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The Curious Hats of Magical Maths

by J.T. Glover

Description

Vedic Mathematics for School offers a fresh and easy approach to learning mathematics. The system was reconstructed from ancient Vedic Sources by the late Bharati Krsna Tirthaji earlier this century and is based on a small collection of sutras. Each Sutra briefly encapsulates a rule of mental working, a principle or guiding maximum. Through simple practice of these methods all may become adept and efficient at mathematics.

Book 1 of the series is intended for primary schools in which many of the fundamental concepts of mathematics are introduced. It has been written from the classroom experience of teachings Vedic methods are used, the rest being introduced at a later stage.

James T. Glover is head of mathematics at St. James Independent School in London. He is director of mathematical studies at the school of Economic 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 for Schools, Book 1 and Foundation Mathematics, Books 1,2 and 3.

“The examples and exercises have been arranged with care and the grading of the latter shows every evidence of the same pedagogic thoroughness. The exercises have been beautifully structured, presenting a mixed bag of fairly straight forword stuff, with may stimulating questions going well beyond the merely routine. For this reason, if no other, the book deserves to serve as a model for text books at this level, in use in this country.”

Vedic Mathematics for Schools is an exceptional book. It is not only a sophisticated pedagogic tool but also introduction to an ancient civilization. It takes us back to many millennia of India’s mathematical heritage. Rooted in the ancient Vedic sources which heralded the dawn of human history and illumined by their erudite exegeses, India’s intellectual, scientific and aesthetic vitality blossomed and triumphed not only in philosophy, physics, ecology and performing 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 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 Arabs first learnt the decimal notation from Indians and introduced it into Europe. The 973 A.D. and travelled to India, testified that the Indian attainments in mathematics were unrivalled and unsurpassed. In Keeping with that ingrained tradition of mathematics in India, S. Ramanujan, “the man who knew infinity”, the genius who was one of the greatest mathematical trails in Cambridge University in the second decade of the twentieth century even though he this not himself possess a university degree.

The real contribution of this book, Vedic Mathematics for Schools, is to demonstrate that Vedic mathematics belongs not only to an hoary 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 from the pioneering work of the late Bharati Krishna Tirthaji, a former Sankarcharya of Puri, who reconstructed a unique system on the basis of ancient Indian mathematics. The book is thus a bridge across centuries, civilizations, linguistic barriers and national frontiers.

Vedic mathematics was traditionally taught aphorisms or sutras. A Sutra is a thread of Knowledge, a theorem, a ground norm, a repository of proof. It is formulated as a proposition to encapsulate a rule or a principle. A single Sutra would encompass a wide and varied range of particular applications and may be linked to a programmed chip of our computer age. These aphorisms of Vedic mathematics have much in common with aphorisms which are contained in Panin’s Ashtadhyayi, that grand edifice of Sanskrit grammar, Both Vedic mathematics and Sanskrit grammar are built on the foundations of rigorous logic and on a deep understanding of how the human mind works. The methodology of Vedic mathematics and of Sanskrit grammar help to hope the human intellect and to guide and groom the human mind into modles 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 mathematics and Sanskrit may eventually be emulated Indian school.

Vedic mathematics is a new and unique system based on simple rules and principles which enable mathematical problems of all kinds to be solved easily and efficiently. The methods and techniques are based on the pioneering work of the late Bharati Krishna Tirthaji, Sankarcarya of Puri, who estabilished the system from the study of ancient Vedic texts coupled with a profound insight into the natural processes of mathematical reasoning.

The characteristics of Vedic mathematics is to present the subject as a unified body of knowledge and so reduce the burden and toil which young students often experience during their studies. It is based on sixteen principles which lie behind short rules of working. Or aphorisms, which are easily remembered. In the Vedic system these aphorisms are called sutras, simple terse statements expressing rules, definations or governing principles. In some topics, the sutras provide rues for special cases as well as for the general case. Understanding their nature and scope is achieved by the practice of their applications.

Experience of teaching the Vedic 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.

This book should be taken as an introductory volume. Many of the methods are developed further at a later stage and so, in the present text, it may not be apparent why a particular method is being given. An important characteristic is that, although there are general methods for calculations and algebraic manipulations, there are also methods for particular methods are introduced at an early stage it is because they relate to more general aspects of the system at a later stage or are simply very quick and easy ways to obtain answers.

The current methods of calculating which have been adopted by most schools are ‘blanket’ methods. For example, with division, only one methods is taught and actually used by the children. Although it will suffice in all cases it may often be difficult to use. The vedic system teachers three basic algorithms for division which are applied sum. The principle is that, if a particular sum can be done by an easier method, then that method should be used. Of course, with children, some mastery of the different methods must be accomplished before this more creative approach can be adopted. A simple example to illustrate this point is the method for finding the product of 19 and 7. The conventional system teaches us to multiply the 7 by 9, to get 63 and then to multiply 7 by 10 to get 70. On summing these we arrive at the answer of 133. Bright children will arrive at this method for themselves but the Vedic mathematics teaches this sort approach systematically.

