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.



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'


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



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.

1000 - 3 5 7
¯ ¯ ¯
From 9 From 9 From 10
= 6 4 3

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

10,000 - 1 0 4 9
From 9 From 9 From 9
From 10
¯ ¯ ¯ ¯
= 8 9 5 1


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) 10000 - 9876
6) 10,000 - 1101
7) 100 - 57
8) 1000 - 57
9) 10,000 - 321
10) 10,000 - 38



Tutorial 2

Using VERTICALLY AND CROSSWISE you do not need to
know 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:

8 2

7 3
_______
5 6 Answer

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

8 2
x 1
7 3
_______
5 6 Answer


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


7 3
x 1
6 4
_______
3 1 2 = 42


Here there is a carry: the 1 in the 12 goes over to make 3 into 4.

Exercise 1 Tutorial 2

Multiply these


1) 8 2) 9 3) 8 4) 7 5) 9 6) 6
x 8 7 9 7 9 6
__ __ __ __ __ __


Here's how to use VERTICALLY AND CROSSWISE
or multiplying numbers close to 100.

Suppose you want to multiply 88 by 98.

Not easy, you might think. But with
VERTICALLY AND CROSSWISE you can give
the answer immediately, using the same method
as on the page.

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:

88 - 12
x 1
98 - 2
_____
86 - 24

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 2) 88 3) 77 4) 93 5) 94 6) 64 7) 98
x 98 97 98 96 92 99 97
__ __ __ __ __ __
__
__ __ __ __ __ __
__



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


Tutorial 3

The easy way to add and subtract fractions.

Use VERTICALLY AND CROSSWISE to write the answer
straight down!


2 + 1 = 10 + 3 = 13
3 5 15 15


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.

So:

5 + 3 = 20+21 = 41
7 4 28 28

Subtracting is just as easy: multiply crosswise as before,
and subtract:

6 - 2 = 18 - 14 = 4
7 3 21 21

Exercise 1 Tutorial 3

Try a few

1) 4 + 1 = 2) 1 + 1 = 3) 2 + 2 =
5 6 3 4 7 3



4) 4 - 1 = 5) 1 - 1 = 6) 8 - 9 =
5 6 4 5 3 5

Tutorial 4

A quick way to square numbers that end in 5 using the formula
BY ONE MORE THAN THE ONE BEFORE.

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

2
7 5 = 5 6 2 5

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



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.

Diagrammatically: (follow same colour links)

2 x 8 = 16

3 2 x 3 8 = 12 16

3 x 4 = 12



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



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 VERTICALLY AND CROSSWISE
formula.

We first put, or imagine, 23 below 21:

2 1
| x |
2 3 x
_______
4 8 3
_______


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 1
| x |
3 1 x
_______
18 9 1
_______


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 2) 22 3) 21 4) 21 5) 32
x 21 31 31 22 21
__ __ __ __ __
__ __ __ __ __





exercise 2a Tutorial 5
Multiply any 2-figure numbers together by mere mental arithmetic!

If you want 21 stamps at 26 pence each you can
easily find the total price in your head.

There were no carries in the method given above.
However, this only involves one small extra step.

21 x 26 = 546



2 1
| x |
2 6 x
_______
4 14 6 = 546
_______ ___


The method is the same as above
except that we get a 2-figure number, 14, in the
middle step, so the 1 is carried over to the left
(4 becomes 5).

So 21 stamps cost £5.46.

Practise a few:



1) 21 2) 23 3) 32 4) 42 6) 71
x 47 43 53 32 72





Exercise 2b Tutorial 5
33 x 44 = 1452

There may be more than one carry in a sum:


3 3
| x |
4 4 x
_______
12 24 12 = 1452
_______ ____


Vertically on the left we get 12.
Crosswise gives us 24, so we carry 2 to the left
and mentally get 144.

Then vertically on the right we get 12 and the 1
here is carried over to the 144 to make 1452.


6) 32 7) 32 8) 31 9) 44 10) 54
x 56 54 72 53 64
__ __ __ __ __
__ __ __ __ __





Any two numbers, no matter how big, can be
multiplied in one line by this method.


TOP <To top of this page



Tutorial 6

Multiplying a number by 11.

To multiply any 2-figure number by 11 we just put
the total of the two figures between the 2 figures.

26 x 11 = 286

Notice that the outer figures in 286 are the 26
being multiplied.

And the middle figure is just 2 and 6 added up.

