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Vedic Mathematics for Schools (Book 2)

by J.T. Glover

Description

Vedic Mathematics for school , Book 2 is intended as a first year text book for senior schools or for children aiming for examination at 11+. It is based on the fundamental principles of Vedic mathematics which were reconstructed earlier this century by Sri Sankaracarya Bharti Krsna Tirthaji . Although the sutras may well be very ancient , practice and experience have shown that they are highly relevant and useful to the modern –day teaching of mathematics. They are entirely applicable to modern approaches to mathematics.

Topics covered include the four rules of number, fractions and decimals, simplifying and solving in algebra, perimeters and areas, ratio and proportion , percentages, averages, constructions. The book contains step by -step worked examples with explanatory notes together with over two hundred practice exercises.

The material in this book is currently used at school around the world associated with the education Renaissance Trust.

It is assumed that pupils using this book already have a degree of mathematical ability. In particular, the times tables need to be fully established I should also be stated that regular practice of mental arithmetic is an essential accompaniment to this course.

Each method used for numerical calculation is separately and exercises are carefully graded to enable the district development steps to be mastered. Each techniques is denoted by one or more of the sutras. The text incorporates explanations and worked examples of all the methods used and includes description of how to set out written work.

The structure of the book is such that at the end of each term a bright pupil should be able to complete about eight chapters. There are three revision chapters, the last which contains practice papers. It is not necessarily indeed that teachers rigidly adhere to the order of chapters as presented. Nevertheless, there are certain topics that should be covered before moving on to more advanced work.

The Course has been written in conjunction with teaching a group of ten and eleven years old s. The main emphasis at this stage is on developing numeracy and its principal fields of applications, since this is the most essential aspects of mathematics. The text concentrates on these areas of mathematics and treats them as the core curriculum of the subjects.

Experience has shown that children benefit most from their own practice and experience rather than being continually provided with explanations of mathematical concepts. The explanations given in this text show the pupil how to practise so that they may development their own understanding.

It is to be hoped that teacher s may provide their own practical ‘ways of demonstrating this system or enabling children to practise and experience the full benefits of Vedic Mathematics unless one gets immersed in the techniques, leaving behind all previous personal paradigms and prejudices about mathematics.

**James T.** Glovers is head of mathematics at st. James Independent School in London where he has been a teacher for eighteen years .He is director of mathematical studies at the School of Economics Science and a Fellow of the Institute of Mathematics and its Applications. He has been researching Vedic Mathematics and its use in education for more than twenty years and has run public courses in London on the subject . other books by the author are : An Introductory Course in Vedic Mathematics, Vedic Mathematics, Vedic Mathematics for schools, Book 1 and Foundation Mathematics, Books 1, 2 and 3.

**His Excellency Dr L.M. Singhvi High commissioner for Indian in the UK**

Vedic mathematics for schools is an exceptional book. It is not only a sophisticated pedagogic tool but also an introduction to an ancient civilisation. It takes us back to any millennia of India’s mathematical heritage. Rooted in the ancient Vedic sources which heralded the dawn of human history and illumined ancient vedic sources which heralded the dawn of human history and illumined by their erudite exegesis, India’s intellectual , scientific and aesthetic vitality blossomed arts but also in geometry, algebra and arithmetic. Indian mathematicians gave the world the numerals now in universal use. The crowning glory of Indian mathematics was the invention of zero and the introduction of decimal notation without which mathematics as a scientific discipline could not have made much headway. It is noteworthy that the ancient Greeks and Romans did not have the decimal notation and, therefore, did not make much progress in the numerical sciences. The Arbas first learnt the decimal notation n from Indians Raihan , who was born in 973 A.D. and travelled to India , testified that the Indian attainment s in mathematics were unrivalled and unsurpassed . In keeping with that ingrained tradition of mathematics were unrivalled and unsurpassed. In keeping with that ingrained tradition of mathematics in India , S. Rmanujan , “the man who knew infinity “, the genius who was one of the greatest mathematicians of our time and the mystic for whom “a mathematical equation trials in cambridge University in the second decade of the twentieth century even though he did not himself possess a university in the university degree.

The real contribution of this book, Vedic Mathematic s for Schools , is to demonstrate that Vedic mathematics belongs not only to an horary antiquity but is any day as modern as the day after tomorrow. What distinguishes it particularly is that it has been fashioned by british teachers for use at st James Independent the pioneering work of the late Bharati Krishna Tirathaji , a former Sankracharya of puri, who reconstructed a unique system on the basis of ancient Indian mathematics. The Book is thus a bridge across centuries , civilisation , linguistic barriers and national frontiers.

