key to master mathematics / innovative way to learn math
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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
→∞
+
−
−
=
∀
≠
= ≥∫
∑
∪
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