So 72 x 11 = 792

Exercise 1 Tutorial 6

Multiply by 11:

1) 43 2) 81 3) 15 4) 44 5) 11




77 x 11 = 847

This involves a carry figure because 7 + 7 = 14
we get 77 x 11 = 7147 = 847.

Exercise 2 Tutorial 6

Multiply by 11:

1) 88 2) 84 3) 48 4) 73 5) 56



234 x 11 = 2574

We put the 2 and the 4 at the ends.
We add the first pair 2 + 3 = 5.
and we add the last pair: 3 + 4 = 7.

Exercise 3 Tutorial 6

Multiply by 11:

1) 151 2) 527 3) 333 4) 714 5) 909


Tutorial 7


Method for dividing by 9.

23 / 9 = 2 remainder 5

The first figure of 23 is 2, and this is the answer.
The remainder is just 2 and 3 added up!

43 / 9 = 4 remainder 7

The first figure 4 is the answer
and 4 + 3 = 7 is the remainder - could it be easier?

Exercise 1a Tutorial 7
Divide by 9:

1) 61 2) 33 3) 44 4) 53 5) 80



134 / 9 = 14 remainder 8

The answer consists of 1, 4 and 8.
1 is just the first figure of 134.
4 is the total of the first two figures 1+ 3 = 4,
and 8 is the total of all three figures 1+ 3 + 4 = 8.

Exercise 1b Tutorial 7
Divide by 9:

6) 232 7) 151 8) 303 9) 212 10) 2121




842 / 9 = 812 remainder 14 = 92 remainder 14

Actually a remainder of 9 or more is not usually
permitted because we are trying to find how
many 9's there are in 842.

Since the remainder, 14 has one more 9 with 5
left over the final answer will be 93 remainder 5

Exercise 2 Tutorial 7
Divide these by 9:

1) 771 2) 942 3) 565 4) 555 5) 777 6) 2382 7) 7070





Answers




Answers to exercise 1 Tutorial 1
1) 223
2) 717
3) 495
4) 7655
5) 0124
6) 8899
7) 43
8) 943
9) 9679
10) 9962




Answers to exercise 1 Tutorial 2
1) 64
2) 63
3) 72
4) 49
5) 81
6) 216 = 36

Answers to exercise 2 Tutorial 2



1) 8526
2) 8536
3) 7546
4) 8928
5) 8648
6) 6336
7) 9506 (we put 06 because, like all the others,
we need two figures in each part)




Answers to exercise 3 Tutorial 2


1) 10914
2) 10918
3) 10816
4) 11772
5) 12423
6) 10506 (we put 06, not 6)



Answers to exercise 1 Tutorial 3


1) 29/30
2) 7/12
3) 20/21
4) 19/30
5) 1/20
6) 13/15



Answers to exercise 1 Tutorial 4
1) 2025
2) 4225
3) 9025
4) 1225
5) 225



Answers to exercise 2 (Tutorial 4)



1) 2021
2) 624
3) 4216
4) 221
5) 3009
6) 5621



Answers to exercise 1 Tutorial 5

1) 294
2) 682
3) 651
4) 462
5) 672


Answers to exercise 2a Tutorial 5
1) 987
2) 989
3) 1696
4) 1344
5) 5112




Answers to exercise 2b Tutorial 5

6) 1792
7) 1728
8) 2232
9) 2332
10) 3456


Answers to exercise 1 (Tutorial 6)

1) 473
2) 891
3) 165
4) 484
5) 121



Answers to exercise 2 (Tutorial 6)

1) 968
2) 924
3) 528
4) 803
5) 616



Answers to exercise 3 Tutorial 6



1) 1661
2) 5797
3) 3663
4) 7854
5) 9999



Answers to exercise 1a (Tutorial 7)

1) 6 r 7
2) 3 r 6
3) 4 r 8
4) 5 r 8
5) 8 r 8



Exercise 1b Tutorial 7

Answers to exercise 1b (Tutorial 7)


1) 25 r 7
2) 16 r 7
3) 33 r 6
4) 23 r 5
5) 235 r 6 (we have 2, 2 + 1, 2 + 1 + 2, 2 + 1 + 2 + 1)



Answers to exercise 2 (Tutorial 7)


1) 714 r15 = 84 r15 = 85 r6
2) 913 r 15 = 103 r15 = 104 r6
3) 516 r16 = 61 r16 = 62 r7
4) 510 r15 = 60 r15 = 61 r6
5) 714 r21 = 84 r21 = 86 r3
6) 2513 r15 = 263 r15 = 264 r6
7) 7714 r14 = 784 r14 = 785 r5


Vedic Mathematics

By Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj (1884-1960)

Book ref: ISBN 0 8426 0967 9
Published by Motilal Banarasidas

From the Introduction by Smti Manjula Trivedi 16-03-1965.
An extract:

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.


