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Algebraic Patterns associated with Vedic Mathematics to Shorten Integration of Quadratic Formulae

Dr Vasant V Shastri, Prof. Alex Hankey

The Vedic Mathematics approach has proved effective in helping those preparing for pre-university professional examinations. Here we consider a particular kind of question in such examinations: multiple choice questions requiring students to select the correct answer for an integral with a quadratic denominator. This article shows how to shorten time spent on such questions by using principles associated with Vedic Mathematics. Classifying quadratics according to their first derivatives and discriminants gives an immediate classification of the kinds of result to be expected from the process of integration. The method allows immediate selection of the correct answer in multiple choice questions without entering into details of calculations. These particular kinds of question are among the most difficult in professional mathematics examinations, and can lead to the greatest loss of time, sapping candidate’s confidence to the greatest extent. Providing simple and easily remembered methods to short cut to the correct solution enables candidates to complete this difficult examination section without frustration, or loss of confidence and corresponding loss of emotional energy.

Keywords: Mathematics; Vedic Mathematics; Integration; Quadratic equation

Bharati Krishna’s Special Cases

Kenneth Williams

The numerous Special Cases given by Sri Bharati Krishna Tirthaji and his advocation of their use deserves greater study, not only as regards the many special cases that abound in mathematics but also in educational terms. In this paper these educational advantages are briefly described and three categories in which the special methods can be researched are suggested. Examples are given that show new special cases among these categories and the inclusion of trigonometric functions and derivatives of functions where Bharati Krishna uses only numbers or polynomials.

A Novel Algorithm for Multiplying Four Numbers near Different Bases

Jatinder Kaur, Pavitdeep Singh

Multiplication is one of the most fundamental processes and considered as a laborious operation in mathematics. Vedic maths provides an opportunity to think differently to solve a mathematical problem and makes the process interesting. Currently, there are very few techniques for multiplication of four numbers near different bases.

In this paper, we present a novel approach for performing multiplication of four numbers near different bases. The technique exhibits a special case of multiplication wherein the two numbers are different and may belong to different bases as well. Example, 203*203*305*305 or 506*506*103*103 and so on. The technique makes use of the algebraic equation in solving four numbers near different base which can be considered as a special case for multiplication. Additionally, the technique can be applied in conjunction with doubling & halving technique (sutra) to solve further problems. Different examples will be provided along with proof of the derived approach to facilitate people in learning the new approach.

A Novel Approach of Multiplying Five Numbers Near a Base

Pavitdeep Singh , Jatinder Kaur

Traditional way of performing multiplication is quite labour intensive and boring as against Vedic technique. Vedic Maths inculcates innovation which gives birth to new techniques for solving a problem. Not only it saves a lot of human effort but also makes the process of solving the problem exciting.

In this paper, we would like to present a novel technique for multiplying five different numbers near a base. The technique makes use of the Pascal triangle and algebraic equation in multiplying five numbers near a base. Moreover, the technique can be blended with other sutras in solving the problem. Proof of the derived methodology and a few examples will be discussed to facilitate the learning the new approach.

How the Binomial Theorem Underlies the Working of the Anurupyena and Yavadunam Sutras in the Calculation of Successive Powers of a Number; Application of power Triangles and Calculus

Marianne Fletcher and James Glover

“The binomial theorem is thus capable of practical application and – in its more comprehensive Vedic form – has thus been utilized, to splendid purpose, in the Vedic Sutras. And a huge lot of Calculus work both differential and integral has been and can be facilitated thereby…”

“Vedic Mathematics” Tirthaji

The working of the Anurupyena and Yavadunam sutras, in the calculation of squares, cubes and higher powers of numbers, is demonstrated to be an application of the binomial theorem.

Pascal’s arithmetic triangle (with pinnacle “11”) is conventionally used to help calculate binomial coefficients. The powers of 11 are also directly generated from the successive rows in the arithmetic triangle. The question arose whether a pascal-type triangle, with a number at its pinnacle containing digits other than “11”, might be used to help calculate successive powers of the particular number.

It was found that such a “power triangle” - with the number “ab” at its pinnacle - does not directly generate the powers of “ab”. However, the numbers in its successive rows can, indeed, with the help of the Anurupyena sutra, be employed to calculate any power of “ab”.

