Vedic Mathematics -1

Vedic Mathematics
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
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.
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.
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.

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.
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.
For many more examples, try elsewhere on this page,  the Vedic Maths Tutorial
Mark Gaskell is head of maths at the Maharishi School in Lancashire
'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 Educational Supplement (Curriculum Special)

Books on Vedic Maths
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)/p

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./p

Master Multiplication tables, division and lots more!
We recommed you check out this ebook, it's packed with tips,
tricks and tutorials that will boost your math ability, guaranteed!
Mainly on recurring decimals.
Author: B R Baliga, 1979.

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./p

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./p

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./p

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/p

Author: S.K. Kapoor, 1988. Hardback, 78 pages, A4 size./p

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/p

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./p.

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./p

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

Authors T. G. Unkalkar, S. Seshachala Rao, 1997
Pub: Dandeli Education Socety, Karnataka-581325

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.

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

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

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.

Authors T. G. Unkalkar, 2001
Pub: Dandeli Education Socety, Karnataka-581325

The third book in this series.
Author J.T. Glover , 2002.
Published by Motilal Banarsidass.

Three textbooks plus Teacher's Guide plus Answer Book.
Authors Kenneth Williams and Mark Gaskell, 2002.
Published by Motilal Banarsidass.

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.

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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Index Alphabetical [Index to Pages]
Vedic Maths Tutorial
Vedic Maths is based on sixteen Sutras or principles. These principles are general in nature and can be applied in many ways. In practice many applications of the sutras may be learned and combined to solve actual problems. These tutorials will give examples of simple applications of the sutras, to give a feel for how the Vedic Maths system works.
These tutorials do not attempt to teach the systematic use of the sutras.
For more advanced applications and a more complete coverage of the basic uses of the sutras, we recommend you study one of the texts available at

N.B. The following tutorials are based on examples and exercises given in the book 'Fun with figures' by Kenneth Williams, which is a fun introduction to some of the applications of the sutras for children.

Tutorial 1
Tutorial 2
Tutorial 3
Tutorial 4
Tutorial 5
Tutorial 6
Tutorial 7
Tutorial 8  (By Kevin O'Connor)

Tutorial 1
Use the formula ALL FROM 9 AND THE LAST FROM 10 to
perform instant subtractions.

For example 1000 - 357 = 643
We simply take each figure in 357 from 9 and the last figure from 10.
So the answer is 1000 - 357 = 643
And thats all there is to it!
This always works for subtractions from numbers consisting of a 1 followed by noughts: 100; 1000; 10,000 etc.
Similarly 10,000 - 1049 = 8951
For 1000 - 83, in which we have more zeros than
figures in the numbers being subtracted, we simply
suppose 83 is 083.
So 1000 - 83 becomes 1000 - 083 = 917

Exercise 1 Tutorial 1
Try some yourself:

1)  1000 - 777       =    
2)  1000 - 283       =    
3)  1000 - 505       =    
4)  10,000 - 2345  =    
5)  10,000 - 9876  =    
6)  10,000 - 1011  =    
7) 100 - 57            =    
8) 1000 - 57          =    
9) 10,000 - 321     =    
10) 10,000 - 38     =    
Tutorial 2
not need the multiplication tables beyond 5 X 5.

Suppose you need 8 x 7
8 is 2 below 10 and 7 is 3 below 10.
Think of it like this:

The answer is 56.
The diagram below shows how you get it.

You subtract crosswise 8-3 or 7 - 2 to get 5,
the first figure of the answer.
And you multiply vertically: 2 x 3 to get 6,
the last figure of the answer.

That's all you do:
See how far the numbers are below 10, subtract
one number's deficiency from the other number,
and multiply the deficiencies together.

7 x 6 = 42
Here there is a carry: the 1 in the
12 goes over to make 3 into 4.

Exercise 1 Tutorial 2
Multply These:
1)  8 x 8 =
2)  9 x 7 =
3)  8 x 9 =
4)  7 x 7 =
5)  9 x 9 =
6)  6 x 6 =
Answers to exercise 1 tutorial 2

for multiplying numbers close to 100.