From the Preface by the author Jagadguru
Swami Sri Bharati Krsna Tirthaji Maharaj
Extracts:

We may however, at this point draw the earnest attention of every one concerned
to the following salient items thereof:


1.. The Sutras (aphorisms) apply to and cover each and every part of each and
every chapter of each and every branch of mathematics (including Arithmetic,
Algebra, Geometry - plane and solid, Trigonometry - plane and spherical, Conics
- geometrical and analytical, Astronomy, Calculus - differential and integral
etc.) In fact, there is no part of mathematics, pure or applied, that is beyond
their jurisdiction.

b.. The Sutras are easy to understand, easy to apply and easy to remember, and
the whole work can be truthfully summarised in one word 'Mental'!

c.. Even as regards complex problems involving a good number of mathematical
operations (consecutively or even simultaneously to be performed), the time
taken by the Vedic method will be a third, a fourth, a tenth, or even a much
smaller fraction of the time required according to modern (i.e. current) Western
methods.

d.. And in some very important and striking cases, sums requiring 30, 50, 100
or even more numerous and cumbrous 'steps' of working (according to the current
Western methods) can be answered in a single and simple step of work by the
Vedic method! And little children (of only 10 or 12 years of age) merely look
at the sums written on the blackboard and immediately shout out and dictate the
answers. And this is because, as a matter of fact, each digit automatically
yields its predecessor and its successor! And the children have merely to go on
tossing off (or reeling off) the digits one after another (forwards or
backwards) by mere mental arithmetic (without needing pen or pencil, paper,
slate etc.).

e.. On seeing this kind of work actually being performed by the little
children, the doctors, professors and other 'big-guns' of mathematics are
wonder-struck and exclaim: 'Is this mathematics or magic'? And we invariably
answer and say: 'It is both. It is magic until you understand it; and it is
mathematics thereafter'. And then we proceed to substantiate and prove the
correctness of this reply of ours!

f.. As regards the time required by the students for mastering the whole
course of Vedic Mathematics as applied to all its branches, we need merely state
from our actual experience that 8 months (or 12 months) at an average rate of 2
or 3 hours per day should suffice for completing the whole course of
mathematical studies on these Vedic lines instead of 15 or 20 years required
according to the existing systems of the Indian and also of foreign
universities.

g.. And we were agreeably astonished and intensely gratified to find that
exceedingly tough mathematical problems (which the mathematically most advanced
present day Western scientific world had spent huge amount of time, energy, and
money on and which even now it solves with the utmost difficulty and that also
after vast labour involving large numbers of difficult, tedious and cumbersome
'steps' of working) can be easily and readily solved with the help of these
ultra-easy Vedic Sutras (or mathematical aphorisms) contained in the Parisista
(the appendix portion) of the Atharvaveda in a few simple steps and by methods
that can be conscientiously described as mere 'mental arithmetic'.

h.. It is thus in the fitness of things that the Vedas include 1. Ayurveda
(anatomy, physiology, hygiene, sanitary science, medical science, surgery etc.),
not for the purpose of achieving perfect health and strength in the after-death
future but in order to attain them here and now in our present physical bodies.
2.Dhanurveda (archery and other military sciences), not for fighting with one
another after our transportation to heaven but in order to quell and subdue all
invaders from abroad and all insurgents from within. 3. Gandharva Veda (the
science of art and music) and 4. Sthapatya Veda (engineering, architecture etc.
and all branches of mathematics in general). All these subjects, be it noted,
are inherent parts of the Vedas i.e., are reckoned as 'spiritual' studies and
catered for as such therein.

i.. Similar is the case with Vedangas (i.e., grammar, prosody, astronomy,
lexicography etc.) which according to the Indian cultural conceptions, are also
inherent parts and subjects of Vedic (i.e. religious) study.


From the Foreward by Swami Pratyagatmananda Saraswati
Varanasi, 22-03-1965
An extract:

Vedic Mathematics by the late Shankaracharya (Bharati Krsna Tirtha) of Govardhan
Pitha is a monumental work. In his deep-layer explorations of cryptic Vedic
mysteries relating especially to their calculus of shorthand formulae and their
neat and ready application to practical problems, the late Shankaracharya shows
the rare combination of the probing insight of revealing intuition of a Yogi
with the analytic acumen and synthetic talent of a mathematician.