The study of number begins at one which is an expression of unity. From here all the other numbers arise and if it were not for the number one we would not have any numbers at all. If there is any fear of large numbers it is always comforting to remember that there are really only nine together with nought which stands for nothing. All other numbers are just repetitions of these nine. It is useful to treat these nine numbers as friends. In fact , they are universal friends because everybody uses them every day in one way or another.

Vedic mathematics readily acknowledges the importance of the number one. Many calculations are made simple and easy by relating the numbers involved back to one. The very first sutra or formula in Vedic Mathematics does just this. It realtes every number to unity.

In Vedic mathematics there are sixteen or formulae and about thirteen subsutras. The word sutra (pronounced ‘sootra’) is from ancient India and means a thread of knowledge. The English word ‘suture’ comes from sutra and a suture is short and simple statements which give formulae for how to answer mathematical problems. Each sutra has a large number of uses at all levels of mathematics.

In the research work which has resulted in this course there have been two guiding maxims. The first is that there are only nine numbers, together with a nought, and that these numbers represent the nine Elements as described in the ancient scriptural texts of India. It is well known that the nine numerals and the nought originated in India but the philosophical tradition to the Hindus also ascribes a universal significance to each of the numbers. The second is that the whole of mathematics is governed by the sixteen sutras, or short formula-like aphorisms, which are both objective and subjective in their character. They are objective in that they may be applied to solve everyday problems. The subjective aspect is that a sutra may also describe the way the human mind naturally works. The whole emphasis of the system is on the process and movement taking place in the mind at the time that a problem is being solved. The effect of this is to bring the attention into the present moment.

Vedic Mathematics for Schools Book 1 is a first text designed for the young mathematics student of about eight years of age. The text introduces new and quick methods in numerical calculation and comprehension.

New algorithms used for numerical calculations are introduced and exercises are carefully graded to enable the distinct development steps of each methods to be mastered. Each algorithm is denoted by a simple rule which, when applied and practiced, provides a high standard of mathematical capability. The text incorporates explanations and worked examples of all the methods used and includes description of how to set out written work.

The course has been written for children who, at the age of about eight, have mastered the basics four rules including times tables. Although this is assumed, it is also clear that at this stage the stage the child needs a good deal of revision work in the basics as an on-going practice and this has been taken into account in the composing of exercises. Older children and even adults may also find the techniques interesting and useful. The text provides introductory steps to each Vedic algorithm which may be followed by pupils of the intended age level with some help from an adult. The main emphasis at this stage is on developing numeracy which is the most essential aspect of mathematics. The text concentrates on these areas of mathematics and treats them as the core curriculum of the subject the main Vedic methods used in this book are those for multiplication, division and subtractions to vulgar and decimal fractions, elementary algebra and vinculums are are also given. Topics in geometry, weights and measures and statistics are not included in this text.

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 practice so that they may develop their own understanding. It is also felt that teachers might provide their own understanding. It is also felt that teachers might provide their own practical ways of demonstrating this system or of enabling children to practice and experience the various methods and concepts.

It is assumed that pupils using this book already have a degree of mathematical ability. In particular, the times tables need to be fully estabilished. The Vedic system relies on and develops mental capabilities and many of the answers to questions are obtained only one line. This reliance is greatly aided by regular practice of mental arithmetic.

Only five of the sixteen sutras and thirteen sub-sutras are used in this book others will be introduced in later volumes.

1 All from nine and the last from ten

Nikhilam Navatascaramam Dasatah

2 Vertically and crosswise

Urdhva Tiryagbhyam

3 Transpose and Adjust

Paravartya Yojayet

4 By Elimination and Retention

Lopana Sthapanabhyam

5 By one more than the one before

Ekadhikena Purvena

Preface by His Excellency Dr. L.M. Singhvi, High commissioner for India in the UK | v |

Introduction | vii |

Chapter 1 Simple Practice of number | 1 |

Chapter 2 Multiplication by Nikhilam | 9 |

Chapter 3 Division | 19 |

Chapter 4 Digital Roots | 27 |

Chapter 5 Multiplication by Vertically and Crosswise | 32 |

Chapter 6 Subtraction by Nikhilam | 39 |

Chapter 7 Vulgar Fractions | 46 |

Chapter 8 Decimal Fractions | 58 |

Chapter 9 The Meaning of Number | 72 |

Chapter 10 vinculums | 83 |

Chapter 11 Algebra | 91 |

Answers | 101 |

Item Code:

NAG037

Cover:

Paperback

Edition:

2015

Publisher:

ISBN:

9788120839731

Language:

English

Size:

8.5 inch X 5.5 inch

Pages:

145

Other Details:

Weight of the Book: 435 gms

Vedic Mathematics for School offers a fresh and easy approach to learning mathematics. The system was reconstructed from ancient Vedic Sources by the late Bharati Krsna Tirthaji earlier this century and is based on a small collection of sutras. Each Sutra briefly encapsulates a rule of mental working, a principle or guiding maximum. Through simple practice of these methods all may become adept and efficient at mathematics.

