Vedic
Mathematics
(Preface, introduction and foreward from
the original book on Vedic Mathematics)
Revered Guruji used to say that he had reconstructed the sixteen
mathematical formulae from the Atharvaveda after assiduous research and ‘Tapas’
(austerity) for about eight years in the forests surrounding Sringeri.
Obviously these formulae are not to be found in the present recensions of
Atharvaveda. They were actually reconstructed, on the basis of intuitive
revelation, from materials scattered here and there in the Atharvaveda.
History of Mathematics in India
Vedic Mathematics is the name
given to the ancient system of Mathematics which was rediscovered from the
Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960).
According to his research all of mathematics is based on sixteen Sutras or
word-formulae. For example, 'Vertically and Crosswise` is one of these Sutras.
These formulae describe the way the mind naturally works and are therefore a
great help in directing the student to the appropriate method of solution.
Perhaps the most striking feature
of the Vedic system is its coherence. Instead of a hotch-potch of unrelated
techniques the whole system is beautifully interrelated and unified: the
general multiplication method, for example, is easily reversed to allow
one-line divisions and the simple squaring method can be reversed to give
one-line square roots. And these are all easily understood. This unifying
quality is very satisfying, it makes mathematics easy and enjoyable and
encourages innovation.
In the Vedic system 'difficult'
problems or huge sums can often be solved immediately by the Vedic method.
These striking and beautiful methods are just a part of a complete system of
mathematics which is far more systematic than the modern 'system'. Vedic
Mathematics manifests the coherent and unified structure of mathematics and the
methods are complementary, direct and easy.
The simplicity of Vedic
Mathematics means that calculations can be carried out mentally (though the
methods can also be written down). There are many advantages in using a
flexible, mental system. Pupils can invent their own methods, they are not
limited to the one 'correct' method. This leads to more creative, interested
and intelligent pupils.
Interest in the Vedic system is
growing in education where mathematics teachers are looking for something
better and finding the Vedic system is the answer. Research is being carried
out in many areas including the effects of learning Vedic Maths on children;
developing new, powerful but easy applications of the Vedic Sutras in geometry,
calculus, computing etc.
But the real beauty and
effectiveness of Vedic Mathematics cannot be fully appreciated without actually
practising the system. One can then see that it is perhaps the most refined and
efficient mathematical system possible.
The
Vedic Mathematics Sutras
This list of sutras is taken from
the book Vedic Mathematics, which includes a full list of the sixteen Sutras in
Sanskrit, but in some cases a translation of the Sanskrit is not given in the
text and comes from elsewhere.
This formula 'On the Flag' is not in the list given in Vedic
Mathematics, but is referred to in the text.
The
Main Sutras
By one more than the one before.
|
All from 9 and the last from 10.
|
Vertically and Cross-wise
|
Transpose and Apply
|
If the Samuccaya is the Same it is Zero
|
If One is in Ratio the Other is Zero
|
By Addition and by Subtraction
|
By the Completion or Non-Completion
|
Differential Calculus
|
By the Deficiency
|
Specific and General
|
The Remainders by the Last Digit
|
The Ultimate and Twice the Penultimate
|
By One Less than the One Before
|
The Product of the Sum
|
All the Multipliers
|
The Sub
Sutras
Proportionately
|
The
Remainder Remains Constant
|
The
First by the First and the Last by the Last
|
For
7 the Multiplicand is 143
|
By
Osculation
|
Lessen
by the Deficiency
|
Whatever
the Deficiency lessen by that amount and
set up the Square of the Deficiency
|
Last
Totalling 10
|
Only
the Last Terms
|
The
Sum of the Products
|
By
Alternative Elimination and Retention
|
By
Mere Observation
|
The
Product of the Sum is the Sum of the Products
|
On
the Flag
|
Try a
Sutra
Mark
Gaskell introduces an alternative system of calculation
based on Vedic philosophy
At
the Maharishi School
in Lancashire we have developed a course on
Vedic mathematics for key stage 3 that covers the national curriculum. The
results have been impressive: maths lessons are much livelier and more fun, the
children enjoy their work more and expectations of what is possible are very
much higher. Academic performance has also greatly improved: the first class to
complete the course managed to pass their GCSE a year early and all obtained an
A grade.
Vedic
maths comes from the Vedic tradition of India. The Vedas are the most
ancient record of human experience and knowledge, passed down orally for
generations and written down about 5,000 years ago. Medicine, architecture,
astronomy and many other branches of knowledge, including maths, are dealt with
in the texts. Perhaps it is not surprising that the country credited with
introducing our current number system and the invention of perhaps the most
important mathematical symbol, 0, may have more to offer in the field of maths.
