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## Description

This book is designed for teachers of students in grades 9 to 14. It shows how Vedic Mathematics can be used in a school course but does not cover all school topics (see contents). The book can be used for teachers who wish to learn the Vedic system or to teach courses on Vedic mathematics for this level. Non-teachers who have a background knowledge of mathematics may also find it appropriate (see contents).

Topics included: calculus, series, logs and exponentials, trigonometry (including solving trig equations, proving identities), solution of equations (special types, quadratics, cubics, transcendental), complex numbers, coordinate geometry, transformation geometry, Simple Harmonic Motion, projectile motion, forces in equilibrium, work & moments, etc.

## Reviews

*I have got the book - advanced teachers manual; it is really amazing. It explains each method so easily covering basic concepts and that too within minimal space. :)**Hats off to you.*

-Pallavi

## Preface

This Manual is the third of three self-contained Manuals (elementary, intermediate and advanced) and is designed for adults with a good understanding of basic mathematics to learn or teach the Vedic system. So teachers could use it to learn Vedic Mathematics, or it could be used to teach a course on Vedic Mathematics. It is suitable for teachers of children aged about 13 to 18 years.

The eighteen lessons of this course are based on a series of one week summer courses given at Oxford University by the author to Swedish mathematics teachers between 1990 and 1995. Those courses were quite intensive consisting of eighteen, one and a half hour, lessons. Some of the material here is more advanced than would be given to the average 18 year old student but this is what the teachers wanted on the courses and so the same is given here.

The lessons in this book however probably contain more material than could be given in a one and a half hour lesson. The teacher/reader may wish to omit some sections, go through the material in a different sequence to that shown here or break up some sections.

All techniques are fully explained and proofs and explanations are given, the relevant Sutras are indicated throughout (these are listed at the end of this Manual) and, for convenience, answers are given after each exercise. Cross-references are given showing what alternative topics may be continued with at certain points.

It should also be noted that the Vedic system encourages mental work so we always encourage students to work mentally as long as it is comfortable. In the Cosmic Calculator Course pupils are given a short mental test at the start of most or all lessons, which makes a good start to the lesson, revises previous work and introduces some of the ideas needed in the current lesson. In the Vedic system pupils are encouraged to be creative and use whatever method they like.

Some topics will not be found in this text: for example, there is no section on area and volume. This is because the actual methods are the same as currently taught so that the only difference would be to give the relevant Sutra(s).

## Introduction

Vedic Mathematics is an ancient system of mathematics which was rediscovered early last century by Sri Bharati Krsna Tirthaji (henceforth referred to as Bharati Krsna).

The Sanskrit word “veda” means “knowledge”. The Vedas are ancient writings whose date is disputed but which date from at least several centuries BC. According to Indian tradition the content of the Vedas was known long before writing was invented and was freely available to everyone. It was passed on by word of mouth. The writings called the Vedas consist of a huge number of documents (there are said to be millions of such documents in India, many of which have not yet been translated) and these have recently been shown to be highly structured, both within themselves and in relation to each other (see Reference 2). Subjects covered in the Vedas include Grammar, Astronomy, Architecture, Psychology, Philosophy, Archery etc., etc.

A hundred years ago Sanskrit scholars were translating the Vedic documents and were surprised at the depth and breadth of knowledge contained in them. But some documents headed “Ganita Sutras”, which means mathematics, could not be interpreted by them in terms of mathematics. One verse, for example, said “in the reign of King Kamse famine, pestilence and unsanitary conditions prevailed”. This is not mathematics they said, but nonsense.

Bharati Krsna was born in 1884 and died in 1960. He was a brilliant student, obtaining the highest honours in all the subjects he studied, including Sanskrit, Philosophy, English, Mathematics, History and Science. When he heard what the European scholars were saying about the parts of the Vedas which were supposed to contain mathematics he resolved to study the documents and find their meaning. Between 1911 and 1918 he was able to reconstruct the ancient system of mathematics which we now call Vedic Mathematics.

He wrote sixteen books expounding this system, but unfortunately these have been lost and when the loss was confirmed in 1958 Bharati Krsna wrote a single introductory book entitled “Vedic Mathematics”. This is currently available and is a best-seller (see Reference 1).

There are many special aspects and features of Vedic Mathematics which are better discussed as we go along rather than now because you will need to see the system in action to appreciate it fully. But the main points for now are:

1) The system rediscovered by Bharati Krsna is based on sixteen formulae (or Sutras) and some sub-formulae (sub-Sutras). These Sutras are given in word form: for example Vertically and Crosswise and By One More than the One Before. In this text they are indicated by italics. These Sutras can be related to natural mental functions such as completing a whole, noticing analogies, generalisation and so on.