Book 1 of the series is intended for primary schools in which many of the fundamental concepts of mathematics are introduced. It has been written from the classroom experience of teachings Vedic methods are used, the rest being introduced at a later stage.

James T. Glover is head of mathematics at St. James Independent School in London. He is director of mathematical studies at the school of Economic 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 for Schools, Book 1 and Foundation Mathematics, Books 1,2 and 3.

“The examples and exercises have been arranged with care and the grading of the latter shows every evidence of the same pedagogic thoroughness. The exercises have been beautifully structured, presenting a mixed bag of fairly straight forword stuff, with may stimulating questions going well beyond the merely routine. For this reason, if no other, the book deserves to serve as a model for text books at this level, in use in this country.”

Vedic Mathematics for Schools is an exceptional book. It is not only a sophisticated pedagogic tool but also introduction to an ancient civilization. It takes us back to many millennia of India’s mathematical heritage. Rooted in the ancient Vedic sources which heralded the dawn of human history and illumined by their erudite exegeses, India’s intellectual, scientific and aesthetic vitality blossomed and triumphed not only in philosophy, physics, ecology and performing 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 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 Arabs first learnt the decimal notation from Indians and introduced it into Europe. The 973 A.D. and travelled to India, testified that the Indian attainments in mathematics were unrivalled and unsurpassed. In Keeping with that ingrained tradition of mathematics in India, S. Ramanujan, “the man who knew infinity”, the genius who was one of the greatest mathematical trails in Cambridge University in the second decade of the twentieth century even though he this not himself possess a university degree.

The real contribution of this book, Vedic Mathematics for Schools, is to demonstrate that Vedic mathematics belongs not only to an hoary 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 from the pioneering work of the late Bharati Krishna Tirthaji, a former Sankarcharya of Puri, who reconstructed a unique system on the basis of ancient Indian mathematics. The book is thus a bridge across centuries, civilizations, linguistic barriers and national frontiers.

Vedic mathematics was traditionally taught aphorisms or sutras. A Sutra is a thread of Knowledge, a theorem, a ground norm, a repository of proof. It is formulated as a proposition to encapsulate a rule or a principle. A single Sutra would encompass a wide and varied range of particular applications and may be linked to a programmed chip of our computer age. These aphorisms of Vedic mathematics have much in common with aphorisms which are contained in Panin’s Ashtadhyayi, that grand edifice of Sanskrit grammar, Both Vedic mathematics and Sanskrit grammar are built on the foundations of rigorous logic and on a deep understanding of how the human mind works. The methodology of Vedic mathematics and of Sanskrit grammar help to hope the human intellect and to guide and groom the human mind into modles 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 mathematics and Sanskrit may eventually be emulated Indian school.

Vedic mathematics is a new and unique system based on simple rules and principles which enable mathematical problems of all kinds to be solved easily and efficiently. The methods and techniques are based on the pioneering work of the late Bharati Krishna Tirthaji, Sankarcarya of Puri, who estabilished the system from the study of ancient Vedic texts coupled with a profound insight into the natural processes of mathematical reasoning.

The characteristics of Vedic mathematics is to present the subject as a unified body of knowledge and so reduce the burden and toil which young students often experience during their studies. It is based on sixteen principles which lie behind short rules of working. Or aphorisms, which are easily remembered. In the Vedic system these aphorisms are called sutras, simple terse statements expressing rules, definations or governing principles. In some topics, the sutras provide rues for special cases as well as for the general case. Understanding their nature and scope is achieved by the practice of their applications.

Experience of teaching the Vedic 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.

This book should be taken as an introductory volume. Many of the methods are developed further at a later stage and so, in the present text, it may not be apparent why a particular method is being given. An important characteristic is that, although there are general methods for calculations and algebraic manipulations, there are also methods for particular methods are introduced at an early stage it is because they relate to more general aspects of the system at a later stage or are simply very quick and easy ways to obtain answers.

The current methods of calculating which have been adopted by most schools are ‘blanket’ methods. For example, with division, only one methods is taught and actually used by the children. Although it will suffice in all cases it may often be difficult to use. The vedic system teachers three basic algorithms for division which are applied sum. The principle is that, if a particular sum can be done by an easier method, then that method should be used. Of course, with children, some mastery of the different methods must be accomplished before this more creative approach can be adopted. A simple example to illustrate this point is the method for finding the product of 19 and 7. The conventional system teaches us to multiply the 7 by 9, to get 63 and then to multiply 7 by 10 to get 70. On summing these we arrive at the answer of 133. Bright children will arrive at this method for themselves but the Vedic mathematics teaches this sort approach systematically.