With the late Shankaracharya we belong to a race, now fast becoming extinct, of
diehard believers who think that the Vedas represent an inexhaustible mine of
profoundest wisdom in matters of both spiritual and temporal; and that this
store of wisdom was not, as regards its assets of fundamental validity and value
at least, gathered by the laborious inductive and deductive methods of ordinary
systemic enquiry, but was direct gift of revelation to seers and sages who in
their higher reaches of Yogic realisation were competent to receive it from a
source, perfect and immaculate.

Whether or not the Vedas are believed as repositories of perfect wisdom, it is
unquestionable that the Vedic race lived not as merely pastoral folk possessing
a half or a quarter developed culture and civilisation. The Vedic seers were,
again, not mere 'navel-gazers' or 'nose-tip gazers'. They proved themselves
adepts in all levels and branches of knowledge, theoretical and practical. For
example, they had their varied objective science both pure and applied.

Let us take a concrete illustration. Suppose in a time of drought we require
rains by artificial means. The modern scientist has his own theory and art
(technique) for producing the result. The old seer scientist had his both also,
but different from these now availing. He had his science and technique, called
Yajna, in which Mantra, Yantra, and other factors must co-operate with
mathematical determinateness and precision. For this purpose, he had developed
the six auxiliaries of the Vedas in each of which mathematical skill and
adroitness, occult or otherwise, play the decisive role. The Sutras lay down the
shortest and surest lines. The correct intonation of the Mantra, the correct
configuration of the Yantra (in the making of the Vedi etc., e.g. the quadrate
of a circle), the correct time or astral conjunction factor, the correct
rhythams etc. All had to be perfected so as to produce the desired results
effectively and adequately. Each of these required the calculus of mathematics.
The modern technician has his logarithmic tables and mechanic's manuals. The old
Yajnik had his Sutras.


Pages from the history of the Indian sub-continent:
Science and Mathematics in India

History of Mathematics in India

In all early civilizations, the first expression of mathematical understanding
appears in the form of counting systems. Numbers in very early societies were
typically represented by groups of lines, though later different numbers came to
be assigned specific numeral names and symbols (as in India) or were designated
by alphabetic letters (such as in Rome). Although today, we take our decimal
system for granted, not all ancient civilizations based their numbers on a
ten-base system. In ancient Babylon, a sexagesimal (base 60) system was in use.

The Decimal System in Harappa

In India a decimal system was already in place during the Harappan period, as
indicated by an analysis of Harappan weights and measures. Weights corresponding
to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 have
been identified, as have scales with decimal divisions. A particularly notable
characteristic of Harappan weights and measures is their remarkable accuracy. A
bronze rod marked in units of 0.367 inches points to the degree of precision
demanded in those times. Such scales were particularly important in ensuring
proper implementation of town planning rules that required roads of fixed widths
to run at right angles to each other, for drains to be constructed of precise
measurements, and for homes to be constructed according to specified guidelines.
The existence of a gradated system of accurately marked weights points to the
development of trade and commerce in Harappan society.

Mathematical Activity in the Vedic Period

In the Vedic period, records of mathematical activity are mostly to be found in
Vedic texts associated with ritual activities. However, as in many other early
agricultural civilizations, the study of arithmetic and geometry was also
impelled by secular considerations. Thus, to some extent early mathematical
developments in India mirrored the developments in Egypt, Babylon and China .
The system of land grants and agricultural tax assessments required accurate
measurement of cultivated areas. As land was redistributed or consolidated,
problems of mensuration came up that required solutions. In order to ensure that
all cultivators had equivalent amounts of irrigated and non-irrigated lands and
tracts of equivalent fertility - individual farmers in a village often had their
holdings broken up in several parcels to ensure fairness. Since plots could not
all be of the same shape - local administrators were required to convert
rectangular plots or triangular plots to squares of equivalent sizes and so on.
Tax assessments were based on fixed proportions of annual or seasonal crop
incomes, but could be adjusted upwards or downwards based on a variety of
factors. This meant that an understanding of geometry and arithmetic was
virtually essential for revenue administrators. Mathematics was thus brought
into the service of both the secular and the ritual domains.