The
remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts
early last century. The system is based on 16 sutras or aphorisms, such as:
"by one more than the one before" and "all from nine and the
last from 10". These describe natural processes in the mind and ways of
solving a whole range of mathematical problems. For example, if we wished to
subtract 564 from 1,000 we simply apply the sutra "all from nine and the last
from 10". Each figure in 564 is subtracted from nine and the last figure
is subtracted from 10, yielding 436.
1,000 - 564
= 436
1,000 - 5 6 4
subtract subtract subtract
from from from
9 9 10
¯ ¯ ¯
4
3 6
This
can easily be extended to solve problems such as 3,000 minus 467. We simply
reduce the first figure in 3,000 by one and then apply the sutra, to get the
answer 2,533. We have had a lot of fun with this type of sum, particularly when
dealing with money examples, such as £10 take away £2. 36. Many of the children
have described how they have challenged their parents to races at home using
many of the Vedic techniques - and won. This particular method can also be
expanded into a general method, dealing with any subtraction sum.
The
sutra "vertically and crosswise" has many uses. One very useful
application is helping children who are having trouble with their tables above
5x5. For example 7x8. 7 is 3 below the base of 10, and 8 is 2 below the base of
10.
7 x
8 = 56
7 3
(3 is the difference from base)
8 2
_________
A 7
3 starting at the left subtract
crosswise either 8-3 or
8
2 7-2 to get 5, the
first figure
__________ of the
answer
5
B 7
3
Multiply vertically
x to get 6 (3 x 2)
8
2
__________
5
6
The
whole approach of Vedic maths is suitable for slow learners, as it is so simple
and easy to use.
The
sutra "vertically and crosswise" is often used in long
multiplication. Suppose we wish to multiply 32 by 44. We multiply vertically
2x4=8. Then we multiply crosswise and add the two results: 3x4+4x2=20, so put
down 0 and carry 2. Finally we multiply vertically 3x4=12 and add the carried 2
=14. Result: 1,408.
32 x 44
= 1,408
A
3
2
Starting from the right
x
multiply vertically
4
4
2 x 4 = 8
B
3 2
Multiply crosswise
3 x 4 = 12 and 2 x 4 = 8
4 4
Add them together
_______
0 8
3 x 4 + 2 x 4 = 20
2
Put down 0 and carry 2
C
3 2
Finally multiply vertically
x
3 x 4 = 12 and add the
4
4
carried over 2 = 14
_______________
14
0 8
2
We
can extend this method to deal with long multiplication of numbers of any size.
The great advantage of this system is that the answer can be obtained in one
line and mentally. By the end of Year 8, I would expect all students to be able
to do a "3 by 2" long multiplication in their heads. This gives
enormous confidence to the pupils who lose their fear of numbers and go on to
tackle harder maths in a more open manner.
All
the techniques produce one-line answers and most can be dealt with mentally, so
calculators are not used until Year 10. The methods are either
"special", in that they only apply under certain conditions, or
general. This encourages flexibility and innovation on the part of the
students.
Multiplication
can also be carried out starting from the left, which can be better because we
write and pronounce numbers from left to right. Here is an example of doing
this in a special method for long multiplication of numbers near a base (10,
100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8
below.
We
can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of
the answer and multiplying the "differences" vertically 4x8=32 gives
the second part of the answer.
96
x 92 =
8,832
A 96
4
(4 is the difference from base)
92
8 (8 is the difference from base)
_____________
B
96
4 Subtract crosswise from the left
92
8 96 - 8 = 88 or 92 - 4 = 88
______________
88
C
96
4 Multiply vertically
x
4 x 8 = 32
92
8
____________
88
32
This
works equally well for numbers above the base: 105x111=11,655. Here we add the
differences. For 205x211=43,255, we double the first part of the answer,
because 200 is 2x100.
We
regularly practise the methods by having a mental test at the beginning of each
lesson. With the introduction of a non-calculator paper at GCSE, Vedic maths
offers methods that are simpler, more efficient and more readily acquired than
conventional methods.
There
is a unity and coherence in the system which is not found in conventional
maths. It brings out the beauty and patterns in numbers and the world around
us. The techniques are so simple they can be used when conventional methods
would be cumbersome.
When
the children learn about Pythagoras's theorem in Year 9 we do not use a
calculator; squaring numbers and finding square roots (to several significant
figures) is all performed with relative ease and reinforces the methods that
they would have recently learned.
Mark Gaskell is head of maths at the Maharishi School
in Lancashire
www.vedicmaths.org
'The Cosmic Computer'
by K Williams and M Gaskell, (also in an bridged
edition), Inspiration Books, 2 Oak Tree Court,
Skelmersdale, Lancs WN8 6SP. Tel: 01695 727 986.
Saturday school for primary teachers at
Manchester Metropolitan University
on
October 7. See website.
