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This Manual is the first 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.
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 Cosmic Calculator course there are also many games that help to establish and promote confidence in the ideas used here.
Some topics will be found to be missing in this text: for example, there is no section on area, only a brief mention. 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).
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 By One More than the One Before and Vertically and Crosswise. 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.
PREFACE
LESSON 1: COMPLETING THE WHOLE
Introduction
The Ten Point Circle
Multiples of Ten
Deficiency from Ten
Deficiency and Completion Together
Mental Addition
Completing the Whole
Columns of Figures
By Addition and By Subtraction
Subtracting Numbers Near a Base
LESSON 2 : DOUBLING AND HALVING
Doubling
Multiplying by 4, 8
Halving
Splitting Numbers
Dividing by 4, 8
Extending your Tables
Multiplying by 5, 50, 25
Dividing by 5, 50, 25
Dividing by 5
Dividing by 50, 25
LESSON 3 : DIGIT SUMS
Adding Digits
The Nine Point Circle
Casting out Nines
Digit Sum Puzzles
More Digit Sum Puzzles
The Digit Sum Check
Multiplication Check
The Vedic square
Patterns from the Vedic Square
Number Nine
LESSON 4 : LEFT TO RIGHT
Addition: Left to Right
Multiplication: Left to Right
Doubling and Halving
Subtraction: Left to Right
Checking Subtraction Sums
More Subtractions
LESSON 5 : ALL FROM 9 AND THE LAST FROM 10
All From 9 and the Last from 10
Subtraction
Adding Zeros
One Less
One More
One Less Again
Money
LESSON 6 : NUMBER SPLITTING
Addition
Subtraction
Multiplication
Division
LESSON 7 : BASE MULTIPLICATION
Times Tables
Numbers just Over Ten
Multiplication Table Patterns
Recurring Decimals
Numbers Close to 100
Mentally
Numbers Over 100
Mental Maths
Russian Peasant Multiplication
Larger Numbers
Numbers Above the Base
Proportionately
Another Application of Proportionately
Multiplying Numbers near Different Bases
Squaring Numbers near a Base
A Summary
LESSON 8 : CHECKING AND DIVISIBILITY
Digit Sum Check for Division
The First by the First and the Last by the Last
The First by the First
The Last by the Last
Divisibility by 4
Divisibility by 11
Remainder after Division by 11
Another Digit Sum Check
LESSON 9 : BAR NUMBERS
Removing Bar Numbers
All from 9 and the Last from 10
Subtraction
Creating Bar Numbers
Using Bar Numbers
LESSON 10 : SPECIAL MULTIPLICATION
Multiplication by 11
Carries
Longer Numbers
By One More than the One Before
Multiplication by Nines
The First by the First and the Last by the Last
Using the Average
Special Numbers
Repeating Numbers
Proportionately
Disguises
LESSON 11 : GENERAL MULTIPLICATION
Revision
Two-Figure Numbers
Carries
Moving Multiplier
Extension
Multiplying Binomials
Multiplying 3-Figure Numbers
Written Calculations
LESSON 12 : SQUARING
Squaring Numbers that end in 5
Squaring Numbers Near 50
General Squaring
The Duplex
Number Splitting
Algebraic Squaring
Digit Sums of squares
Square Roots of Perfect Squares
3 and 4 Figure Numbers
LESSON 13 : EQUATIONS
One-step Equations
Two-step Equations
Three-step Equations
LESSON 14 : FRACTIONS
Vertically and Crosswise
A Simplification
Comparing Fractions
Unification of Operations
LESSON 15 : SPECIAL DIVISION
Division by 9
Longer Numbers
Carries
A Short Cut
Division by 8 etc.
Division by 99, 98 etc.
Divisor Below a Base Number
Two-Figure answers
Divisor Above a Base Number
LESSON 16 : THE CROWNING GEM
Single Figure on the Flag
Short Division Digression
Longer Numbers
Negative Flag Digits
Decimalising the Remainder
SUTRAS AND SUB-SUTRAS
9-POINT CIRCLES
REFERENCES
INDEX OF THE VEDIC FORMULAE
INDEX
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