Athough this method is more laborious than direct application of Tirthaji’s sutras, there are some special cases where the n-th row in a power triangle can be directly used to find the n-th power of a number.

It is of interest to note that the sum of any power triangle and its “complement” (after applying the Nikhilam sutra to the number at the top) always yields Pascal’s original “11” arithmetic triangle again.

The discussion ends with a brief look at how successive differentiations and integrations of the variables and yield the terms in the binomial expansion of . Employing such a method, particularly for fractional and negative indices, can be a useful alternative approach to finding the terms in a binomial expansion. It is easy to remember and easy to apply.

A New Approach to the Teaching of Coordinate Geometry

James Glover

The Vedic Mathematics sutras of Shankaracarya Sri Bharati Krishna Tirtha can be usefully employed in the teaching and learning of coordinate geometry in two ways, described in two parts of this paper. In the first instance, elementary coordinate geometry is used to illustrate how the sutras appear in conventional treatment of a subject. This is due to the fact that the sutras express fundamental ideas, thought-patterns, cognitions and concepts that underlie the forms, strategies and problem-solving mechanisms that are in common use. The simple applications within coordinate geometry also help to reveal the meaning of the sutras. By use of the sutras, common formulae of various types take on a slightly different form in order to facilitate better understanding and greater efficiency. They furnish us with fast techniques for solving problems that students face in their public exams.

The second part of the paper takes one aphorism, or formula, from Vedic Mathematics and applies it to various aspects for finding areas of shapes in the context of coordinate geometry. The formula deals with areas of parallelograms and triangles. It then provides a spectacular one-line proof of Pythagoras’ theorem. Further applications reveal how it also gives simple proofs of compound angle formulae used in trigonometry and a short and easy proof of the formula used for finding the distance from a point to a line. It is believed that these proofs, Pythagoras’ theorem, etc., have not been published before now.

Keywords: Vedic Maths, Coordinate Geometry, Pythagoras’ Theorem, Compound Angle Formulae

A Survey on Implementation of Vedic Mathematics Sutras in

Information Technology

Anil AR

Vedic Mathematics is a powerful tool for fast and efficient computational purposes in the area of information technology. The implementation of various Vedic sutras reduces the time taken for computation in the field of Computer Science, Digital Signal Processing and various applications in Cryptography and Communication. In the field of information technology, the different applications need large computation capacity and large amount of energy. So to reduce the complexity, we need to select appropriate Vedic sutras and then implement them appropriately.

In this survey is aimed at assessing the implications of different Sutras used in various applications of computer science. The result shows that on implementation of appropriate Vedic Ssutras for computational purposes, the accuracy, efficiency, speed and power consumption change drastically in a positive way, compared to the traditional computational methods.

Keywords: Vedic Mathematics, Information Technology

Deeper Reasons Why Students Find Vedic Mathematics So Enjoyable

Alex Hankey and Vasanth Shastri

Vedic Mathematics has proved very popular with school students taught according to its prescribed methods. Many reasons have been offered for this: it is new and different; it gives alternative ways of solving mathematics problems, allowing the student to choose between a variety of possible methods to solve a given problem. In India, in particular, it offers means of learning in accordance with ancient principles of instruction, the sutra method, and so on. Here we offer a completely new approach to understanding its popularity: it stimulates the endorphin system. The reason for suggesting this possibility is that students find the approach genuinely enjoyable, strikingly so. Using the Sutras and Upasutras of Vedic Mathematics often causes joy – to an extent seen in few other subjects learned for final school examinations. This paper reviews what is known about the endorphin system. Its purpose is not entirely clear, but stimulating it makes people happy, exhilarated and joyful, consistent with experiences in classes learning according to Vedic Mathematics. One key observation is that supressing the endorphin system in infants leads to autism. It must therefore play an important role in generating healthy attitudes and mental function. A major use of the brain is to decide what to do in novel situations, or how to achieve some new goal. The paper gives reasons why creative thinking may be stimulated through feedback from the endorphin system, based on the very general hypothesis that endorphin is released to encourage kinds of mental process that will increase competitive success in a competitive environment. Methods used by Vedic Mathematics generate mental activity in this category. If this reasoning holds, the positive feedback given to teachers using Vedic Mathematics is more deeply explained.