Vedic mathematics was traditionally taught through aphorism or sutras. A sutra is a thread of knowledge , a theorem , a ground norm , a repository of [roof. It is formulated as a proposition to encapsulate a rule or a principle . A single sutra would generally encompass a wide and varied range of particular applications and may be linked to a programmed chip of our computer age. These aphorism of Vedic mathematics have much in common with aphorism which are contained in Panini Ashtadhayi , that grad edifice of Sanskrit grammar . Both Vedic mathematic s and Sanskrit grammar are built on the foundations of rigorous logic an d on a deep understanding of how the human mind works . The methodology of Vedic mathematics and of Sanskrit grammar help to hone the human intellectual and guide and groom the human mind into modes of logical reasoning.

I hope that vedic mathematics for schools will prove to be an asset of great value as a pioneering exemplar and will be used and adopted by discerning teachers throughout the world . It is also my prayer and hope that the example of st James Independent Schools in teaching Vedic mathematics and Sanskrit may eventually be emulated in every Indian school.

Vedic Mathematic for schools book 2 is a texts for the first – year at senior schools Book 1 of the series is for primary schools and does not have to be read first since each of the methods in the presents work are introduced from their inceptions. Thesew books are based on the fundamental ancient Hindu system which uses a few simple rules and principles which in Sanskrit are called sutras,. The sutras enable fast and easy methods of calculations.

**On the discovery and nature of Vedic mathematics**

This system of mathematics as recreated by the late sri Bhartai Krishna Tirthaji (1884- 1960) a brilliant scholar and exponent of the spiritual teachings of the Veda (he held the seat of sankaracharya of puri for many years) . The Vedas are the ancient scriptural texts of India written in sanskrit . They are concerned with both the spiritual and secular aspects of life because , in those times, no essentials difference was perceived between the two . the veda deal with many subjects but the texts are frequently difficult to understand . tirthaji made great efforts to dig out the system of mathematics from these texts and came up with sixteen sutras and about thirteen sub- sutras. A sutra I s a pithy statement containing a governing principle , a method or a rule of working and the ones which he discovered relate to mathematics. Not long before he died , tirthaji wrote an illustrative volume on the subject entitled Vedic mathematics which published posthumously in 1965 (motilal Banrasidas , Delhi).

Sri Tirthaji applied the sutras to the mathematics of this day so in his book we find a good deal of arithmetic and algebra . It has since been shown that the system is equally applicable to more up –to-date aspects of mathematics both at an elementary level as well as in more sophisticated fields. The reason that this is possible relies on the nature of the sutras. They frequently describes how the mind approaches, or deals with , a problem in the easiest way. To take a very simple example , consider finding the sum of 267 and 98 . The blanket method involves cumbersome arithmetic. Most people would realise the easiest method is too add 100 and takes off 2. The answer of 365 is then found using a complement . the Vedic system teaches this sort of approaches systematic ally rather than leaving it to chance and hence we find a number of different possible methods for any particular sum. This is of tremendous use because it enhances variety of strategy . It also enables the subjects to be kept alive by directing the attention towards underlying pattern and relationship of course , with children , some mastery of the different methods must be accomplished before this more creative approach can be adopted.

The Sutras, which are used in translation in this book, provide easily remembered world –formulae for solving problems in arithmetic, algebra, geometry and their various applications. The methods are fast and effective because they rely on mental working. In many applications answer are found in one line and for this reason the mathematics can often appear to be intuitively based. Nevertheless, it is all quite logical and systematic.

**Vedic mathematics in education**

In the Untitled kingdom there are growing signs of dissatisfaction with the lack of training in numeracy and the accompanying degeneration of algebraic and other mathematical skills. Some would argue that this is due to misguided educational methods whilst others lay blame on the widespread use of electronic calculating aids. During this century , the treatment of mathematics in educational has seen a decline in rigour and , more recently , a move away from the so called ‘ drudgery “ of sums.

Vedic Mathematics for schools aims at providing a good mathematical training without the necessity of relying on calculating aids. It is also aims to relate mathematics to the natural laws expressed in the Vedic sutras.

