This document provides an introduction to the concepts of Vedic mathematics. It begins with a preface describing the purpose and contents of the book. The introduction then discusses the history of Vedic mathematics, noting that it was developed based on 16 sutras or formulas found in ancient Hindu scriptures. It explains that Vedic mathematics allows for mental calculations and flexibility. Chapter 1 defines Vedic mathematics as a system based on 16 sutras and provides a brief history of mathematics in ancient India. Chapter 2 outlines several key features of Vedic mathematics, including its coherence, flexibility, ability to do calculations mentally or in writing, promotion of creativity, appeal to all levels, increase in mental ability, efficiency, ease of use,

1. AN INTRODUCTION TO VEDIC
MATHEMATICS- MULTIPLICATION
By
SARANYA U N
(B.Sc , Mathematics)

2. Preface
The book “An Introduction to Vedic mathematics-multiplication” is intended for the
students and teachers in any syllabus. In this book all the levels of multiplication
have been deal with in a simple and lucid manner. A sufficiently large number of
problems have been solved. By studying this book , the student is expected to
understand the concept of Vedic mathematics, a little history,features,Vedic
multiplication, etc. Suggestion for the further improvement of this book will be
highly appreciated.
Saranya U N.

3. CONTENTS
Title Page No:
Preface
Introduction
Chapter 1. what is Vedic mathematics 2
A little history 3
Chapter 2. Features 6
Chapter 3. Sixteen Vedic Sutras 10
Chapter 4. Basic terms and laws 12
Chapter 5. Multiplication 16
Reference

4. INTRODUCTION
Vedic mathematics – a gift given to this world by the ancient sages of
India. A system which is far simpler and more enjoyable than modern
mathematics. 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 one method. This creates more creative, interested
and intelligent pupils. Vedic Mathematics refers to the technique of calculations
based on a set of 16 Sutras, or aphorisms, as algorithms and their upa-sutras or
corollaries derived from these Sutras. Any mathematical problems (algebra,
arithmetic, geometry or trigonometry) solved mentally with these sutras. Vedic
Mathematics is more coherent than modern mathematics.
Vedic Mathematics offers a fresh and highly efficient approach to
mathematics covering a wide range – starts with elementary multiplication and
concludes with a relatively advanced topic, the solution of non-linear partial
differential equations. But the Vedic scheme is not simply a collection of rapid
methods; it is a system, a unified approach. Vedic Mathematics extensively
exploits the properties of numbers in every practical application.

5. CHAPTER – 1
What is Vedic Mathematics
 Vedic period begins around 1500 BC and ended after 500 BC
 Vedas (Books of Knowledge) are the most sacred Hindu Scriptures
 Atharvavedam – supposedly contains a set of sixteen sutras that describe all
of mathematics
 Sutra is often translated word formula and is short and easily memorized and
recited
 Vedic Mathematics is a system of mathematics based on these sixteen sutras

6. A LITTLE HISTORY
 Several important mathematical concepts came out of the subcontinent
 Decimal place value system
 Arabic numerals based on symbols used here
 Zero (also discovered independently elsewhere)
 Mathematical astronomy in use by third millennium B.C.
 Mathematics was used during Vedic period for the construction of alters
 Jainism followed the Vedic period and found mathematicians working with
 Cubic and quartic equations,
 Permutations and combinations,
 A rather developed notion of infinity, including multiple “levels” or
“sizes” of infinity.

7. JAGADGURU SWAMI SRI
BHARATI KRSNA TIRTHAJI MAHARAJA
 Born in 1884 to an educated and pious family
 Received top marks in school
 Sat for the M.A. exam of the American College of Sciences (Rochester NY)
in Sanskrit, Philosophy, English, Mathematics, History and Science.
 Became Sankaracharya (major religious leader) of Govardhana Matha (akin
to a monastery) in Puri, a city in the east Indian state of Orissa
 Wrote sixteen volumes based on sixteen Sutras written 1911-1918

8.  Volumes were unaccountably lost without a trace
 Rewrote manuscript from memory in 1956-1957 before touring the USA;
published posthumously in 1965 as “Vedic Mathematics”

9. CHAPTER – 2
FEATURES
Here are many features of the Vedic system which contrast
significantly with conventional mathematics.
Coherence – Perhaps the most striking feature of the Vedic system is its
coherence. Instead of a hotchpotch 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 root. And these
are easily understood. This unifying quality is very satisfying; it makes
mathematics easy and enjoyable and encourages innovation.
Flexibility – In modern teaching you usually have one way of doing a
calculation. This is rigid and boring, and intelligent and creative students
rebel against it. Once you allow variations you get all sorts of benefits.
Children become more creative. The teacher is encouraging innovation and
children respond. In the Vedic system there are general methods that always
work, for example a method of multiplication that can applied to any
numbers. But the Vedic system has many special methods, when a
calculation has some special characteristics that can be used to find the
answer more easily. And it’s great fun when you spot that method.