Arithmetic operations (Ganit) such as addition, subtraction, multiplication,
fractions, squares, cubes and roots are enumerated in the Narad Vishnu Purana
attributed to Ved Vyas (pre-1000 BC). Examples of geometric knowledge
(rekha-ganit) are to be found in the Sulva-Sutras of Baudhayana (800 BC) and
Apasthmaba (600 BC) which describe techniques for the construction of ritual
altars in use during the Vedic era. It is likely that these texts tapped
geometric knowledge that may have been acquired much earlier, possibly in the
Harappan period. Baudhayana's Sutra displays an understanding of basic geometric
shapes and techniques of converting one geometric shape (such as a rectangle) to
another of equivalent (or multiple, or fractional) area (such as a square).
While some of the formulations are approximations, others are accurate and
reveal a certain degree of practical ingenuity as well as some theoretical
understanding of basic geometric principles. Modern methods of multiplication
and addition probably emerged from the techniques described in the Sulva-Sutras.



Pythagoras - the Greek mathematician and philosopher who lived in the 6th C B.C
was familiar with the Upanishads and learnt his basic geometry from the Sulva
Sutras. An early statement of what is commonly known as the Pythagoras theorem
is to be found in Baudhayana's Sutra: The chord which is stretched across the
diagonal of a square produces an area of double the size. A similar observation
pertaining to oblongs is also noted. His Sutra also contains geometric solutions
of a linear equation in a single unknown. Examples of quadratic equations also
appear. Apasthamba's sutra (an expansion of Baudhayana's with several original
contributions) provides a value for the square root of 2 that is accurate to the
fifth decimal place. Apasthamba also looked at the problems of squaring a
circle, dividing a segment into seven equal parts, and a solution to the general
linear equation. Jain texts from the 6th C BC such as the Surya Pragyapti
describe ellipses.



Modern-day commentators are divided on how some of the results were generated.
Some believe that these results came about through hit and trial - as rules of
thumb, or as generalizations of observed examples. Others believe that once the
scientific method came to be formalized in the Nyaya-Sutras - proofs for such
results must have been provided, but these have either been lost or destroyed,
or else were transmitted orally through the Gurukul system, and only the final
results were tabulated in the texts. In any case, the study of Ganit i.e
mathematics was given considerable importance in the Vedic period. The Vedang
Jyotish (1000 BC) includes the statement: "Just as the feathers of a peacock and
the jewel-stone of a snake are placed at the highest point of the body (at the
forehead), similarly, the position of Ganit is the highest amongst all branches
of the Vedas and the Shastras."

(Many centuries later, Jain mathematician from Mysore, Mahaviracharya further
emphasized the importance of mathematics: "Whatever object exists in this moving
and non-moving world, cannot be understood without the base of Ganit (i.e.
mathematics)".)



Panini and Formal Scientific Notation

A particularly important development in the history of Indian science that was
to have a profound impact on all mathematical treatises that followed was the
pioneering work by Panini (6th C BC) in the field of Sanskrit grammar and
linguistics. Besides expounding a comprehensive and scientific theory of
phonetics, phonology and morphology, Panini provided formal production rules and
definitions describing Sanskrit grammar in his treatise called Asthadhyayi.
Basic elements such as vowels and consonants, parts of speech such as nouns and
verbs were placed in classes. The construction of compound words and sentences
was elaborated through ordered rules operating on underlying structures in a
manner similar to formal language theory.



Today, Panini's constructions can also be seen as comparable to modern
definitions of a mathematical function. G G Joseph, in The crest of the peacock
argues that the algebraic nature of Indian mathematics arises as a consequence
of the structure of the Sanskrit language. Ingerman in his paper titled
Panini-Backus form finds Panini's notation to be equivalent in its power to that
of Backus - inventor of the Backus Normal Form used to describe the syntax of
modern computer languages. Thus Panini's work provided an example of a
scientific notational model that could have propelled later mathematicians to
use abstract notations in characterizing algebraic equations and presenting
algebraic theorems and results in a scientific format.

Philosophy and Mathematics

Philosophical doctrines also had a profound influence on the development of
mathematical concepts and formulations. Like the Upanishadic world view, space
and time were considered limitless in Jain cosmology. This led to a deep
interest in very large numbers and definitions of infinite numbers. Infinite
numbers were created through recursive formulae, as in the Anuyoga Dwara Sutra.
Jain mathematicians recognized five different types of infinities: infinite in
one direction, in two directions, in area, infinite everywhere and perpetually
infinite. Permutations and combinations are listed in the Bhagvati Sutras (3rd C
BC) and Sathananga Sutra (2nd C BC).

Jain set theory probably arose in parallel with the Syadvada system of Jain
epistemology in which reality was described in terms of pairs of truth
conditions and state changes. The Anuyoga Dwara Sutra demonstrates an
understanding of the law of indeces and uses it to develop the notion of
logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are used to
denote log base 2, log base 3 and log base 4 respectively. In Satkhandagama
various sets are operated upon by logarithmic functions to base two, by squaring
and extracting square roots, and by raising to finite or infinite powers. The
operations are repeated to produce new sets. In other works the relation of the
number of combinations to the coefficients occurring in the binomial expansion
is noted.