As with the body, exercise and training are required to develop health, strength , agility and skill , so too with the young mind , training is required for development of knowledge, creativity and the ability to reason mathematics and, in particular, the experience of working with number, provide one of the most powerful tools to accomplish this. For this purpose, these Vedic methods are used for basic to accomplish this. For skills. Not only are they enjoyable , but they also encourage the use of numerical skills. Not only are they enjoyable, but they also encourage the uses for basic numerical skills . Not only are they enjoyable, but they also encourage the use of mental arithmetic . For example , there are methods which replace long multiplication and division where by the answer to any such con can be obtained in one line. Experience of teaching these methods to children has shown that a high degree of mathematical ability can be attained from an early age while the subject is enjoyed for its own merits.

I have received criticism that vedic mathematics only provides a few interesting techniques and is not really relevant to a core curriculum. There has also been the comment that children do not understand these methods which are merely tricks. None of these comments has ever come from anyone who has studied and practised the system but only from those who seem to be looking at it from the outside and therefore have little understanding themselves. A soup packet has the name , cooking instructions and even a picture printed on the outside. Until the soup has been tasted, however, there is only peripheral knowledge and certainly no basis from which to make a sound judgement.

Nevertheless , in reply to the first criticism it has to be understood that in the Vedic system there are general methods for all calculations and algebraic manipulations, and also short and easy ways for particular cases , for instance in multiplying or dividing numbers close to a power of ten . where such particular methods are introduced at an early stage it is because they usually relate to more general aspect of the system at a later stage. The current methods of calculating which have been adopted by most schools are ‘blanket’ methods and no short cut methods, or even intuitive approaches, are systematically used. For example, with division, only one method is taught and , although it will suffice in all cases, it may often be difficult to use . The Vedic system in this book teaches three basic methods for division which are applied to meet the particular case although each could be used for any division sum.

The second criticism raises an important issues in relation to school mathematics. This concerns the use and understanding of set formula to solve problems. There are many instances in education where the students are required to learn a formula solve a particular type of problem whilst not understanding the mathematics behind the formula. We could take as an example the well known formula used for solving quadratic equations. When students first learn to use this formula there are few who understand how it has been derived. The usual practice is to gain some familiarity with its application before learning how it arises. Another example which is even more stark concerns the formula for finding the volume of a pyramid wit a polygonal base. The formula is volume equals one third of the base area times the perpendicular height and is learnt quite early on in the school curriculum. But to prove or derive this result the student needs to be adept either in three dimensional geometry or in the calculus. The advanced character. Much of school mathematics is like this and there is nothing amiss. When a student is faced with a problem which can be solved using a simple well – known formula then , by all means , it should be used . We accept the validity of such formulae on trust and , that it works is magical . The magic becomes mathematics when the formulae are understood.

Vedic mathematics appears, at first, to have a magical. When the methods are understood, particularly in relation to one another, than it is unified mathematics.

**On the nature of mathematics**

Mathematics is a practical science as it helps us with daily life. It also help us to understand the mysteries of the universe.

The study of mathematics may be seen as having two directions, an outer and an inner . The outer direction moves us to applying number , order and mathematical relationship s in the world around us . It is practical , useful and beneficial. This includes the everyday activities of shopkeepers, accountants, technicians, engineers, scientists, gamblers, etc., etc. In fact our lives, in the ordinary sense, would be very limited were it the responsibility of those with some understanding of the structure of mathematics to ensure that institution are retained which and disseminate mathematics as a structured body of knowledge.

The other direction in mathematics is an inner one. It takes us back to the very foundation blocks upon which the subject stands. Ultimately it reminds us of our origin , the unity , Supreme self, which is the basis of the entire creation . mathematics returns us to unity in a very simple way and this is because it starts at one. The number one is a simple expressions of this unity . so too is the idea of equality . We speak, read and write ‘ equals’ so frequently that it is all too easy to pass over its philosophic significance. There is so much in mathematics to remind us of the underlying unity in all forms and creatures.

So these are the two directions but they are not exclusive . It is more the case that the inner directions enhances the understanding of the outer directions. Indeed without the inner the outer direction becomes lost, trivial and lifeless. It is the Vedic system that enables these two directions to be studied in mutual harmony and this may be accomplished through correct appreciation of the sutras.