10. Having only one method of, say, multiplying is like a carpenter who uses a
screwdriver for every job. The skilled craftsman selects the tool most
appropriate for the job and gets it done quicker, better and with more
satisfaction.
So there are special methods that apply in special cases and also general
methods. You don’t have to use these special methods but they are there if
you want to.
Calculations can often be carried out from right to left or from left to right.
You can represent numbers in more than one way; we can work 2 or more
figures at a time if we wish.
This flexibility adds to the fun and gives pupils the freedom to choose their
own approach. This in turn leads to the development of creativity and
intuition. The Vedic system does not insist on a purely analytic approach as
many modern teaching methods do. This makes a big difference to the
attitude which children have towards mathematics.
In this rapidly changing world adaptability and flexibility are absolutely
essential for success. For the future we can expect more change and perhaps
at a more rapid pace.
Mental, improves memory – The ease and 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 creates to more creative, interested and intelligent pupils. It
also leads to improved memory and greater mental agility.

11. Bear in mind also that mathematical objects are mental objects. In working
directly with these objects as in mental math you get closer to the objects
and understand them and their properties and relationships much better. Of
course there are times especially early on when physical activities are great
help to understanding.
Promotes creativity – All these features of Vedic math encourage students
to be creative in doing their math. Being naturally creative students like to
devise their own methods of solution. The Vedic system seeks to cultivate
intuition, having a conscious proof or explanation of a method beforehand is
not essential in the Vedic methodology. This appeals to the artistic types
who prefer not to use analytical ways of thinking.
Appeals to every one – The Vedic system appears to be effective over all
ability ranges: the able child loves the choice and freedom to experiment and
less able may prefer to stick to the general methods but loves the simple
patterns they can use. Artistic type love the opportunity to invent and have
their own unique input, while the analytic type enjoy the challenge and
scope of multiple methods.
Increases mental ability – Because the Vedic system uses the ultra-easy
methods mental calculation is preferred and leads naturally to develop
mental ability. And this in turn leads to growth in other subjects.
Efficient and fast – In the Vedic system ‘difficult’ problems or huge sums
can often be solved immediately. These striking and beautiful methods are
just a part of a complete system of mathematics which is far more systematic

12. than the modern ‘system’. Vedic Mathematics manifests the coherent and
unified structure naturally inherent in mathematics and the methods are
direct, easy and complementary.
Easy, fun – The experience of the joy of mathematics is an immediate and
natural consequence of practicing Vedic Mathematics. And this is the true
nature of maths – not the rigid and boring ‘system’ that is currently
widespread.
Methods apply in algebra – another important feature of the Vedic system
is that once an arithmetic method has been mastered the same method can be
applied to algebraic cases of that type – the beautiful coherence between
arithmetic and algebra is clearly manifest in the Vedic system.

13. CHAPTER – 3
SIXTEEN VEDIC SUTRAS (FORMULAE)
The original Sanskrit sutras (formulae) with their generalized mathematical
meaning are as follows:
1. Ekadhikena purvena –
By one more than previous one.
2. Nikhilam Navatascharamam Dashatah –
All from nine and last from ten.
3. Urdhva triyagbhyam –
Vertically & cross wire.
4. Paravarthy yojayet –
Transpose & apply
5 Shunyam samyasamuchchaye
The summation is equal to zero
6. shunyamanyat-
If one is in ratio , other one is zero
7 Sankalanam-vyavakalanam-

14. By addition and subtraction
8. puranapuranaabhyam
By completion and non completion.
9. chalan kalanabhya
Sequential motion
10. Yavadunam-
The deficiency.
11. Vyashtisamashtih-
Whole as one & one as whole.
12. Sheshanyanken charamena-
Remainder by last digit.
13. Sopantyadwayamantyam-
Ultimate and twice the penultimate.
14. Ekanyunena Purvena-
By one less than the previous one.
15. Gunit Samuchchayah-
The whole product is same.
16. Gunak Samuchchayah-
Collectivity of multipliers