Impact of Vedic Mathematics in Education for Development of

Sustainable Technologies

Samrudh J, S G Raghavendra Prasad, Nithyashree S

It is a proven fact that Mathematics plays a major role in various fields. It is an important subjects of study and thereby an integral part of education. Unfortunately, a lot of students face problems with the subject. Some circumstances at a young age may lead to disliking of the subject throughout. Thus, it becomes essential to ensure that alternatives are designed to make sure that the students develop a liking towards the subject. Vedic Mathematics is the best solution for this. Vedic Mathematics is the name given to the ancient system of Indian Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krishna Tirthaji. The most striking feature of this system is its coherence where a set of unrelated techniques is interrelated and unified into one. Interest in the Vedic system is growing in the field of education where the stakeholders are looking for innovative methodologies and have realised that the Vedic system is a primitive answer. Vedic Mathematics is being taught in schools from an elementary level. Brain Development has been observed to be better in students who practice Vedic Mathematics. Research is also being carried out in many areas including the effects of learning Vedic Mathematics on children, developing powerful applications of the Vedic Sutras in different fields etc. Our paper highlights the power of Vedic Mathematics and also provides a survey on the impact of Vedic Mathematics in Education. Further, we stress on the fact that Vedic Mathematics, if made an integral part of education from an elementary level, can help in enormous growth of technology. At the same time, performance of students becomes more efficient thereby becoming a pathway for creating Sustainable Technologies.

Teaching Vedic Math to Non-Traditional Audiences

Richard Blum, A.S.A., M.A.A.A.

This paper will be an expansion of a paper presented on-line to a group of Vedic Math (VM) practitioners. Over the last 30 years, I have taught and held workshops for many different types of groups interested in learning VM. The groups have run the gamut from elementary school students to PhD candidates, parents, teachers and the elderly, unemployed individuals needing better math skills to obtain jobs, prison inmates in a correctional institution and to the deaf. It is the experiences that I had with the last two groups that I will be discussing today.

This paper will concentrate on the special approaches I needed to utilize and results that were achieved with the prison inmates and the deaf. I found these experiences quite a bit different from all of my other teaching engagements and feel they would have value to all VM teachers now and in the future.

Increasing Human Potential of Computation through Indian Intellectual

Dr. Smitha S

The ultimate aim of Education is the overall development of the learner. Education is the divine force that leads us from darkness to light, from ignorance to knowledge, from the unreal to the real.

The present-day world places high demands on human potential: qualities such as efficiency, speed, accuracy, decision-making power, logical reasoning, high-powered thinking, and confidence of personality, are all required. Even though education has undergone a tremendous change with the development of modern science and technology, much still needs to be improved upon for the holistic development of the learner. It is the need of the hour to revisit our ancient treasure of rich intellectual traditions found in the Vedic algorithms.

This paper attempts to create awareness of the vast potential of Vedic/Ancient Indian Mathematics. It presents and discusses the results of a study that was conducted on a sample of Secondary School Students (Class VIII) of select schools of Thiruvananthapuram and Kollam districts (N=240) of Kerala, India. The sample for the study was selected on the basis of the Random Sampling Technique. The study aimed at testing the effectiveness of a prepared Learning Package of Vedic Mathematics Sutras for enhancing a Positive Mathematics Attitude and Computational Speed among Secondary School students in certain select areas. The investigator selected the Non-equivalent Pre-test/Post-test Control Group Design (Gay, 1987) which is one of the strongest of the Quasi Experimental Designs. The required data was collected using appropriate tools which were standardized by the investigator. The effectiveness of the package was tested by comparing the pre-test/post-test scores using appropriate statistical techniques including ANCOVA, Repeated ANOVA, LSD test of post hoc comparison, and were interpreted accordingly. The findings of the study emphasize the desirability of immediate inclusion of Vedic Mathematics applications in the present curriculum for grooming our students in order to achieve success in their future life by completely reducing their Mathematics anxiety. This anxiety plays a key role in their skill in computation and decision making.