I has been found that in India today there is widespread belief that the Vedas are merely historical texts consisting of various hymns and injunction for the performance of sacrifices, etc. A more accurate understanding of Veda is found in the idea of true knowledge . The nature of knowledge is that it is a living aspect of the human mind. This is the spirit in which these sutras are to be regarded when you are faced with a problem and the method of solution comes to mind the law by which the solution is found is expressed in one of the sutras . Thus the sutras have to do with the way the mind works in the presents.

**Contents**

Preface by his Excellence Dr. L.M. Singhavi, High Commissioner for India in the UK | v | |

Introduction | vii | |

1 | Multipying by all from 9 and from 10 | 1 |

2 | Multiplication by Vertically and Crosswise | 13 |

3 | division | 23 |

4 | Subtraction by all from 9 and the last from 10 | 36 |

5 | Prime and Composite Numbers | 43 |

6 | Fractions | 51 |

7 | Algebra | 61 |

8 | Practice and Revision 1 | 75 |

9 | Geometry 1 | 83 |

10 | Digital Roots | 100 |

11 | Divisibility | 106 |

12 | Addition and subtraction of fractions | 112 |

13 | Decimal frcations | 121 |

14 | Perimeters and areas | 128 |

15 | Straight Division | 140 |

16 | Practice and Revision 2 | 147 |

17 | Working Base Multiplication | 153 |

18 | Ratio and proportion | 159 |

19 | Geometry 2 - The Rectangle Proportion | 168 |

20 | Order of Operation | 180 |

21 | Multiplication and Division of Decimals | 186 |

22 | Percentages | 194 |

23 | Averages | 200 |

24 | Graphs | 205 |

25 | Calculation Using Vinculums | 213 |

26 | Geometry 3 - Angles | 218 |

27 | Practice and Revisions 3 | 226 |

Appendix - Tables of Weights and Measures | 231 | |

Answers | 235 |

Item Code:

NAJ189

Cover:

Paperback

Edition:

2013

Publisher:

ISBN:

9788120816701

Language:

English

Size:

8.5 inch X 5.5 inch

Pages:

300

Other Details:

Weight of the Book: 380 gms

Vedic Mathematics for school , Book 2 is intended as a first year text book for senior schools or for children aiming for examination at 11+. It is based on the fundamental principles of Vedic mathematics which were reconstructed earlier this century by Sri Sankaracarya Bharti Krsna Tirthaji . Although the sutras may well be very ancient , practice and experience have shown that they are highly relevant and useful to the modern –day teaching of mathematics. They are entirely applicable to modern approaches to mathematics.

Topics covered include the four rules of number, fractions and decimals, simplifying and solving in algebra, perimeters and areas, ratio and proportion , percentages, averages, constructions. The book contains step by -step worked examples with explanatory notes together with over two hundred practice exercises.

The material in this book is currently used at school around the world associated with the education Renaissance Trust.

It is assumed that pupils using this book already have a degree of mathematical ability. In particular, the times tables need to be fully established I should also be stated that regular practice of mental arithmetic is an essential accompaniment to this course.

Each method used for numerical calculation is separately and exercises are carefully graded to enable the district development steps to be mastered. Each techniques is denoted by one or more of the sutras. The text incorporates explanations and worked examples of all the methods used and includes description of how to set out written work.

The structure of the book is such that at the end of each term a bright pupil should be able to complete about eight chapters. There are three revision chapters, the last which contains practice papers. It is not necessarily indeed that teachers rigidly adhere to the order of chapters as presented. Nevertheless, there are certain topics that should be covered before moving on to more advanced work.

The Course has been written in conjunction with teaching a group of ten and eleven years old s. The main emphasis at this stage is on developing numeracy and its principal fields of applications, since this is the most essential aspects of mathematics. The text concentrates on these areas of mathematics and treats them as the core curriculum of the subjects.

Experience has shown that children benefit most from their own practice and experience rather than being continually provided with explanations of mathematical concepts. The explanations given in this text show the pupil how to practise so that they may development their own understanding.

It is to be hoped that teacher s may provide their own practical ‘ways of demonstrating this system or enabling children to practise and experience the full benefits of Vedic Mathematics unless one gets immersed in the techniques, leaving behind all previous personal paradigms and prejudices about mathematics.

**James T.** Glovers is head of mathematics at st. James Independent School in London where he has been a teacher for eighteen years .He is director of mathematical studies at the School of Economics Science and a Fellow of the Institute of Mathematics and its Applications. He has been researching Vedic Mathematics and its use in education for more than twenty years and has run public courses in London on the subject . other books by the author are : An Introductory Course in Vedic Mathematics, Vedic Mathematics, Vedic Mathematics for schools, Book 1 and Foundation Mathematics, Books 1, 2 and 3.