Since Jain epistemology allowed for a degree of indeterminacy in describing
reality, it probably helped in grappling with indeterminate equations and
finding numerical approximations to irrational numbers.

Buddhist literature also demonstrates an awareness of indeterminate and infinite
numbers. Buddhist mathematics was classified either as Garna (Simple
Mathematics) or Sankhyan (Higher Mathematics). Numbers were deemed to be of
three types: Sankheya (countable), Asankheya (uncountable) and Anant
(infinite).


Philosophical formulations concerning Shunya - i.e. emptiness or the void may
have facilitated in the introduction of the concept of zero. While the zero
(bindu) as an empty place holder in the place-value numeral system appears much
earlier, algebraic definitions of the zero and it's relationship to mathematical
functions appear in the mathematical treatises of Brahmagupta in the 7th C AD.
Although scholars are divided about how early the symbol for zero came to be
used in numeric notation in India, (Ifrah arguing that the use of zero is
already implied in Aryabhatta) tangible evidence for the use of the zero begins
to proliferate towards the end of the Gupta period. Between the 7th C and the
11th C, Indian numerals developed into their modern form, and along with the
symbols denoting various mathematical functions (such as plus, minus, square
root etc) eventually became the foundation stones of modern mathematical
notation.


The Indian Numeral System

Although the Chinese were also using a decimal based counting system, the
Chinese lacked a formal notational system that had the abstraction and elegance
of the Indian notational system, and it was the Indian notational system that
reached the Western world through the Arabs and has now been accepted as
universal. Several factors contributed to this development whose significance is
perhaps best stated by French mathematician, Laplace: "The ingenious method of
expressing every possible number using a set of ten symbols (each symbol having
a place value and an absolute value) emerged in India. The idea seems so simple
nowadays that its significance and profound importance is no longer appreciated.
It's simplicity lies in the way it facilitated calculation and placed arithmetic
foremost amongst useful inventions."


Brilliant as it was, this invention was no accident. In the Western world, the
cumbersome roman numeral system posed as a major obstacle, and in China the
pictorial script posed as a hindrance. But in India, almost everything was in
place to favor such a development. There was already a long and established
history in the use of decimal numbers, and philosophical and cosmological
constructs encouraged a creative and expansive approach to number theory.
Panini's studies in linguistic theory and formal language and the powerful role
of symbolism and representational abstraction in art and architecture may have
also provided an impetus, as might have the rationalist doctrines and the
exacting epistemology of the Nyaya Sutras, and the innovative abstractions of
the Syadavada and Buddhist schools of learning.

Influence of Trade and Commerce, Importance of Astronomy

The growth of trade and commerce, particularly lending and borrowing demanded an
understanding of both simple and compound interest which probably stimulated the
interest in arithmetic and geometric series. Brahmagupta's description of
negative numbers as debts and positive numbers as fortunes points to a link
between trade and mathematical study. Knowledge of astronomy - particularly
knowledge of the tides and the stars was of great import to trading communities
who crossed oceans or deserts at night. This is borne out by numerous references
in the Jataka tales and several other folk-tales. The young person who wished to
embark on a commercial venture was inevitably required to first gain some
grounding in astronomy. This led to a proliferation of teachers of astronomy,
who in turn received training at universities such as at Kusumpura (Bihar) or
Ujjain (Central India) or at smaller local colleges or Gurukuls. This also led
to the exchange of texts on astronomy and mathematics amongst scholars and the
transmission of knowledge from one part of India to another. Virtually every
Indian state produced great mathematicians who wrote commentaries on the works
of other mathematicians (who may have lived and worked in a different part of
India many centuries earlier). Sanskrit served as the common medium of
scientific communication.



The science of astronomy was also spurred by the need to have accurate calendars
and a better understanding of climate and rainfall patterns for timely sowing
and choice of crops. At the same time, religion and astrology also played a role
in creating an interest in astronomy and a negative fallout of this irrational
influence was the rejection of scientific theories that were far ahead of their
time. One of the greatest scientists of the Gupta period - Aryabhatta (born in
476 AD, Kusumpura, Bihar) provided a systematic treatment of the position of the
planets in space. He correctly posited the axial rotation of the earth, and
inferred correctly that the orbits of the planets were ellipses. He also
correctly deduced that the moon and the planets shined by reflected sunlight and
provided a valid explanation for the solar and lunar eclipses rejecting the
superstitions and mythical belief systems surrounding the phenomenon. Although
Bhaskar I (born Saurashtra, 6th C, and follower of the Asmaka school of
science, Nizamabad, Andhra ) recognized his genius and the tremendous value of
his scientific contributions, some later astronomers continued to believe in a
static earth and rejected his rational explanations of the eclipses. But in
spite of such setbacks, Aryabhatta had a profound influence on the astronomers
and mathematicians who followed him, particularly on those from the Asmaka
school.