2) Not only does the system give many striking general and special methods, previously unknown to modern mathematics, but it is far more coherent and integrated as a system.

3) Vedic Mathematics is a system of mental mathematics (though it can also be written down).

Many of the Vedic methods are new, simple and striking. They are also beautifully interrelated so that division, for example, can be seen as an easy reversal of the simple multiplication method (similarly with squaring and square roots). This is in complete contrast to the modern system. Because the Vedic methods are so different to the conventional methods, and also to gain familiarity with the Vedic system, it is best to practice the techniques as you go along.

## Contents

PREFACE **LESSON 1 LEFT TO RIGHT CALCULATIONS**

1.1 INTRODUCTION

1.2 ADDITION

1.3 MULTIPLICATION

Advantages OF LEFT TO RIGHT CALCULATION

1.4 WRITING LEFT TO RIGHT SUMS

1.5 SUBTRACTION

1.6 DIGIT SUMS

1.7 CHECKING DEVICES

CHECKING SUBTRACTION SUMS

1.8 ALL FROM 9 AND THE LAST FROM 10

1.8a SUBTRACTION FROM A BASE

1.8b BAR NUMBERS

ADVANTAGES OF BAR NUMBERS

1.8c GENERAL SUBTRACTION

**LESSON 2 SPECIAL METHODS**

2.1 MULTIPLICATION NEAR A BASE

2.1a Numbers just below the base

2.1.b Above the base

2.1c Above and below

2.1d Proportionately

2.1e With different bases

2.2 MENTAL CALCULATIONS

2.3 SPECIAL NUMBERS

2.3a Repeating numbers

2.3b Proportionately

2.3c Disguises

2.4 DIVISION BY NINE

2.4a Adding Digits

2.4b A Short Cut

2.4c Dividing by 8

2.4d Algebraic Division

2.4e Dividing by 11, 12 etc.