**His Excellency Dr L.M. Singhvi High commissioner for Indian in the UK**

Vedic mathematics for schools is an exceptional book. It is not only a sophisticated pedagogic tool but also an introduction to an ancient civilisation. It takes us back to any millennia of India’s mathematical heritage. Rooted in the ancient Vedic sources which heralded the dawn of human history and illumined ancient vedic sources which heralded the dawn of human history and illumined by their erudite exegesis, India’s intellectual , scientific and aesthetic vitality blossomed arts but also in geometry, algebra and arithmetic. Indian mathematicians gave the world the numerals now in universal use. The crowning glory of Indian mathematics was the invention of zero and the introduction of decimal notation without which mathematics as a scientific discipline could not have made much headway. It is noteworthy that the ancient Greeks and Romans did not have the decimal notation and, therefore, did not make much progress in the numerical sciences. The Arbas first learnt the decimal notation n from Indians Raihan , who was born in 973 A.D. and travelled to India , testified that the Indian attainment s in mathematics were unrivalled and unsurpassed . In keeping with that ingrained tradition of mathematics were unrivalled and unsurpassed. In keeping with that ingrained tradition of mathematics in India , S. Rmanujan , “the man who knew infinity “, the genius who was one of the greatest mathematicians of our time and the mystic for whom “a mathematical equation trials in cambridge University in the second decade of the twentieth century even though he did not himself possess a university in the university degree.

The real contribution of this book, Vedic Mathematic s for Schools , is to demonstrate that Vedic mathematics belongs not only to an horary antiquity but is any day as modern as the day after tomorrow. What distinguishes it particularly is that it has been fashioned by british teachers for use at st James Independent the pioneering work of the late Bharati Krishna Tirathaji , a former Sankracharya of puri, who reconstructed a unique system on the basis of ancient Indian mathematics. The Book is thus a bridge across centuries , civilisation , linguistic barriers and national frontiers.

Vedic mathematics was traditionally taught through aphorism or sutras. A sutra is a thread of knowledge , a theorem , a ground norm , a repository of [roof. It is formulated as a proposition to encapsulate a rule or a principle . A single sutra would generally encompass a wide and varied range of particular applications and may be linked to a programmed chip of our computer age. These aphorism of Vedic mathematics have much in common with aphorism which are contained in Panini Ashtadhayi , that grad edifice of Sanskrit grammar . Both Vedic mathematic s and Sanskrit grammar are built on the foundations of rigorous logic an d on a deep understanding of how the human mind works . The methodology of Vedic mathematics and of Sanskrit grammar help to hone the human intellectual and guide and groom the human mind into modes of logical reasoning.

I hope that vedic mathematics for schools will prove to be an asset of great value as a pioneering exemplar and will be used and adopted by discerning teachers throughout the world . It is also my prayer and hope that the example of st James Independent Schools in teaching Vedic mathematics and Sanskrit may eventually be emulated in every Indian school.

Vedic Mathematic for schools book 2 is a texts for the first – year at senior schools Book 1 of the series is for primary schools and does not have to be read first since each of the methods in the presents work are introduced from their inceptions. Thesew books are based on the fundamental ancient Hindu system which uses a few simple rules and principles which in Sanskrit are called sutras,. The sutras enable fast and easy methods of calculations.

**On the discovery and nature of Vedic mathematics**

This system of mathematics as recreated by the late sri Bhartai Krishna Tirthaji (1884- 1960) a brilliant scholar and exponent of the spiritual teachings of the Veda (he held the seat of sankaracharya of puri for many years) . The Vedas are the ancient scriptural texts of India written in sanskrit . They are concerned with both the spiritual and secular aspects of life because , in those times, no essentials difference was perceived between the two . the veda deal with many subjects but the texts are frequently difficult to understand . tirthaji made great efforts to dig out the system of mathematics from these texts and came up with sixteen sutras and about thirteen sub- sutras. A sutra I s a pithy statement containing a governing principle , a method or a rule of working and the ones which he discovered relate to mathematics. Not long before he died , tirthaji wrote an illustrative volume on the subject entitled Vedic mathematics which published posthumously in 1965 (motilal Banrasidas , Delhi).