19th May 2000 Times Education Supplement (Curriculum
Special)
____________________________________________________
Books
on Vedic Maths
VEDIC MATHEMATICS
Or Sixteen Simple Mathematical Formulae from the Vedas
The original introduction to Vedic
Mathematics.
Author: Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja, 1965 (various
reprints).
Paperback, 367 pages, A5 in size.
ISBN 81 208 0163 6 (cloth)
ISBN 82 208 0163 4 (paper)
MATHS OR MAGIC?
This is a popular book giving a brief outline of some
of the Vedic Mathematics methods.
Author: Joseph Howse. 1976
ISBN 0722401434
Currently out of print.
A PEEP INTO VEDIC MATHEMATICS
Mainly on recurring decimals.
Author: B R Baliga, 1979.
Pamphlet.
INTRODUCTORY
LECTURES ON VEDIC MATHEMATICS
Following various lecture courses in London an interest arose for printed material
containing the course material. This book of 12 chapters was the result covering
a range topics from elementary arithmetic to cubic equations.
Authors: A. P. Nicholas, J. Pickles, K. Williams, 1982. Paperback, 166 pages,
A4 size.
DISCOVER VEDIC MATHEMATICS
This has sixteen chapters each of which focuses on one
of the Vedic Sutras or sub-Sutras and shows many applications of each. Also
contains Vedic Maths solutions to GCSE and 'A' level examination questions.
Author: K. Williams, 1984, Comb bound, 180 pages, A4.
ISBN 1 869932 01 3.
VERTICALLY AND CROSSWISE
This is an advanced book of sixteen chapters on one
Sutra ranging from elementary multiplication etc. to the solution of non-linear
partial differential equations. It deals with (i) calculation of common
functions and their series expansions, and (ii) the solution of equations, starting
with simultaneous equations and moving on to algebraic, transcendental and
differential equations.
Authors: A. P. Nicholas, K. Williams, J. Pickles (first published 1984), new
edition 1999. Comb bound, 200 pages, A4.
ISBN 1 902517 03 2.
TRIPLES
This book shows applications of Pythagorean Triples
(like 3,4,5). A simple, elegant system for combining these triples gives
unexpected and powerful general methods for solving a wide range of
mathematical problems, with far less effort than conventional methods use. The
easy text fully explains this method which has applications in trigonometry
(you do not need any of those complicated formulae), coordinate geometry (2 and
3 dimensions) transformations (2 and 3 dimensions), simple harmonic motion,
astronomy etc., etc.
Author: K. Williams (first published 1984), new edition 1999. Comb bound.,168
pages, A4.
ISBN 1 902517 00 8
VEDIC MATHEMATICAL CONCEPTS OF SRI VISHNU SAHASTRANAMA
STOTRAM
Author: S.K. Kapoor, 1988. Hardback, 78 pages, A4
size.
ISSUES IN VEDIC MATHEMATICS
Proceedings of the National workshop on Vedic
Mathematics 25-28 March 1988 at the University
of Rajasthan, Jaipur.
Paperback, 139 pages, A5 in size.
ISBN 81 208 0944 0
THE NATURAL CALCULATOR
This is an elementary book on mental mathematics. It
has a detailed introduction and each of the nine chapters covers one of the
Vedic formulae. The main theme is mental multiplication but addition,
subtraction and division are also covered.
Author: K. Williams, 1991. Comb bound ,102 pages, A4 size.
ISBN 1 869932 04 8.
VEDIC MATHEMATICS FOR SCHOOLS BOOK 1
Is a first text designed for the young mathematics
student of about eight years of age, who have mastered the four basic rules
including times tables. The main Vedic methods used in his book are for multiplication,
division and subtraction. Introductions to vulgar and decimal fractions,
elementary algebra and vinculums are also given.
Author: J.T,Glover, 1995. Paperback, 100 pages + 31 pages of answers, A5 in
size.
ISBN 81-208-1318-9.
JAGATGURU SHANKARACHARYA SHRI BHARATI KRISHNA TEERTHA
An excellent book giving details of the life of the
man who reconstructed the Vedic system.
Dr T. G. Pande, 1997
B. R. Publishing Corporation, Delhi-110052
INTRODUCTION
TO VEDIC MATHEMATICS
Authors T. G. Unkalkar, S. Seshachala Rao, 1997
Pub: Dandeli Education Socety, Karnataka-581325
THE
COSMIC COMPUTER COURSE
This covers Key Stage 3 (age 11-14 years) of the
National Curriculum for England and Wales. It consists of three books each of
which has a Teacher's Guide and an Answer Book. Much of the material in Book 1
is suitable for children as young as eight and this is developed from here to
topics such as Pythagoras' Theorem and Quadratic Equations in Book 3. The
Teacher's Guide contains a Summary of the Book, a Unified Field Chart (showing
the whole subject of mathematics and how each of the parts are related),
hundreds of Mental Tests (these revise previous work, introduce new ideas and
are carefully correlated with the rest of the course), Extension Sheets (about
16 per book) for fast pupils or for extra classwork, Revision Tests, Games,
Worksheets etc.