Suppose you want to multiply 88 by 98.
Not easy,you might think. But with
you can give the answer immediately,
using the same method as above

Both 88 and 98 are close to 100.
88 is 12 below 100 and 98 is 2 below 100.

You can imagine the sum set out like this:
As before the 86 comes from subtracting crosswise:
88 - 2 = 86 (or 98 - 12 = 86: you can subtract either way,
you will always get the same answer).
And the 24 in the answer is just 12 x 2: you
multiply vertically.
So 88 x 98 = 8624

Exercise 2 Tutorial 2
This is so easy it is just mental arithmetic.
Try some:
1)  87 x 98 =
2)  88 x 97 =
3)  77 x 98 =
4)  93 x 96 =
5)  94 x 92 =
6)  64 x 99 =
7)  98 x 97 =
Answers to Exercise 2 Tutorial 2  < click
Multiplying numbers just over 100.
103 x 104 = 10712
The answer is in two parts: 107 and 12,
107 is just 103 + 4 (or 104 + 3),
and 12 is just 3 x 4.

Similarly 107 x 106 = 11342
107 + 6 = 113 and 7 x 6 = 42

Exercise 3 Tutorial 2
Again, just for mental arithmetic
Try a few:
1)  102 x 107 =
2)  106 x 103 =
3)  104 x 104 =
4)  109 x 108 =
5)  101 x123 =
6)  103 x102 =
Answers to exercise 3 Tutorial 2  < click

Tutorial 3
The easy way to add and subtract fractions.
to write the answer straight down!

Multiply crosswise and add to get the top of the answer:
2 x 5 = 10 and 1 x 3 = 3. Then 10 + 3 = 13.
The bottom of the fraction is just 3 x 5 = 15.
You multiply the bottom number together.

Subtracting is just as easy: multiply
crosswise as before, but the subtract:

Exercise 1 Tutorial 3
Try a few:






Answers to Exercise 1 Tutorial 3   < click

Tutorial 4

A quick way to square numbers that end in 5 using

752 = 5625
75² means 75 x 75.
The answer is in two parts: 56 and 25.
The last part is always 25.
The first part is the first number, 7, multiplied
by the number "one more", which is 8:
so 7 x 8 = 56


Similarly 852 = 7225 because 8 x 9 = 72.
Exercise 1 Tutorial 4
Try these:
1) 452 =
2) 652 =
3) 952 =
4) 352 =
5) 152 =
Answers to Exercise 1 Tutorial 4 < click
Method for multiplying numbers where the first
figures are the same and the last figures add up to 10.

32 x 38 = 1216
Both numbers here start with 3 and the last figures (2 and 8) add up to 10.
So we just multiply 3 by 4 (the next number up)
to get 12 for the first part of the answer.

And we multiply the last figures: 2 x 8 = 16 to
get the last part of the answer.



And 81 x 89 = 7209
We put 09 since we need two figures as in all the other examples.
Exercise 2 Tutorial 4
Practise some:
1)  43 x 47 =
2)  24 x 26 =
3)  62 x 68 =
4)  17 x 13 =
5)  59 x 51 =
6)  77 x 73 =
Answers to Exercise 2 Tutorial 4
Tutorial 5
An elegant way of multiplying numbers using a simple pattern
21 x 23 = 483
This is normally called long multiplication but actually
the answer can be written straight down using the
We first put, or imagine, 23 below 21:

There are 3 steps:
a) Multiply vertically on the left: 2 x 2 = 4.
This gives the first figure of the answer.
b) Multiply crosswise and add: 2 x 3 + 1 x 2 = 8
This gives the middle figure.
c) Multiply vertically on the right: 1 x 3 = 3
This gives the last figure of the answer.

And thats all there is to it.
Similarly 61 x 31 = 1891

6 x 3 = 18; 6 x 1 + 1 x 3 = 9; 1 x 1 = 1
Exercise 1 Tutorial 5
Try these, just write down the answer:
1) 14 x 21
2) 22 x 31
3) 21 x 31
4) 21 x 22
5) 32 x 21

Om Tat Sat

 (My humble salutations to  Swamy jis , Vedic Mathematicians  and Hinduism com  for the collection)


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