AN INTRODUCTION TO VEDIC

MATHEMATICS- MULTIPLICATION

By

SARANYA U N

(B.Sc , Mathematics)

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.

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

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.

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

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.

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

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”

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.

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.

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

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.

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-

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

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)

Ekadhica of 25 = 25+1= 26

b) Ekanyuna (one less)

e.g*1) Ekanyuna of 9 = 9-1 = 8 2)

Ekanyuna of 17= 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

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 w o 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) Addit ion of one posi t ive and negat ive d igi t means their

difference with the sign of higher digit.

e.g : 5+3 = 5-3 = 2

5+3 = -5+3 = -2 or 2

(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 posi t ive digi ts or two negative digi ts

(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

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

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,

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

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

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.

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.

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

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

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

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