i

MODERN APPROACH

TO

SPEED MA+H SECRE+ Key to Master Speed Mathemagic

By

Vitthal B. JadhavVitthal B. JadhavVitthal B. JadhavVitthal B. Jadhav

vii

Foreword

The growing appeal of ‘speed math ’ is a result of new fast calculation methods

that have arisen over recent years. This is due to the fact that methods now exist

which are so easy that mental calculation becomes an attraction: the ability to

perform involved calculations easily is seen as a challenge and a delight.

Doing math purely for the joy of it is not the only attraction though. Quick on-the-

spot calculations are often required in everyday life, for example in business

meetings, while trying to establish the amount of paint or number of tiles for a project

etc, while in the shop. Calculation can also be used to impress and as stimulation to

the mind. Mental mathematics has many benefits in improving ones mental agility,

memory and so on.

Previously, calculations were carried out on paper, or with a calculator or similar

device and this were understandably seen as a drudge. Now people are beginning to

see that such work can be done easily, using the mind only, and therefore with a

much greater sense of achievement, and often quicker than the alternatives.

The power of these methods is due to the use of Vedic mathematics, which is a

system of mathematics reconstructed by Bharati Krsna Tirthaji between 1911 and

1918 and published in his book ‘Vedic Mathematics’ in 1965.

This system uses the natural functions of the mind to create the most refined and

effective devices possible, and since it is the mind which creates mathematics this

inevitably results in the most pleasing and efficient mental system possible. On the

basis that the Sutras of Vedic Mathematics describe all the natural functions of mind

we can be assured that the Vedic system must give the ultimate in efficiency in

mental mathematics.

viii

In this book the author shows various devices that can be used to perform speed

calculations. Many are totally new, but in any case practice will be needed in order to

appreciate the effectiveness of the methods and to get skilled using them.

One of the most brilliant and useful ideas of all time is also discussed here – the

invention of zero. Giving a name and a symbol to nothing by the ancient Indians was

truly a stroke of genius and was the key to developing the extraordinarily powerful

number system which is used the world over today.

In the innovative techniques shown here the reader will find much to think about

and study. Many areas of mathematics are examined in an illuminating and

informative way.

MathematicianKenneth R. William

( Author of Triple , Cosmic Calculator , Vertically and Crosswise )

PREFACE

Nature is dressed up with different types of treasure and secrets.

Some events in nature occur randomly (arbitrarily) while other events occur

according to specific pattern. There is reason behind each past, present or

future event. Curious mind had always tried to find reason behind this event

which leads to birth of sciences like astronomy, economics, statistics,

physics, chemistry, geography, geology, mathematics etc. Roots of all these

scientific branches track back to only one subject known as mathematics !

Due to this reason - Carl Friedrich Gauss (Prince of mathematics) called

mathematics as “Queen of science”.

Arithmetic, economics, astronomy, physics , chemistry etc scientific

branches had played important role in human progress. In ancient time ,

mathematics was mainly used to count object , area. Tally mark or rope

with knot is used to record number of object, but this method was not so

effective. Further symbol were used to track increment or decrement in

product. Different counting system like Roman number system , decimal

number system were used in different parts of world. Decimal (Hindu-Arabic)

number system – invented by Indian mathematicians lived in north-west

India at the beginning of the Vedic period- was supreme among all

counting (number) systems. Decimal number system is boon to scientific

world.

Decimal number system is introduced to whole world by Indian

mathematician and astronomer- Brahmagupta - born in 598 A.D. He wrote

‘Bramhasphut Sidhant’ and ‘Khandakhadyaka’ at age of 30 and 65

respectively. In 10th century- Al Mamuna translated these two novels into

Arabic language as 'Sind Hind' and ‘Al-Arkada’ respectively in order to

introduce decimal number system to Arab. In 1202 Italian mathematician

Leonardo of Pisa (Fibonacci) wrote ‘Liber Abacci’ to spread decimal number

system in Europe. However, almost up to 18th century - decimal system was

not fully due to misleading opinion or opposition posed by POP and church.

Revolution begins in European mathematics after fully accepting the decimal

system.