Key Words: Vedic Mathematics, Mathematics Attitude, Computational Speed

A Teacher’s Viewpoint of Vedic Mathematics

Sumita Bansal

Vedic Mathematics is the name given to an ancient system of calculation which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krishna Tirthaji Maharaj. Vedic Mathematics is based on 16 sutras or formulas and their 13 sub-sutras or corollaries. These sutras are able to solve complex calculations with ease and simplicity. Complex mathematical problems can be solved in less time compared to conventional mathematical procedures. Vedic Mathematics is said to manifest a coherent and unified structure of arithmetic. Its methods are direct and easy. The sutras not only develop our logical thinking, but also encourage innovation. But Vedic Mathematics is not without its critics. In this paper the strengths and perceived weaknesses of Vedic Mathematics are assessed. In conclusion, an attempt is made to answer the question whether or not Vedic Mathematics should form part of a school curriculum.

Vedic Mathematics: Its impact on Children with Special Needs

Pooja Rani and Sukhwinder Kaur

Vedic mathematics is a system applied in the field of mathematics, which efficiently solves a number of the tricky mathematical problems. It is a tool used by some teachers to reduce long calculations and make mathematics an interesting subject. Students who have a good grasp of this can do extremely well in their exams. It is a branch of mathematics which was not commonly known but the late Jagadguru Sri Bharati Krishan Tirthaji Maharaj is famous for popularising the approach. Children with special needs (SEN) can also be taught Vedic mathematics. As these children face problems in solving long mathematical sums, Vedic mathematics can help them extensively. The present paper is an attempt to know to what extent Vedic mathematics can improve the mathematical skills of children with special needs. Thirty SEN students of Grade IX were taken as a sample from Government Schools of Chandigarh. After providing twenty days intervention to the experimental group and twenty days intervention to the control group, improvement was seen in the marks scored by children with special needs (experimental group) in the post-test as compared to the pre-test.

The Side of a Regular Polygon – Lilavati’s Approach

Anant Vyawahare, Sanjay Deshpande

The text Lilavati is one of the finest contributions of a great mathematician Bhaskara (also known as Bhaskaracharya of 12th Century) and one of the most popular books on ancient mathematics. Lilavati was the name of his daughter and Bhaskara taught her topics of mathematics relevant to the12th century. The year 2014 was the 900th birth year of Bhaskara and was celebrated thought India. Lilavati is the only text in ancient Indian mathematics which contains arithmetic, algebra, geometry, combinatorics, trigonometry and astronomy using interesting puzzles and games. The original script is in Sanskrit.

In this paper, we present the Lilavati approach of finding the side of a regular polygon of n sides (n from 3 to 9) inscribed in a circle with known diameter. As a matter of fact, any triangle can be circumscribed, but not all polygons can be circumscribed. Bhaskara knew this and hence he dealt only with circumscribable polygons. It is to be appreciated how Bhaskara obtained the results with no tools available in 12th century.

Lilavati contains 261 stanzas (shlokas). Of those, stanzas 144 to 223 are devoted to geometry. After 223, trigonometry and other topics are discussed.

From shloka 195, start the problems on figures circumscribed in a circle with known diameter. This part contains length of a chord, angle made by the arc at the center, etc. Our focus in on shlokas 202, 203 and 204.

Bharathi Krishna Tirtha’s Vedic Maths and Early Indian mathematics:

Comparison of the Fundamental Arithmetic Operations

Indian mathematicians of the past millennia describe twenty operations of arithmetic (additionally 8 determinations). Of these, Addition, Subtraction, Multiplication, Division, Squaring, etc. are considered fundamental operations. In a recently published work (Prasad, 2015), Multiplication techniques from early Indian mathematics were compared against the Multiplication techniques contained in Bharathi Krishna Tirtha’s methods in Vedic Mathematics (henceforth VM). Several similarities were found, including one-to-one match with certain techniques, notably, the ‘Vertically and Crosswise’ method in VM called Urdhva Tiryagbhyam, and named Tastha by the early Indian mathematicians. The current paper is an extension of this comparative analysis to additional fundamental arithmetic operations viz. Addition, Subtraction and Division. Furthermore, the operation of Squaring is also included in the current analysis. As in the previous work, these arithmetic operations are compared and contrasted within the two systems, viz. early Indian mathematics and VM. The comparison reveals some similarities in the two systems, such as Left-to-Right and Right-to-Left processes for Squaring, Addition and Subtraction operations. Furthermore, the idea of the operation of Division being an inverse of Multiplication is seen in both systems. Following these comparisons, some conclusions are drawn.