Authors: K. Williams and M. Gaskell, 1998.
All Textbooks and Guides are A4 in size, Answer Books are A5.
GEOMETRY FOR AN ORAL TRADITION
This book demonstrates the kind of system that could
have existed before literacy was widespread and takes us from first principles
to theorems on elementary properties of circles. It presents direct, immediate
and easily understood proofs. These are based on only one assumption (that magnitudes
are unchanged by motion) and three additional provisions (a means of drawing
figures, the language used and the ability to recognise valid reasoning). It
includes discussion on the relevant philosophy of mathematics and is written
both for mathematicians and for a wider audience.
Author: A. P. Nicholas, 1999. Paperback.,132 pages, A4 size.
ISBN 1 902517 05 9
THE CIRCLE REVELATION
This is a simplified, popularised version of
"Geometry for an Oral Tradition" described above. These two books
make the methods accessible to all interested in exploring geometry. The
approach is ideally suited to the twenty-first century, when audio-visual forms
of communication are likely to be dominant.
Author: A. P. Nicholas, 1999. Paperback, 100 pages, A4 size.
ISBN 1902517067
VEDIC MATHEMATICS FOR SCHOOLS BOOK 2
The second book in this series.
Author J.T. Glover , 1999.
ISBN 81 208 1670-6
Astronomica;
Applications of Vedic Mathematics
To
include prediction of eclipses and planetary positions,
spherical trigonometry etc.
Author Kenneth Williams, 2000.
ISBN 1 902517 08 3
Vedic Mathematics, Part 1
We found this book to be well-written, thorough and
easy to read. It covers a lot of the basic work in the original book by B. K.
Tirthaji and has plenty of examples and exercises.
Author S. Haridas
Published by Bharatiya Vidya Bhavan, Kulapati K.M. Munshi Marg, Mumbai - 400
007, India.
INTRODUCTION TO VEDIC MATHEMATICS – Part II
Authors T. G. Unkalkar, 2001
Pub: Dandeli Education Socety, Karnataka-581325
VEDIC
MATHEMATICS FOR SCHOOLS BOOK 3
The third book in this series.
Author J.T. Glover , 2002.
Published by Motilal Banarsidass.
THE COSMIC CALCULATOR
Three textbooks plus Teacher's Guide plus Answer Book.
Authors Kenneth Williams and Mark Gaskell, 2002.
Published by Motilal Banarsidass.
TEACHER’S
MANUALS – ELEMENTARY & INTERMEDIATE
Designed for teachers (of children aged 7 to 11 years,
9 to 14 years respectively) who wish to teach the Vedic system.
Author: Kenneth Williams, 2002.
Published by Inspiration Books.
TEACHER’S
MANUAL – ADVANCED
Designed for teachers (of children aged 13 to 18
years) who wish to teach the Vedic system.
Author: Kenneth Williams, 2003.
Published by Inspiration Books.
FUN WITH FIGURES (subtitled: Is it Maths or Magic?)
This is a small popular book with many illustrations,
inspiring quotes and amusing anecdotes. Each double page shows a neat and quick
way of solving some simple problem. Suitable for any age from eight upwards.
Author: K. Williams, 1998. Paperback, 52 pages, size A6.
ISBN 1 902517 01 6.
Please note the Tutorial below is based on material from this book 'Fun with
Figures'
Book review of 'Fun with Figures'
From 'inTouch', Jan/Feb 2000, the Irish National Teachers Organisation (INTO)
magazine.
"Entertaining, engaging and
eminently 'doable', Williams' pocket volume reveals many fascinating and useful
applications of the ancient Eastern system of Vedic Maths. Tackling many number
operations encountered between First and Sixth class, Fun with Figures offers
several speedy and simple means of solving or double-checking class activities.
Focusing throughout on skills associated with mental mathematics, the author
wisely places them within practical life-related contexts."
"Compact, cheerful and liberally
interspersed with amusing anecdotes and aphorisms from the world of maths,
Williams' book will help neutralise the 'menace' sometimes associated with
maths. It's practicality, clear methodology, examples, supplementary exercises
and answers may particularly benefit and empower the weaker student."
"Certainly a valuable investment
for parents and teachers of children aged 7 to 12."
Reviewed by Gerard Lennon, Principal,
Ardpatrick NS, Co Limerick.
The Tutorial below is based on material
from this book 'Fun with Figures'
___________________________________
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