Sri Tirthaji applied the sutras to the mathematics of this day so in his book we find a good deal of arithmetic and algebra . It has since been shown that the system is equally applicable to more up –to-date aspects of mathematics both at an elementary level as well as in more sophisticated fields. The reason that this is possible relies on the nature of the sutras. They frequently describes how the mind approaches, or deals with , a problem in the easiest way. To take a very simple example , consider finding the sum of 267 and 98 . The blanket method involves cumbersome arithmetic. Most people would realise the easiest method is too add 100 and takes off 2. The answer of 365 is then found using a complement . the Vedic system teaches this sort of approaches systematic ally rather than leaving it to chance and hence we find a number of different possible methods for any particular sum. This is of tremendous use because it enhances variety of strategy . It also enables the subjects to be kept alive by directing the attention towards underlying pattern and relationship of course , with children , some mastery of the different methods must be accomplished before this more creative approach can be adopted.

The Sutras, which are used in translation in this book, provide easily remembered world –formulae for solving problems in arithmetic, algebra, geometry and their various applications. The methods are fast and effective because they rely on mental working. In many applications answer are found in one line and for this reason the mathematics can often appear to be intuitively based. Nevertheless, it is all quite logical and systematic.

**Vedic mathematics in education**

In the Untitled kingdom there are growing signs of dissatisfaction with the lack of training in numeracy and the accompanying degeneration of algebraic and other mathematical skills. Some would argue that this is due to misguided educational methods whilst others lay blame on the widespread use of electronic calculating aids. During this century , the treatment of mathematics in educational has seen a decline in rigour and , more recently , a move away from the so called ‘ drudgery “ of sums.

Vedic Mathematics for schools aims at providing a good mathematical training without the necessity of relying on calculating aids. It is also aims to relate mathematics to the natural laws expressed in the Vedic sutras.

As with the body, exercise and training are required to develop health, strength , agility and skill , so too with the young mind , training is required for development of knowledge, creativity and the ability to reason mathematics and, in particular, the experience of working with number, provide one of the most powerful tools to accomplish this. For this purpose, these Vedic methods are used for basic to accomplish this. For skills. Not only are they enjoyable , but they also encourage the use of numerical skills. Not only are they enjoyable, but they also encourage the uses for basic numerical skills . Not only are they enjoyable, but they also encourage the use of mental arithmetic . For example , there are methods which replace long multiplication and division where by the answer to any such con can be obtained in one line. Experience of teaching these methods to children has shown that a high degree of mathematical ability can be attained from an early age while the subject is enjoyed for its own merits.

I have received criticism that vedic mathematics only provides a few interesting techniques and is not really relevant to a core curriculum. There has also been the comment that children do not understand these methods which are merely tricks. None of these comments has ever come from anyone who has studied and practised the system but only from those who seem to be looking at it from the outside and therefore have little understanding themselves. A soup packet has the name , cooking instructions and even a picture printed on the outside. Until the soup has been tasted, however, there is only peripheral knowledge and certainly no basis from which to make a sound judgement.

Nevertheless , in reply to the first criticism it has to be understood that in the Vedic system there are general methods for all calculations and algebraic manipulations, and also short and easy ways for particular cases , for instance in multiplying or dividing numbers close to a power of ten . where such particular methods are introduced at an early stage it is because they usually relate to more general aspect of the system at a later stage. The current methods of calculating which have been adopted by most schools are ‘blanket’ methods and no short cut methods, or even intuitive approaches, are systematically used. For example, with division, only one method is taught and , although it will suffice in all cases, it may often be difficult to use . The Vedic system in this book teaches three basic methods for division which are applied to meet the particular case although each could be used for any division sum.

The second criticism raises an important issues in relation to school mathematics. This concerns the use and understanding of set formula to solve problems. There are many instances in education where the students are required to learn a formula solve a particular type of problem whilst not understanding the mathematics behind the formula. We could take as an example the well known formula used for solving quadratic equations. When students first learn to use this formula there are few who understand how it has been derived. The usual practice is to gain some familiarity with its application before learning how it arises. Another example which is even more stark concerns the formula for finding the volume of a pyramid wit a polygonal base. The formula is volume equals one third of the base area times the perpendicular height and is learnt quite early on in the school curriculum. But to prove or derive this result the student needs to be adept either in three dimensional geometry or in the calculus. The advanced character. Much of school mathematics is like this and there is nothing amiss. When a student is faced with a problem which can be solved using a simple well – known formula then , by all means , it should be used . We accept the validity of such formulae on trust and , that it works is magical . The magic becomes mathematics when the formulae are understood.