Keywords: Indian mathematics, Vedic Mathematics, History, Arithmetic operations

Lilavati and Vedic Mathematics

Jayanta Acharya

Lilavati and Vedic Mathematics are two treatises on mathematics written in different time periods. They seem to be independent and have different interpretations of mathematical rules. But both of them are written from the same foundation of the Vedic tradition. So, in reality they have many similarities. For an instance, the squaring rules of any number given by them are very similar. Although the forms of formulae seem to be different they follow the same philosophy of mathematics.

Key words: Lilavati, Ista, Vedic Mathematics.

Contributions of Kerala to Mathematics

P.Devaraj

The discovery of general algorithms of calculus and the development and application of infinite series techniques are major tools for modern mathematics. The astronomers and mathematicians from Kerala made major contributions during 14th to 16th centuries. And these contributions are 300 years prior to European mathematicians’ statements in these fields.

Social background and political situations helped in these contributions. Mathematics was an important and helpful tool to the study of Jyotisha sastra. The result was that those mathematical topics like properties of a circle, sphere, mensuruation, etc., needed in the study of astronomy got prime attention and gained enormous development. The Bhutha Samkhya and Katapayadi were the numeration systems used in ancient texts. The major mathematical texts from Kerala can be classified into two periods, 7th and 14th centuries. Their works were mainly commentaries, improvements on the works of earlier scholars. The period 1350-1600 was the glorious period when Kerala Mathematics attained great heights known as Golden age of Kerala Mathematics. The study of the circle and its chords broke the finite barrier and searched for the infinite. The significant period in the history of Kerala Mathematics started with Sangama Gramma Madhavan (1340-1425). Madhavan gave the value of Pi(π) correct to eleven decimal places. He described it in the Bhuta Samkhya system. Sangama Gramma Madhavan also stated

π/4 =1-1/3 + 1/5 - 1/7 +….

He also made sine tables from zero degrees to ninety at the intervals of 3.75 degrees.

Madhavan’s main contributions came to be known through the works of his followers, such as Neelakanda Somayaji, Jeyshtadevan, Sankaravaryar etc. The knowledge might have been transmitted to other parts of the world through foreigners that came to Kerala for trading of spices, etc. Many of the results discovered by Kerala Mathematicians became known as the works of a few Europeans.

Relevance of Shulbasūtras of the Yajurveda: Modern Context

Dr Daya Shankar Tiwary

The Yajurveda (1200 BCE to 1000 BCE) contains valuable knowledge in the realm of mathematics which has eternal value. The science of mathematics with all its branches such as arithmetic, algebra, geometry and trigonometry, etc., was so well developed in ancient India that many modern scholars find to their dismay that some of the European discoveries were discovered in previous centuries. It is necessary to integrate this valuable treasure of Vedic mathematical knowledge with that of modern mathematics. Vedic people were fully acquainted with mathematical knowledge and had knowledge of applied geometry as well. The origin of geometry is from Shulbashutra. Vedic people used to make sacrificial altars in definite prescribed shapes and sizes using special types of bricks. There are eight Shulbashutras- Baudhayana, Manava, Apastamba, Katyayana, Satyashadha, Vadhula, Varaha and Maitrayani. The first four of these are available as independent texts. Seven Shulbashutras belong to the Krishna Yajurveda and one, namely Katyayana Shulbashutra, belongs to the Shukla Yajurveda. Shulbashutras of vedanga taught the important rules to construct triangles, rectangles, squares, parallelograms and circles by explaining their properties.

The equation c2=a2+b2 , which is known as Pythogoras theorem, was first given by  Baudhayana (800 BCE) before Pyhthagoras (580-500 BCE) . This is basically Shulba theorem (Baudhayana Shulbashutra: 1-48) .

This research paper is mainly focused at Shulbashutras in determining the special blend of scientific approach of Vedic seers in the field of modern mathematics.