Decimal system gives easy and abstract way to represent any

number. Unlike other counting system, this system use only finite number of

symbol. The decimal system can express any number by using only ten

symbols i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . It is efficient over other counting

system (like Roman number system) due to its compact and proper

representation. It gives way to represent number so that rules for primary

operation like addition, subtraction, multiplication, division becomes simple

and faster. In short, easy and proper representation of number is key to

simplify primary operation ! Viz. 12452334 * 125 , 1234 * 9,999,998

apparently these two multiplication seems difficult, but represent 125 as

1000

8 and 9,999,998 as 10,000,002 . Then you will come up with answers

as 1556541750 and 12339997532 instantly and amazed by yourself ! Thus

multiplication can be simplified by representing number into alternate /

equivalent form containing more number of zeroes.

Similarly, same secret (principle) lies behind Trachtenberg speed

mathematics method, Vedic Mathematics or Booth's Multiplier responsible for

boosting multiplication speed in computer. For convenience, let us call easy

and proper representation of number as Emultiplier (Easy Multiplier).

Generally its difficult to accurately define Emultiplier. Alternate equivalent

form or representation of number containing more number of zeroes than

original number which can speed up given primary operation is called as

optimal (E-multiplier) form. According to this definition, Emultiplier of the

1,999,989 and 4,444,444 will be 2,000,011 representation and 64 4 *

9 910 −

form respectively.

In Binary Number System, if we need to multiply any number by

1,111,111 (decimal 127) , then we have to carry out process of adding 6

times . But if the same product is computed by optimal / Emultiplier form of

1,111,111 = 10,000,00 1 , then less number of addition (here just one

addition) is required to get answer. In computer science, 10000001 is called

as Booth’s multiplier of 1,111,111. This explains how Emultiplier boost speed

of multiplication in computer. Thus zero is secret key to master speed

mathematics. So understand zero, you will understand whole speed math.

No doubt- Indian will feel proud once again due to all above significance of

zeroes.

Book necessity

Today Vedic mathematics is becoming popular due to its simple, faster

and coherent methods. Vedic mathematics also gives an emphasis to zero.

(refer chapter named global number system to understand it) . Like a game,

there are very few rules or basic principle. Once one understand these rule,

then there is no need to remember thousand’s of shortcut. These rules help

to understand nature of number and awake number sense within us. Today

most of the speed math book does not reveal these rules. Such book tries

to teach shortcut as mathematics. Often true mathematics is not taught to

student. This book tries to teach fundamental concepts in speed math and

try to remove math phobia in student.

Why it is essential to study mathematics? How to understand it ?

Language like English, Marathi, Hindi etc has important role in progress

of society. Language assist us for sharing information, feeling etc .

Mathematics is universal language to understand nature, way to hunt

problem and reach desired goal. Secular mathematics tries to develop logical

thinking or intuitive ability for solving problems and give new insight to its

reader. If one need to understand nature / universe / surrounding , interpret

business - economic moves or decide better strategy to dominate over

market or protect oneself from exploitation carried out under the fancy name

of business or politics, understand recent trend in market – then one need

understand math. Quantitative or numerical data - like number of user in

particular area who use specific brand helps to many companies to boost

their profit, decide next strategy, taking decision regarding amount of product

to be supplied etc.

Presently used arithmetic notations were not used upto 16th the century.

Mathematics was mainly expressed by using linguistic phrases. There was

no significant progress due to ambiguity and less expressive power posed

by linguistic phrases. Modern notation is very brief. Few mathematical

notations can express huge information. Like western musical notation, math

notation follow strict rules and it is impossible to express information in very

few world by using language other than mathematics. Shortly, mathematics

is most abstract and unambiguous language. Modern mathematical language

is convenient for mathematician and experts but more complex to understand

for dummies. Present book tries to avoid use of mathematical notation as far

as possible due to complexity posed by mathematical notation and tries to

explain each formula (method) in informal spoken language. Like a love,

mathematics is universal language. Mathematics is present everywhere. In

nature, mathematics is found in form of Fibonacci sequence which is

considered as fingerprint of God. Word are building blocks of language. Any

spoken statement is formed by word. So it is essential to familiar with these

linguistic words in order to learn any language. In same way, to learn

mathematics one need to understand building block of math i.e. number.

