Vedic Mathsindia

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Preface

A- A DESCRIPTIVE PREFATORY NOTE ON THE ASTOUNDING WONDERS OF ANCIENT INDIAN VEDIC MATHEMATICS

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.

VEDIC MATHEMATICS TEACHER'S MANUAL 3 ADVANCED LEVEL - 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
6.6 e   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
7.4 a   Left to Right
7.4 b    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

8.5 a Gradients
8.5 b Length of Perpendicular
8.5 c Circle Problems
8.5 d 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 ALGEBRAIC DIVISION

LESSON 11 : SQUARE ROOTS

11.1 Squaring
11.2 Square Root of a Perfect square
11.2 a Preamble
11.2 b Two-Figure Answer
            Reversing Squaring
11.2 c 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

12.1COMPOUND ANGLES
12.2INVERSE FUNCTIONS
12.3THE GENERAL TRIPLE
12.4 TRIGONOMETRIC EQUATIONS
12.4aSIMPLE EQUATIONS
12.4b A SPECIAL TYPE

LESSON 13 : COMBINED OPERATIONS
13.1 Algebraic 

13.2 Arithmetic
13.2 a SUMS OF PRODUCTS
13.2 b ADDITION AND DIVISION
13.2 c  STRAIGHT DIVISION
13.2 d MEAN AND MEAN DEVIATION
13.2 e DIVIDING SUMS OF PRODUCTS
13,2 f 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 x 2 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
INDEX