The study of number begins at one which is an expression of unity. From here all the other numbers arise and if it were not for the number one we would not have any numbers at all. If there is any fear of large numbers it is always comforting to remember that there are really only nine together with nought which stands for nothing. All other numbers are just repetitions of these nine. It is useful to treat these nine numbers as friends. In fact , they are universal friends because everybody uses them every day in one way or another.

Vedic mathematics readily acknowledges the importance of the number one. Many calculations are made simple and easy by relating the numbers involved back to one. The very first sutra or formula in Vedic Mathematics does just this. It realtes every number to unity.

In Vedic mathematics there are sixteen or formulae and about thirteen subsutras. The word sutra (pronounced ‘sootra’) is from ancient India and means a thread of knowledge. The English word ‘suture’ comes from sutra and a suture is short and simple statements which give formulae for how to answer mathematical problems. Each sutra has a large number of uses at all levels of mathematics.

In the research work which has resulted in this course there have been two guiding maxims. The first is that there are only nine numbers, together with a nought, and that these numbers represent the nine Elements as described in the ancient scriptural texts of India. It is well known that the nine numerals and the nought originated in India but the philosophical tradition to the Hindus also ascribes a universal significance to each of the numbers. The second is that the whole of mathematics is governed by the sixteen sutras, or short formula-like aphorisms, which are both objective and subjective in their character. They are objective in that they may be applied to solve everyday problems. The subjective aspect is that a sutra may also describe the way the human mind naturally works. The whole emphasis of the system is on the process and movement taking place in the mind at the time that a problem is being solved. The effect of this is to bring the attention into the present moment.

Vedic Mathematics for Schools Book 1 is a first text designed for the young mathematics student of about eight years of age. The text introduces new and quick methods in numerical calculation and comprehension.

New algorithms used for numerical calculations are introduced and exercises are carefully graded to enable the distinct development steps of each methods to be mastered. Each algorithm is denoted by a simple rule which, when applied and practiced, provides a high standard of mathematical capability. The text incorporates explanations and worked examples of all the methods used and includes description of how to set out written work.

The course has been written for children who, at the age of about eight, have mastered the basics four rules including times tables. Although this is assumed, it is also clear that at this stage the stage the child needs a good deal of revision work in the basics as an on-going practice and this has been taken into account in the composing of exercises. Older children and even adults may also find the techniques interesting and useful. The text provides introductory steps to each Vedic algorithm which may be followed by pupils of the intended age level with some help from an adult. The main emphasis at this stage is on developing numeracy which is the most essential aspect of mathematics. The text concentrates on these areas of mathematics and treats them as the core curriculum of the subject the main Vedic methods used in this book are those for multiplication, division and subtractions to vulgar and decimal fractions, elementary algebra and vinculums are are also given. Topics in geometry, weights and measures and statistics are not included in this text.

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 practice so that they may develop their own understanding. It is also felt that teachers might provide their own understanding. It is also felt that teachers might provide their own practical ways of demonstrating this system or of enabling children to practice and experience the various methods and concepts.

It is assumed that pupils using this book already have a degree of mathematical ability. In particular, the times tables need to be fully estabilished. The Vedic system relies on and develops mental capabilities and many of the answers to questions are obtained only one line. This reliance is greatly aided by regular practice of mental arithmetic.

Only five of the sixteen sutras and thirteen sub-sutras are used in this book others will be introduced in later volumes.

1 All from nine and the last from ten

Nikhilam Navatascaramam Dasatah

2 Vertically and crosswise

Urdhva Tiryagbhyam

3 Transpose and Adjust

Paravartya Yojayet

4 By Elimination and Retention

Lopana Sthapanabhyam

5 By one more than the one before

Ekadhikena Purvena

Preface by His Excellency Dr. L.M. Singhvi, High commissioner for India in the UK | v |

Introduction | vii |

Chapter 1 Simple Practice of number | 1 |

Chapter 2 Multiplication by Nikhilam | 9 |

Chapter 3 Division | 19 |

Chapter 4 Digital Roots | 27 |

Chapter 5 Multiplication by Vertically and Crosswise | 32 |

Chapter 6 Subtraction by Nikhilam | 39 |

Chapter 7 Vulgar Fractions | 46 |

Chapter 8 Decimal Fractions | 58 |

Chapter 9 The Meaning of Number | 72 |

Chapter 10 vinculums | 83 |

Chapter 11 Algebra | 91 |

Answers | 101 |

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