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 second of three (elementary, intermediate and advanced) Manuals which are designed for adults with a basic understanding of mathematics to learn or teach the Vedic system. So teachers could use it to learn Vedic Mathematics, though it is not suitable as a text for children (for that the Cosmic Calculator Course is recommended). Or it could be used to teach a course on Vedic Mathematics.

The sixteen 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.

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 (e.g. recurring decimals).

All techniques are fully explained and proofs are given where appropriate, 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 2 INTERMEDIATE LEVEL - Contents

PREFACE

LESSON 1 : Basic Devices

Introduction
Digit Sums
Left to Right
Addition
Multiplication
Advantages of left to Right Calcn
Writing Left to Right Sums
Checking Devices
Subtraction
Subtraction from Left to Right
Checking Subtraction Sums

LESSON 2 : MORE BASIC DEVICES

Number Splitting
Addition 14 / Subtraction
Multiplication 16 / Division
All from 9 and the Last from 10
Subtraction from a Base
Calculations Involving Money
First Extension
Second Extension
Combining the Extensions
Bar Numbers
Advantages of Bar Numbers
General Subtraction

LESSON 3 : SPECIAL METHODS

Proportionately
Doubling and Halving
Extending the Multiplication Tables
Multiplying by 5, 50, 25
All from 9 and the Last from 10: Multiplication
Numbers just below 100
Geometrical Proof
Algebraic Proof
Other Bases
Numbers above the Base
One Number ABOVE and one below the Base
Proportionately
Multiplying Numbers near Different Bases
Squaring Numbers near a Base
Mental Calculations
Special methods

LESSON 4 : BY ONE MORE THAN THE ONE BEFORE

Special Multiplications
Squaring Numbers that End in 5 br> A Variation
Multiplication Summary
Recurring Decimals
Denominator Ending in 9
Proof
A Short Cut
Proportionately
Longer Numerators

LESSON 5 : AUXILIARY FRACTIONS

Auxiliary Fractions: First Type
Denominators Ending in 8, 7, 6
Auxiliary Fractions: Second Type
Denominators Ending in 1
Alternative Method
Denominators Ending in 2, 3, 4
Working 2, 3 etc. Figures at a Time ;

LESSON 6 : VERTICALLY AND CROSSWISE

Fractions
Adding & Subtracting Fractions
Proof
A Simplification
Comparing Fractions
Multiplication and Division
General Multiplication
Revision
Multiplying 2-Figure Numbers
Explanation
Explanation of earlier special method
The Digit Sum Check
Multiplying 3-Figure Numbers
Moving Multiplier
Algebraic Multiplications
The Digit Sum Check
Three-Figure Numbers
Four-Figure Numbers
Writing Left to Right Sums
From Right to Left
setting the sums out
Using Bar Numbers

LESSON 7 : SQUARES AND SQUARE ROOTS

General Squaring
Two-Figure Numbers
Number Splitting
Algebraic Squaring
Squaring Longer Numbers
Written Calculations – Left to Right
Written Calculations – Right to Left
Square Roots of Perfect Squares

LESSON 8 : SPECIAL MULTIPLICATION METHODS

Special Numbers
Repeating Numbers
Proportionately
Disguises
Using the Average
PROOF
Multiplication by Nines
Multiplication by 11
Percentages
Increasing
Reducing

LESSON 9 : TRIPLES

Definitions
Triples for 45°, 30° and 60°
Triple Addition
Double Angle
Variations of 3,4,5
Quadrant Angles
Rotations

LESSON 10 : SPECIAL DIVISION

Division by Nine
Adding Digits
A Short Cut
Dividing by 8
Algebraic Division
Dividing by 11, 12 etc.
Larger Divisors
Divisor just Below a Base
A Simplification
Divisor just Above a Base
Proportionately

LESSON 11 : STRAIGHT DIVISION

Single Figure on the Flag
Short Division Digression
Longer Numbers
Multiplication Reversed
Decimalising the Remainder
Negative Flag Digits
Larger Divisors
ALGEBRAIC DIVISION

LESSON 12 : EQUATIONS

Linear
One x Term
Two x Terms
Quadratic Equations
Simultaneous Equations
By Addition and By Subtraction
A Special Type
General Method
Another Special Type

LESSON 13 : APPLICATIONS OF TRIPLES

Triple Subtraction br> Triple Geometry
Angle Between Two Lines
Half Angle
Coordinate Geometry
Gradients
Circle Problems
Length of Perpendicular

LESSON 14 : SQUARE ROOTS

Squaring
Square Root of a Perfect square
Preamble
Two-Figure Answer
Reversing Squaring
Three-Figure Answer
Reversing Squaring
General Square Roots
Changing the Divisor
Heuristic Proof

LESSON 15 : DIVISIBILITY

Elementary Parts
The Ekadhika
Osculation
Explanation
Testing Longer Numbers
Other Divisors
The Negative Osculator

LESSON 16 : COMBINED CALCULATIONS

Algebraic
Arithmetic
Pythagoras Theorem




SUTRAS AND SUB-SUTRAS

9-POINT CIRCLES

REFERENCES

INDEX OF THE VEDIC FORMULAE

INDEX