15. CHAPTER- 4
BASIC TERMS AND LAWS
It is essential to know certain important terms before proceeding with Vedic
Mathematics. The terms are as follows
a) Ekadhika (one more)
e. g 1) Ekadhika of 4 = 4+1 = 5 2)
Ekadhicaof25=25+1=26
b) Ekanyuna (one less)
e.g*1) Ekanyuna of 9 = 9-1 = 8 2)
Ekanyunaof17=17-1=16
c) Purak (complement)
e.g 1) Purak of 6 from 10 is 4
2) Purak of 8 from 9 is 1
3) Purak of 1 from 10 is 9
d) Rekhank
Rekhank means a digit bar on its head. Usually Purak is a
Rekhank.
Purak of 7 from 10 is (7-10)= -3 or 3
Purak of 7 from 9 is (7-09)= -2 or 2

16. e) If we write 1 to 9 in ascending order, we get the first vedic formula- Ekadhika
Purvena. But if we write 9 to 10 in descending order , we get the fourteenth
formula — Ekanyunen Purvena.
Thus , Formula 1: 1 2 3 4 5 6 7 8 9
Formula 2: 9 S 7 6 5 4 3 2 1
Formula 1&14 as in above order , are the complements of 10 (i.e. 1 & 2, 2 &
S, 3 & 7 & so on).
f) Any negative digit or number can written as a bar on the top of that
digit or number.
e.g : -8 can also be written as 8
-34 can also be written as 3 4 or 34
g)Addition and Subtraction:
(1) Addition of t wo positive digits or two negative digits
(Rekhanks) means their addition with the sign of these digits.
e.g : 3+5 = +8 or 8
(-3)+(-5) = 3+5 = -8 or 8
(2) Addition of one positive and negative digit means their
difference with the sign of higher digit.
e.g : 5+3 = 5-3 = 2
5+3 = -5+3 = -2 or 2

17. (3) Subtraction of Rekhank from a positive digit means changing the
sign of Rekhank followed by their addition.
e.g: 5-3 = 5 - (-3) =5+3 = 8
2-6 = 2 -- (-6) =2+6 = 8
(4) Subtraction of a positive digit from a Rekhank means changing
the sign of positive digit followed by their addition.
e.g: 3-4 = 3 + (4) = 7
8-14 = 8 + 14 = 22 or 2 2
h) Multiplication and Division :
(1) The product of two positive digits or two negative digits
(Rekhanks) is always positive
e.g: 3 * 5 = 15; 4 * 7 = 28
i.e. ( + ) * ( + ) = (+) and (-)* H = (+)
(2) The product of one positive digit and one rekhank is always a
rekhank or negative number.
e.g: 5 * 5 = 25; 7 * 3 = 21
i.e. (+) * (-) = (-); and (-) * (+) = (-)
(3) The division of two positive digits or two Rekhanks is always
positive
e . g : 8 / 2 = 4 ; 6 / 3 = 2

18. i . e . ( + ) / ( + ) ; a n d ( - ) / ( - ) = ( + )
Beejank
It means the conversion of any number into a single digit. It is done by the
addition of all digits of the number. If the addition is a two digit number , these digits
of addition are further added to get a single digit.
e.g. Beejank of 125 = 1 + 2 + 5 = 8
Beejank of 31426= 3 + 1 + 4 + 2 + 6 = 16
And again 1 + 6 = 7

19. CHAPTER 5
MULTIPLICATION
BY NIKHILAM SUTRA
We now pass on to a systematic exposition of certain sailent , interesting,
important an d necessary of the at most value and utility in connection with
arithmetical calculations and beginning with the processes and methods
described in the Vedic mathematical sutras.
Suppose we have to multiply 9 by 7 (10)
(1)We should take, as base for our calculations, that 9-1
Power of 10 which is nearest to the numbers to be 7-3
Multiplied. In this case 10 itself is that of power; --------
(2) Put the two numbers 9 an d 7 above and below on 6/3
The left-hand side
(3)Subtract of each item from the base (10) and write down the reminders (1and 3) on the
right hand side with a connecting minus sign (-) between them, to show that the numbers
to be multiplied are both of them less than 10.
(4) The product will have two parts, one of the left side and one of the right. A vertical
dividing line may be drawn for the purpose of demarcation of the two parts.
(5) Now, the left-hand-side digit can be arrived on the way,

20. Subtract the sum of two deficiencies
(1+3+=4) from the base (10). You get
the same answer (6) again. 10-1-3=6
(6) Now vertically multiply the two deficit figures(1 and 3). The product is 3. And this is
the right-hand-side proportion of the answer.
(10) 9-1
7-3
6/3
(7) Thus 9*7=63
More examples
Special case-1
The base now required is 100.
1)91*91 2)93*92 3)89*95
91-9 93-7 89-11
91-9 92-8 95-5
82/81 85/56 84/55
4)88*88 5)25*98
88-12 25-75
88-12 98-2
76/144=77/44 23/150=24/50