The main aim of this book is to awake number sense and remove math

phobia in student. In addition, almost all methods in book are novel and

wherever necessary they are explained with plenty of example. No doubt-

reader will like to it. Just try to read it with open mind and observe beauty

of numerical pattern hidden in specific method. Like scientist or curious kid -

try to understand subject by two types of questions -“Why ? and ‘How ?’ –

which are mothers of every invention. At last, remember there are no such

thing exist which is called as possible-impossible, easy-difficult , beautiful –

disgusting , wise – foolish , modern – ancient , dirty – clean , general-specific

like – unlike , applicable – non applicable , gain-loss , real - virtual etc .

It lies in ones mentality (thought). Beauty lies in observer’s thought (or eye).

So change your mentality (thought) and tell yourself - “math is easy subject”

before starting to read this book.

‘Remember- zero is the equation of life. There is nothing to lose nor

to gain. So free from fear (because fear is worst quality in mankind that

refrain people from doing great things), be independent, entrepreneur and

legend, It doesn’t matter what world thinks about you but it matters what

you think about yourself !! So it doesn’t matter, how math feels to world.

Don’t listen to those who tell math is difficult. Tell your mind math is easy

and change your thinking. Soon you will experience the magic. According to

psychologist 50 % disease or any phobia can be easily cured just by

changing thoughts. So if one wants to become expert in specific subject,

then one need get rid of phobia and develop interest in desired subject by

directing thought accordingly.

At last, I dedicate this gem in golden garland to all readers.

Vitthal B. Jadhav

Jadhavvitthal1989@gmail.com

v

ContentForeword by Mathematician Kenneth R. William Vii

1. Inter Base Conversion Method 1

Multiplication / Division by 10n 13

Multiplication / Division by factor of 10n 15

Square of number ending with 5n 18

Splitting Principle 20

2. Monodigit Number 25

3. Faster division by 10n − 1 or monodigit number 49

4. Vertically Crosswise Multiplication Method 54

5. Global Number System 66

6. Ripple operator 82

7. Derivation of Trachtenberg Formulae to Multiplyany Number with 3-12

89

8. Square of number close to 10n having tens digitas x=7, 8 or 9

97

9. Square, Square Root and Cube of Specific Number 98

10. Recursive Square Method 102

11. Sliding Ruler Multiplication Method( Unification of Vertically Crosswise and Trachtenberg

Multiplication Method )

105

12. Duplex square made easy 114

13. Squaring and cubing of any number 117

14. Osculation based divisibility Test 121

15. Divisibility by 10n ± 1 and Its Application 129

16. Remainder Corollary 139

17. VJ’s Universal Divisibility Test 144

18. Divisibility Chart 160

19. Nth power of two digit number made easy 161

20. Novel Approach for Multinomial Expansion 171

vi

Content21. Computing mth Root of n digit number 178

22. Magical Game 183

23. Calendar calculation made easy 204

24. Why 0.999...... =1 ? 209

25. Common Balance Puzzle 211

26. Shift Add Representation and its Application 214

27. VJ’s Multiplication Method 219

28. Arithmetic Checking 229

Remainder computation 230

Algebraic Simplification 234

29. Modified Quine-McCluskey Method

( For engineering student )

235

30. VJ’s Matrix (Rectangular) Method

( One line method to compute nth root any number )

243

31. Vedic Division Method 270

32. Golden Lemma and Golden Pattern 278

33. Fun with Recurring Decimal 308

34. Principle behind Proportionately Sutra 335

Global Number System (Slides) 339

Bibliography 380

35 | Modern Approach to Speed Math Secret

Monodigit Number

i) Then m+ n = 9 gaps will be filled as follow.

i.e. L M R

2 8 14 20 24 22 16 10 4

+ES=6 +6 + 6 (m*ES) + 6 + 6 + ES=6

ii) Ripple carry addition by taking single digit as sum.

( 2 ) ( 0 8 ) ( 1 4) (2 0 ) (2 4 ) (2 2 ) (1 6 ) (1 0 ) (0 4 )

0 1 2 2 2 1 1 Carry = 0

2 9 6 2 6 3 7 0 4

4444*66666 296263704∴ =

3) 2222 * 77 = ?