Mathematics played a vital role in Aryabhatta's revolutionary understanding of
the solar system. His calculations on pi, the circumferance of the earth
(62832 miles) and the length of the solar year (within about 13 minutes of the
modern calculation) were remarkably close approximations. In making such
calculations, Aryabhatta had to solve several mathematical problems that had not
been addressed before, including problems in algebra (beej-ganit) and
trigonometry (trikonmiti).



Bhaskar I continued where Aryabhatta left off, and discussed in further detail
topics such as the longitudes of the planets; conjunctions of the planets with
each other and with bright stars; risings and settings of the planets; and the
lunar crescent. Again, these studies required still more advanced mathematics
and Bhaskar I expanded on the trigonometric equations provided by Aryabhatta,
and like Aryabhatta correctly assessed pi to be an irrational number. Amongst
his most important contributions was his formula for calculating the sine
function which was 99% accurate. He also did pioneering work on indeterminate
equations and considered for the first time quadrilaterals with all the four
sides unequal and none of the opposite sides parallel.

Another important astronomer/mathematician was Varahamira (6th C, Ujjain) who
compiled previously written texts on astronomy and made important additions to
Aryabhatta's trigonometric formulas. His works on permutations and combinations
complemented what had been previously achieved by Jain mathematicians and
provided a method of calculation of nCr that closely resembles the much more
recent Pascal's Triangle. In the 7th century, Brahmagupta did important work in
enumerating the basic principles of algebra. In addition to listing the
algebraic properties of zero, he also listed the algebraic properties of
negative numbers. His work on solutions to quadratic indeterminate equations
anticipated the work of Euler and Lagrange.



Emergence of Calculus

In the course of developing a precise mapping of the lunar eclipse, Aryabhatta
was obliged to introduce the concept of infinitesimals - i.e. tatkalika gati to
designate the infinitesimal, or near instantaneous motion of the moon, and
express it in the form of a basic differential equation. Aryabhatta's equations
were elaborated on by Manjula (10th C) and Bhaskaracharya (12th C) who derived
the differential of the sine function. Later mathematicians used their intuitive
understanding of integration in deriving the areas of curved surfaces and the
volumes enclosed by them.



Applied Mathematics, Solutions to Practical Problems

Developments also took place in applied mathematics such as in creation of
trigonometric tables and measurement units. Yativrsabha's work Tiloyapannatti
(6th C) gives various units for measuring distances and time and also describes
the system of infinite time measures.

In the 9th C, Mahaviracharya ( Mysore) wrote Ganit Saar Sangraha where he
described the currently used method of calculating the Least Common Multiple
(LCM) of given numbers. He also derived formulae to calculate the area of an
ellipse and a quadrilateral inscribed within a circle (something that had also
been looked at by Brahmagupta) The solution of indeterminate equations also drew
considerable interest in the 9th century, and several mathematicians contributed
approximations and solutions to different types of indeterminate equations.

In the late 9th C, Sridhara (probably Bengal) provided mathematical formulae for
a variety of practical problems involving ratios, barter, simple interest,
mixtures, purchase and sale, rates of travel, wages, and filling of cisterns.
Some of these examples involved fairly complicated solutions and his
Patiganita is considered an advanced mathematical work. Sections of the book
were also devoted to arithmetic and geometric progressions, including
progressions with fractional numbers or terms, and formulas for the sum of
certain finite series are provided. Mathematical investigation continued into
the 10th C. Vijayanandi (of Benares, whose Karanatilaka was translated by
Al-Beruni into Arabic) and Sripati of Maharashtra are amongst the prominent
mathematicians of the century.