21. We can extend this multiplication rule to numbers consisting of a larger number of
digits, thus:
1) 888*998 2) 879*999 3) 888*991
888-112 879-121 888-12
998-002 999-001 991-009
886/224 878/121 879/1008=888/008
4)988*998 5)99979*99999
988-012 99979-00021
998-002 99999-00001
986/024 99978/00021
6) 999999997*999999997
999999997-000000003
999999997-000000003
999999994/000000009
Special case-2
1)12*8 2)107*93
12+2 107+7
8-2 93-7
10/4 =96 100/ 49=99/51

22. 3)1033*997 4)10006*9999
1033+33 10006+6
997-3 9999+1
1030/099=1029/901 10005/0006=10004/9994
MULTIPLES AND SUBMULTIPLES
Suppose we have to multiply 41 by 41. Both of these are away from the base 100
that by our adopting that as our actual base, we shall get 59 and 59 as the deficiency from
the base
1) our chart will take this shape:
100/2 =50
(1) We take 50 as working base.
(2) By cross multiplication we get 32 on the left hand side.
41-9
41-9
32/81
16/81
(3) As 50 is a half of 100, we therefore divide32 by 2and put 16 down as
the real left-hand-side portion of the answer.
(4) The right-hand –side portion 81 remain unaffected.
(5)Therefore answer is 1681.

23. OR
2) Instead of taking 100 as our theoretical base and its multiple 50 as our working base
and dividing 32 by 2, we may take 10, as our theoretical base and its multiple 50 as our
working base and ultimately multiply 32 by 5 and get 160 for the left-hand-side. And as
10 as our theoretical base and we are therefore untitled to only one digit on the right hand
side, we retain one of the 81 on the right hand side, “carry” the 8 of the 81 over to the left,
add it to the 160already and thus obtain 168 as our left hand side portion of the answer.
The product of 41 and 41 is thus found to be 1681.
10*5=50
41- 9
41-9
32/8 1
*5
160/81=1681
OR
3) Instead of taking 100 or 10as our theoretical base and 50 a sub-multiple, we may
take 10 and 40 as the bases respectively and work at the multiplication as shown
below. And we find that the product is 1681 the same as we obtained by the first
and the second methods.

24. 10* 4= 40
41+ 1
41+1
42/1
*4/
168/1
Examples
1) (1)59*59 OR (2)59*59
Working base 10*6=60 Working base 10*5=50
59 – 1 59 + 9
59 – 1 59+9
58/1 68/81
*6/ *5/
348/ 1 348/ 1
2) 23* 23 3)48*49
Working base=10*2=20 Working base=10*5=50
23+3 48 - 2
23+3 49 - 1
26/9 47/ 2
*2/ *5/
52/9 235/2

25. 4) 249*245 5)46*44
Working base = 1000/4=250 Working base=100/2
249-1 46-4
245-5 44-6
244/005 40/24
61/005 20/24
Suppose we have to find the square of 9.
The following will be successful stage of mental working:
(1) We should take up the nearest power of 10, I.e., 10 itself as our base.
(2) As 9 is 1 less than 10, we should decrease it still further by 1 and set 8 down as
our left side portion of the answer.
8/
(3) And, on the right hand, we put down the square of that deficiency 12
(4) Thus 92
=81
E.g. 72
= (7-3) / 32
=4/9=49
122
=(12+2)/22
=14/4=144
152
=(15+5)/ 52
=20/(2)5=225
192
=(19+9)/92
=28/(8)1=361
912
=(91-9)/92
=82/81=8281

26. 1082
=(108+8)/82
=116/64=11664
Special cases
Squaring of numbers ending with 5
E.g. Square of 15
Here last digit is 5 and previous one is 1. So, one more than that is 2. Now, Sutra in
this context tells us to multiply the previous digit by more than itself, i.e. by 2. So
the left hand side digit is 1*2: and the right-hand side digit is the vertical-
multiplication product, i.e. 25.
Thus 152
=1*2/25=22/5=225
Similarly
252
=2*3/25=62/5=625
652
=6*7/25=42/25=4225
1152
=11*12/25=132/25=13225

27. REFERENCE
1)Vedic mathematics
-Jagadguru Swami sri Bharati Krsna Tirthaji Maharaja
Sankaracharya of govt. ardhana matha, puri
Published by
Motilal Banarsidass publishers
PTE LTD.
2) Magical world of mathematics
-T S Unkalkar
3)www.Wikipedia.org