**LESSON 3 RECURRING DECIMALS**

3.1 DENOMINATOR ENDING IN 9

3.2 A SHORT CUT

3.3 PROPORTIONATELY

3.4 LONGER NUMERATORS

3.5 DENOMINATORS ENDING IN 8, 7, 6

3.6 DENOMINATORS ENDING IN 1

3.7 DENOMINATORS ENDING IN 2, 3, 4

3.8 WORKING 2, 3 ETC. FIGURES AT A TIME

**LESSON 4 TRIPLES**

4.1 Definitions

4.2 Triples for 45°, 30° and 60°

4.3 Triple Addition

4.4 Double Angle

4.5 Variations of 3,4,5

4.6 Quadrant Angles

4.7 Rotations

**LESSON 5 GENERAL MULTIPLICATION**

5.1 TWO-Figure Numbers

Explanation/ The Digit Sum Check

5.2 Moving Multiplier

5.3 Algebraic PRODUCTS

The Digit Sum Check

5.4 Three-Figure Numbers

5.5 Four-Figure Numbers

5.6 Writing Left to Right Sums

5.7 From Right to Left

setting the sums out

5.8 Using Bar Numbers

**LESSON 6 SOLUTION OF EQUATIONS **

6.1 TRANSPOSE AND APPLY

6.1a SIMPLE EQUATIONS

6.1b MORE THAN ONE X TERM

6.2 SIMULTANEOUS EQUATIONS

6.2a GENERAL SOLUTION

6.2b Special Types

6.3 QUADRATIC EQUATIONS

6.4 ONE IN RATIO THE OTHER ONE ZERO

6.5 MERGERS

6.6 WHEN THE SAMUCCAYA IS THE SAME IT IS ZERO

6.6a Samuccaya as a common factor

6.6b Samuccaya as the Product of the Independent Terms

6.6c Samuccaya as the Sum of the Denominators

6.6d Samuccaya as a Combination or Total

Proof/ Extension

6.6e other types

6.7 THE ULTIMATE AND TWICE THE PENULTIMATE

6.8 ONLY THE LAST TERMS

6.9 SUMMATION OF SERIES

6.10 FACTORISATION

**LESSON 7 SQUARES AND SQUARE ROOTS**

7.1 Squaring 2-FIGURE NUMBERS

7.2 Algebraic Squaring

7.3 Squaring Longer Numbers

7.4 Written Calculations

7.4a Left to Right

7.4b Right to Left

7.5 Square Roots of Perfect Squares

**LESSON 8 APPLICATIONS OF TRIPLES**

8.1 Triple Subtraction

8.2 Triple Geometry

8.3 Angle Between Two Lines

8.4 Half Angle

8.5 Coordinate Geometry

8.5a Gradients

8.5b Length of Perpendicular

8.5c Circle Problems

8.5d Equation of a Line

8.6 Complex Numbers

**LESSON 9 DIVISIBILITY**

9.1 Elementary Parts

9.2 The Ekadhika

9.3 Osculation

Explanation

9.4 Testing Longer Numbers

9.5 Other Divisors

9.6 The Negative Osculator

9.7 OSCULATING WITH GROUPS OF DIGITS

**LESSON 10 STRAIGHT DIVISION**

10.1 Single Figure on the Flag

10.2 Short Division Digression

10.3 Longer Numbers

Multiplication Reversed

10.4 Decimalising the Remainder

10.5 Negative Flag Digits

10.6 Larger Divisors

10.7 Alebraic Division

**LESSON 11 SQUARE ROOTS**

11.1 Squaring

11.2 Square Root of a Perfect square

11.2a Preamble

11.2b Two-Figure Answer

Reversing Squaring

11.2c Three-Figure Answer

Reversing Squaring

11.3 General Square Roots

11.4 Changing the Divisor

Heuristic Proof

11.5 Algebraic Square Roots

**LESSON 12 TRIPLE TRIGONOMETRY**

12.1 COMPOUND ANGLES

12.2 INVERSE FUNCTIONS

12.3 THE GENERAL TRIPLE

12.4 TRIGONOMETRIC EQUATIONS

12.4a SIMPLE EQUATIONS

12.4b A SPECIAL TYPE

**LESSON 13 COMBINED OPERATIONS**

13.1 Algebraic

13.2 Arithmetic

13.2a SUMS OF PRODUCTS

13.2b ADDITION AND DIVISION

13.2c STRAIGHT DIVISION

13.2d MEAN AND MEAN DEVIATION

13.2e DIVIDING SUMS OF PRODUCTS

13,2f VARIANCE

13.3 Pythagoras’ Theorem

**LESSON 14 SOLUTION OF POLYNOMIAL EQUATIONS**

14.1 QUADRATIC EQUATIONS

14.1a x > 1

PROOF

14.1b x < –1

14.1c 0 < x < 1

14.1d 0 < x < 1 and x2 Coefficient > 1

14.1e –1 < x < 0

14.1f x LARGE

14.2 HIGHER ORDER EQUATIONS

14.2a CUBE ROOT

14.2b CUBIC EQUATIONS

A SIMPLIFICATION

A CUBIC WITH 0 < x < 1

14.2c QUINTICS

**LESSON 15 CALCULUS METHODS**

15.1 Partial Fractions

15.2 Integration by ‘Parts’

TRUNCATING

15.3 BINOMIAL AND MACLAURIN THEOREMS

15.4 Derivatives of a Product

15.5 Derivative of A Quotient

15.6 Differential Equations – 1

15.7 Differential Equations – 2

15.8 Limits 220

**LESSON 16 APPLIED MATHEMATICS**

16.1 SIMPLE HARMONIC MOTION

16.2 PROJECTILES

16.3 FORCES IN EQUILIBRIUM

16.4 WORK AND MOMENT

**LESSON 17 TRIGONOMETRIC FUNCTIONS**

17.1 DERIVATIVES

17.2 SERIES EXPANSIONS

17.3 INVERSE TRIGONOMETRIC FUNCTIONS

17.3a DERIVATIVES

17.3b SERIES

17.4 EVALUATING TRIGONOMETRIC FUNCTIONS

17.4a COSINE

17.4b SINE

17.4c INVERSE TANGENT

**LESSON 18 TRIGONOMETRIC AND TRANSCENDENTAL EQUATIONS**

18.1 POLYNOMIAL EQUATIONS

18.2 TRIGONOMETRIC EQUATIONS

18.3 TRANSCENDENTAL EQUATIONS

KEPLER’S EQUATION

**REFERENCES ****SUTRAS AND SUB-SUTRAS****INDEX OF THE VEDIC FORMULAE ****INDEX**

## Back Cover

¯ Vedic Mathematics was reconstructed from ancient Vedic texts early last century by Sri Bharati Krsna Tirthaji (1884-1960). It is a complete system of mathematics which has many surprising properties and applies at all levels and areas of mathematics, pure and applied.

¯ It has a remarkable coherence and simplicity that make it easy to do and easy to understand. Through its amazingly easy methods complex problems can often be solved in one line.

¯ The system is based on sixteen word-formulae (Sutras) that relate to the way in which we use our mind.

¯ The benefits of using Vedic Mathematics include more enjoyment of maths, increased flexibility, creativity and confidence, improved memory, greater mental agility and so on.

This Advanced Manual is the third of three self-contained Manuals designed for teachers who wish to teach the Vedic system, either to a class or to other adults/teachers. It is also suitable for anyone who would like to teach themselves the Vedic methods.