence, architects are not using traditional architecture techniques now that are based on climatic conditions. Building materials has different categories from mud- plastered to reed-thatched to timber-framed in accordance with the availability of local material. Some houses are built to withstand earthquakes, while others can be built quickly if washed away by heavy monsoon rains.In some areas where there are limitations of building material, natural materials such as mud, grass, bamboo, thatch or sticks are used, instead of transporting materials from far place which is a blot on sustainability practices, for semi-permanent structures which require regula r maintenance and replacement. The advantages of such traditional architecture are the construction materials are cheap and easily available and relatively little labor is required. As the needs and resources of the people change, traditional architecture evolve to include more durable materials such as tones, clay tiles, metals etc.Though they are more expensive to build, they are very durable structures. In Asia climate has a major influence on traditional architecture. High thermal mass or significant amounts of insulation characterize buildings in cold climates. Lighter materials are used to build buildings in warm climates and designed for sufficient cross-ventilation through openings in the fabric of the building. In areas which have high levels of rainfall, flat roofs are avoided, even in areas with flat roofs, water harvesting techniques are being used. (Traditional Architecture In Asia . 2010)The overall effect of traditional houses is like walking through a well-curates ar t exhibit, where people can admire the buildings. The density of different buildings and stores satisfies the pedestrian's need for visual interest. It is a key part of what we call â€Å"walkabout'. This is what made historic downtowns beautiful in a way that no government or philanthropist could recreate today, and why historic preservationists nurse a broken heart with every lost structure. Traditional and modern architecture have mostly been seen as antitheses, impossible to reconcile, especially in Africa.They appear to belong to efferent ages, utilize different materials and methods, and encourage or support different lifestyles. This essay aims at seeking points where a merging of principles may be attempted between the two positions. Compare from both of them, modern building has very good facilities including toilets, kitchen etc. And more over the design is very different. They are designed according to the requirements and also the life would be much easier there in the modern building. (Traditional vs. Modern Architecture 2011). But on other hand, traditional house have great design too.It is graceful and warm and inviting. It is also beautiful. Of course traditional house can't guarantee that the roof isn't going to leak, the windows are properly sealed and the kitchen appliance is in the working order. Traditional house cannot guarantee for it. ‘Modern apartment building or traditional house ? (Teenage. 2011) The fact that modern buildings are prevalent proves modern style has its own advantages. In my country, population explosion has been a headache and the following problem is where to settle those extra citizens.Since the land is limited, one good solution is replacing those old buildings which occupy large space with tall and thin modern buildings. Also, modern buildings usually have the same and simple structure so that they can be finished in a relatively short time, compared to the traditional ones. As a result, modern buildings au gment the efficiency and make it possible to meet the increasing large demand of house nowadays. Furthermore, as modern buildings are always applied with advanced technology and theories, people can gain more security when living in such environment.But, there are many people still strongly recommend the traditional style. Specifically, unlike the modern style which can be seen everywhere, traditional buildings representing unique cantonal culture only exist in certain countries. In this way, those building can be built for special use like tourist attractions. This would bring a great profit and earn the country a good fame. In addition, buildings with traditional sense are a good way to memorize the past history and display the ancient scenes. As a result of this, some new buildings are necessary to be built in traditional style but not all the buildings.Modern buildings still play the key role in today's society and will gradually expand its affect zone. ‘Some people think all the new buildings should be built in traditional style? (Elise. 010). However, modern buildings often use steel infrastructure, where the interior columns carry most of the loading. Since this type of construction is lighter per floor, they can be built higher, cheaper, and quicker. What are the differences between ancient and modern buildings? Monsoon. 2008). For the opposite, most ancient buildings had load bearing walls, which limited their height, and accounted for the thicker walls.This also resulted in a lot less available window space. In fact in today society, one of the most significant problems accompanying with the population exploration is house problem, so more and more KY-scrapers instead of traditional buildings are built. As far as this phenomenon is concerned, some people think that we should construct much more buildings in traditional styles. Admittedly, there are some reasons for those people who stand for constructing building in traditional way. First of al l, the traditional buildings may possess more aesthetic values and historical meanings.Compared with the modern ones, the traditional buildings contain paintings or characters relating to the past certain age or dynasties; which endow more value to the buildings. Secondly, he traditional buildings often provide more spaces to house owners or renters; thereby making the living condition much easier and more comfortable. (Modern and traditional architecture 2010). However, maybe we do not think that we should build our building mainly in traditional way. Firstly, it is decided by the present social phenomenon that the number of population living in the planet nowadays has never appeared even before.Correspondingly, we have to build most our living houses in a way that never come before. Besides that, constructing our building in a modern way is also an integral part of sustaining ecosystem. Let us try to imagine that if we all build our house in traditional way, take china for example , which traditional buildings are usually one or two layers, and can it accommodate the present 1. 3 billion population . The might be a possible we could build a few numbers of buildings in traditional style which in order to hand down the traditional culture.But based on the social condition, most of our buildings should still construct in modern way. But , can modern and traditional architecture coexist? In today's world anything is possible for example Instead of painting beautiful designs on the wall, en can Just use wall paper instead which can be replaced or removed at any time. The thin line between modern architecture and traditional architecture is that Modern architecture explores mainly with the interior features whereas traditional architecture is mainly worked on the exterior features.Therefore modern architecture and traditional are definitely able to coexist. ‘To what extent do you agree or disagree? -modern & traditional building(Cathy. 2009). There is also a vast difference between modernity as an attitude and modernism as an architectural style. Modernity as an attitude, according to me, can co-exist with tradition. Modernity deals with transformation and change in the present and tries to incorporate it in buildings. Thus, it keeps changing with time. The standard steel frame and glass construction which was ‘modern' during the early 20th century is no longer modern today.