Vedic mathematics appears, at first, to have a magical. When the methods are understood, particularly in relation to one another, than it is unified mathematics.

**On the nature of mathematics**

Mathematics is a practical science as it helps us with daily life. It also help us to understand the mysteries of the universe.

The study of mathematics may be seen as having two directions, an outer and an inner . The outer direction moves us to applying number , order and mathematical relationship s in the world around us . It is practical , useful and beneficial. This includes the everyday activities of shopkeepers, accountants, technicians, engineers, scientists, gamblers, etc., etc. In fact our lives, in the ordinary sense, would be very limited were it the responsibility of those with some understanding of the structure of mathematics to ensure that institution are retained which and disseminate mathematics as a structured body of knowledge.

The other direction in mathematics is an inner one. It takes us back to the very foundation blocks upon which the subject stands. Ultimately it reminds us of our origin , the unity , Supreme self, which is the basis of the entire creation . mathematics returns us to unity in a very simple way and this is because it starts at one. The number one is a simple expressions of this unity . so too is the idea of equality . We speak, read and write ‘ equals’ so frequently that it is all too easy to pass over its philosophic significance. There is so much in mathematics to remind us of the underlying unity in all forms and creatures.

So these are the two directions but they are not exclusive . It is more the case that the inner directions enhances the understanding of the outer directions. Indeed without the inner the outer direction becomes lost, trivial and lifeless. It is the Vedic system that enables these two directions to be studied in mutual harmony and this may be accomplished through correct appreciation of the sutras.

I has been found that in India today there is widespread belief that the Vedas are merely historical texts consisting of various hymns and injunction for the performance of sacrifices, etc. A more accurate understanding of Veda is found in the idea of true knowledge . The nature of knowledge is that it is a living aspect of the human mind. This is the spirit in which these sutras are to be regarded when you are faced with a problem and the method of solution comes to mind the law by which the solution is found is expressed in one of the sutras . Thus the sutras have to do with the way the mind works in the presents.

**Contents**

Preface by his Excellence Dr. L.M. Singhavi, High Commissioner for India in the UK | v | |

Introduction | vii | |

1 | Multipying by all from 9 and from 10 | 1 |

2 | Multiplication by Vertically and Crosswise | 13 |

3 | division | 23 |

4 | Subtraction by all from 9 and the last from 10 | 36 |

5 | Prime and Composite Numbers | 43 |

6 | Fractions | 51 |

7 | Algebra | 61 |

8 | Practice and Revision 1 | 75 |

9 | Geometry 1 | 83 |

10 | Digital Roots | 100 |

11 | Divisibility | 106 |

12 | Addition and subtraction of fractions | 112 |

13 | Decimal frcations | 121 |

14 | Perimeters and areas | 128 |

15 | Straight Division | 140 |

16 | Practice and Revision 2 | 147 |

17 | Working Base Multiplication | 153 |

18 | Ratio and proportion | 159 |

19 | Geometry 2 - The Rectangle Proportion | 168 |

20 | Order of Operation | 180 |

21 | Multiplication and Division of Decimals | 186 |

22 | Percentages | 194 |

23 | Averages | 200 |

24 | Graphs | 205 |

25 | Calculation Using Vinculums | 213 |

26 | Geometry 3 - Angles | 218 |

27 | Practice and Revisions 3 | 226 |

Appendix - Tables of Weights and Measures | 231 | |

Answers | 235 |

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Heramba Ganapati arrived safely today and was shipped promptly. Another fantastic find from Exotic India with perfect customer service. Thank you. Jai Ganesha Deva

Marc, UK

I ordered Padmapani Statue. I have received my statue. The delivering process was very fast and the statue looks so beautiful. Thank you exoticindia, Mr. Vipin (customer care). I am very satisfied.

Hartono, Indonesia

Very easy to buy, great site! Thanks

Ilda, Brazil

Our Nandi sculpture arrived today and it surpasses all expectations - it is wonderful. We are not only pleasantly surprised by the speed of international delivery but also are extremely grateful for the care of your packaging. Our sculpture needed to travel to an off-lying island of New Zealand but it arrived safely because of how well it had been packaged. Based upon my experience of all aspects of your service, I have no hesitation in recommending Exotic India.

BWM, NZ

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