2 4(7) *(2) ( ) *( )

7, 2,

2, 4

* 7*2 14 10

1, 4, 1 4 5

m na b

a m

b n

a b T U

T U ES

=

⇒ = =

= =

= = = +

⇒ = = = + =

i) L M R

1 06 10 10 09 04

+5 (2*5=10) + 5

= 1 7 1 0 9 4 (By using ripple carry addition)

77*2222 171094∴ =

Exercise

i) 222 * 5555 ii) 4444* 66666 iii) 777*777 iv) 8888*8888 v) 11111* 66666

v) 333*44 vii) 66 * 88 viii) 666*7777 ix) 666*555 x) 44444*5555

54 | Modern Approach to Speed Math Secret

Vertically Crosswise Multiplication Method

4. Vertically Crosswise Multiplication Method

Introduction

Vertically crosswise method is general multiplication method. We can calculate any

multiplication of two number or polynomial in one line by using vertically crosswise method. Let us

understand this method by following examples.

I. Multiplication by 2 Digit Number

1) 2 Digit × 2 Digit Multiplication

Example 1) 24 * 27 = ?

Steps

1) Multiply unit digit of both number vertically.

1 2

2 7

2*7 = 14 => 14

2) Cross multiply last 2 digit of both number and add them.

1 2

2*2 + 1 * 7 = 11 2 7

11 + 1 (Carry from 14) = 12 => 12 4

3) Multiply first digit of two number vertically

1 2

2 7

2*1 + 1 (carry) = 3 => 3 2 4

273 | Modern Approach to Speed Math Secret

Golden Lemma / Golden Pattern

GOLDEN LEMMA / GOLDEN PATTERN

BASIC TERM / OPERATION

1) Ascendant and Descendant

Suppose we need to expand multinomial

1 2 20 1 2 2 1....... +m m m

mm m

na x a x a x a x a x a

, then we

call coefficient of 1kx as descendant of coefficient of kx .

Thus , for above multinomial

0a is called descendant of 1a

1a is called descendant of 2a

1ma is called descendant of ma

Let us call 0 1 2 1 , , , ... ma a a a as descendants of multinomial. descendants

present in given term are called as descendants of term.

Similarly ,

ma is called ascendant of 1ma

1ma is called ascendant of 2ma

1a is called ascendant of 0a .

2) Operation on set

Let 0 1 2 1 , , , , mmkU T T T T T be set of terms, then

differential , integration and summation operation on set are defined as follow

0 1 2 1

0 1 2 1

0 1 2 1

, , , , ,

dx dx , dx , dx , , dx , dx

Sum of all members = ......

= SuAlso

mmk

mmk

mmk

d d d d d d

dx dx dx dx dx dxU T T T T T

U T T T T T

U T T T T T

m of all member in empty set = 0

274 | Modern Approach to Speed Math Secret

Golden Lemma / Golden Pattern

3) n

k

! for

! ( ) !

( 1) ( 2) ..... ( 1) for

!

( 1) ( 2) .... ( 2) ( 1) for ( )

( ) !

W , k W

R , k W

R , W

n

k n k

n n n n k

k

n n n k kn k

n k

n

nn

k

n

− − − − +

=

− − + +−

−

∈ ∈

∈ ∈

∈ ∈

VJ’s GOLDEN LEMMA

( )

( )

,

k , 0

0 0

0

0

0

, ,

* Then

Where

Highest power of x in ( ) =

P

0

U

Let k, l , m n W

( )

( )

l l

k

k

k

k

kk l

mn m l

lk

h ll

l

n

l

P

P

f x m p

n

k

R

f x a x

f x

h

a

S x

−

==

∞−

=

=

=

=

=

=

∈ ∈

=

∑∏

∑

∏

∑

( )

, 1 ,

, 1

1

present in

m

( )

U for 1

S = Sum of all members ( terms) in set U = U

lim

d

k

hx

l rl r

l r k

i

k k

l l l

a U

r

f x

x

dU ka

da

→∞

+

−

−

=

∀

≠

= ≥∫

∑

∪