The leading light of 12th C Indian mathematics was Bhaskaracharya who came from
a long-line of mathematicians and was head of the astronomical observatory at
Ujjain. He left several important mathematical texts including the Lilavati and
Bijaganita and the Siddhanta Shiromani, an astronomical text. He was the
first to recognize that certain types of quadratic equations could have two
solutions. His Chakrawaat method of solving indeterminate solutions preceded
European solutions by several centuries, and in his Siddhanta Shiromani he
postulated that the earth had a gravitational force, and broached the fields of
infinitesimal calculation and integration. In the second part of this treatise,
there are several chapters relating to the study of the sphere and it's
properties and applications to geography, planetary mean motion, eccentric
epicyclical model of the planets, first visibilities of the planets, the
seasons, the lunar crescent etc. He also discussed astronomical instruments
and spherical trigonometry. Of particular interest are his trigonometric
equations: sin(a + b) = sin a cos b + cos a sin b; sin(a - b) = sin a cos b -
cos a sin b;



The Spread of Indian Mathematics

The study of mathematics appears to slow down after the onslaught of the Islamic
invasions and the conversion of colleges and universities to madrasahs. But this
was also the time when Indian mathematical texts were increasingly being
translated into Arabic and Persian. Although Arab scholars relied on a variety
of sources including Babylonian, Syrian, Greek and some Chinese texts, Indian
mathematical texts played a particularly important role. Scholars such as Ibn
Tariq and Al-Fazari (8th C, Baghdad), Al-Kindi (9th C, Basra), Al-Khwarizmi (9th
C. Khiva), Al-Qayarawani (9th C, Maghreb, author of Kitab fi al-hisab al-hindi),
Al-Uqlidisi (10th C, Damascus, author of The book of Chapters in Indian
Arithmetic), Ibn-Sina (Avicenna), Ibn al-Samh (Granada, 11th C, Spain),
Al-Nasawi (Khurasan, 11th C, Persia), Al-Beruni (11th C, born Khiva, died
Afghanistan), Al-Razi (Teheran), and Ibn-Al-Saffar (11th C, Cordoba) were
amongst the many who based their own scientific texts on translations of Indian
treatises. Records of the Indian origin of many proofs, concepts and
formulations were obscured in the later centuries, but the enormous
contributions of Indian mathematics was generously acknowledged by several
important Arabic and Persian scholars, especially in Spain. Abbasid scholar
Al-Gaheth wrote: " India is the source of knowledge, thought and insight".
Al-Maoudi (956 AD) who travelled in Western India also wrote about the
greatness of Indian science. Said Al-Andalusi, an 11th C Spanish scholar and
court historian was amongst the most enthusiastic in his praise of Indian
civilization, and specially remarked on Indian achievements in the sciences and
in mathematics. Of course, eventually, Indian algebra and trigonometry reached
Europe through a cycle of translations, travelling from the Arab world to Spain
and Sicily, and eventually penetrating all of Europe. At the same time, Arabic
and Persian translations of Greek and Egyptian scientific texts became more
readily available in India



Yet, few modern compendiums on the history of mathematics have paid adequate
attention to the often pioneering and revolutionary contributions of Indian
mathematicians. But as this essay amply demonstrates, a significant body of
mathematical works were produced in the Indian subcontinent. The science of
mathematics played a pivotal role not only in the industrial revolution but in
the scientific developments that have occurred since. No other branch of science
is complete without mathematics. Not only did India provide the financial
capital for the industrial revolution (see the essay on colonization) India also
provided vital elements of the scientific foundation without which humanity
could not have entered this modern age of science and high technology.



Notes:

Mathematics and Music: Pingala (3rd C AD), author of Chandasutra explored the
relationship between combinatorics and musical theory anticipating Mersenne
(1588-1648) author of a classic on musical theory.



Mathematics and Architecture: Interest in arithmetic and geometric series may
have also been stimulated by (and influenced) Indian architectural designs -
(as in temple shikaras, gopurams and corbelled temple ceilings). Of course, the
relationship between geometry and architectural decoration was developed to it's
greatest heights by Central Asian, Persian, Turkish, Arab and Indian architects
in a variety of monuments commissioned by the Islamic rulers.



Transmission of the Indian Numeral System: Evidence for the transmission of the
Indian Numeral System to the West is provided by Joseph (Crest of the Peacock):-

· Quotes Severus Sebokht (662) in a Syriac text describing the "subtle
discoveries" of Indian astronomers as being "more ingenious than those of the
Greeks and the Babylonians" and "their valuable methods of computation which
surpass description" and then goes on to mention the use of nine numerals.

· Quotes from Liber abaci (Book of the Abacus) by Fibonacci
(1170-1250): The nine Indian numerals are ...with these nine and with the sign 0
which in Arabic is sifr, any desired number can be written. (Fibonaci learnt
about Indian numerals from his Arab teachers in North Africa)



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