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Description
This book describes applications of a new mathematical device: the osculator. Osculation involves applying the osculator to a number or algebraic expression repeatedly until the end of the number/expression is reached. They have many applications: in factorisation (of numbers and polynomials), divisibility, number bases, substitutions, recurring decimals, continued fractions etc.
Details
156 pages.
Size: 24.5cm by 19cm.
Paperback. 2024
Author: Kenneth Williams
ISBN 978-1-902517-47-6.
Introduction
INTRODUCTION
Osculation is an important part of the system of Vedic mathematics, a system which was reconstructed from the ancient Vedas by Sri Bharati Krishna Tirthaji about a century ago. Osculation is a simple but powerful process with an extensive range of applications and variations; it is like a key that is repeatedly applied to some other value in a consistent way. There are many types of osculator and many types of osculation.
In its simplest form osculation involves multiplying the last digit of a number by a number (the osculator) and adding the next digit. This process is repeated until the other end of the number is reached.
We may osculate:
right to left or left to right,
with groups of digits,
with variables instead of numbers.
Tirthaji devoted two chapters in his book1 to divisibility devices and two chapters to recurring decimals (discounting the first chapter) - 81 pages in all – suggesting that he saw these two topics as significant, and in fact they are closely linked. The Illustrative Examples that follow this Introduction will give an idea of some the applications of osculation.
This material is unusual and probably unique. Tirthaji’s osculators are a theme in mathematics, with applications in many areas. Many aspects of the Vedic system surprise, and need to be practised for their full advantages to be appreciated:
- like bar numbers (introduced here in Appendix 1 for those not familiar with them),
- numbers with more than one digit in a place,
- working both right to left as well as left to right
- and the close and natural relationship between numbers and algebra.
The ease with which the Vedic methods can be assimilated, understood and carried out also leads to a preference for mental work which in turn promotes creativity.
The processes introduced here are simple, easily understood and mentally attractive. And the appealing osculation pattern has so many applications that it is assimilated quickly and naturally.
The techniques shown are explained with many examples, and also algebraically so as to give a precise exposition of their working and range of application. Proofs are given too as we go along. Many exercises are included, with answers following immediately below them.
Osculators can be positive, negative, fractional, variable, and even imaginary; they have a special notation and can be combined in various ways.
This book (which is a shortened version of the original) will be useful to teachers who wish to offer new insights in their lessons, to those interested in original ideas or in the Vedic system, or to the general reader who likes new, unifying ideas.
Contents
INTRODUCTION vi
ILLUSTRATIVE EXAMPLES viii
1 Special Numbers 1
Disguises 3
Special Numbers – Two Types 5
The Osculators 5
Summary 6
2 Osculation 7
The Positive Osculator, P 8
Osculation 9
Modulo Arithmetic 10
A Simplification 11
Osculator Notation 12
Summary 14
3 The Negative Osculator 15
Generating Negative Osculators, Q 15
P + Q = D 17
Osculating with Q 18
Reverse Osculation, Using P1 or Q1 20
Halving the Work 23
Special Numbers and Osculators 23
Summary 24
4 Rapid Osculation Devices 25
Proof of Earlier Results 25
The Product Law 26
Other Devices 29
Osculator Wheels 30
Summary 30
5 Symmetries 31
The Mid-Point 31
The Symmetrical Products Formula 32
The Quarter Point 36
The Three Quarter Point 36
Application 37
When L is odd 37
Summary 39
6 The Last Osculator 40
Beginning and End of a Cycle 41
Expressions of Unity 41
Neighbours of P0 44
Summary 46
7 All Numbers are Special 47
Coefficient of Pn and Qn 47
Extension of Osculator Notation 49
Immediate Osculators 50
The Osculator Notation 51
Finding the Best Osculators 53
Unlimited Special Numbers 56
Summary 57
8 Divisibility Testing 58
At-Sight Solutions 58
Divisibility by a number ending in 9 60
Divisor ending in 1, 3, 7 or 9 63
Divisor not ending in 1, 3, 7, 9 64
The Negative Osculator 65
Groups of Digits 67
Summary 69
9 Crosswise Factorisation 70
Common Factors 70
Factorisation 74
General Factorisation 76
Summary 81
10 The Remainder Osculators 82
The Remainder Osculators 82
Opposing Power Sequences 83
Special Numbers 84
Summary 88
11 Using the Remainder Osculators 87
Osculating from the Left 87
Useful Remainder Osculators 91
Checking 93
Summary 94
12 Number Bases 95
Place Value 95
Converting into Base Ten 95
Converting From Base 98
Summary 101
13 Osculating with 10 102
Base 10 102
Useful Factorisation Facts 103
Quadratics 104
Cubics 106
Higher Order Polynomials 108
Specific and General 109
Summary 110
14 Polynomial Osculation 111
Polynomials and Numbers 111
Translating Polynomial Form 112
Substitution and Division 112
Factorising Cubics 114
Summary 116
15 Factorising Polynomials 117
A Polynomial in x is a No. in Base x 117
Quadratics 118
Cubics 120
Higher Order Polynomials 123
Summary 124
16 Recurring Decimals 125
Rec. Dec., Osculators & Rems 125
Getting Recurring Decimal Digits 127
Equivalent Fractions 131
Summary 133
APPENDICES - Bar Numbers 134
Osculator Tables 140
GLOSSARY 145
REFERENCES 145
INDEX 146
Illustrative Examples
- Find 23 × 101 and 93 × 67. (Chapter 1, Examples 1 and 8))
23 × 101 = 2323
93 × 67 = 31 × 201 = 6231.
- Find the position of the point a quarter of the way through the osculator cycle for
D = 17. (Chapter 5, Example 6)
Q42 = 42 = -1. So, n = 4.
- Is 123451 divisible by 41? (Chapter 8)
|
12 3 4 5 1
|
Osculating with -4 we arrive at zero. So, Yes.
- Find the smallest number, divisible by 19, that ends in four 7s. (Chapter 8)
Osculating four 7s by 2 we arrive at 1, so 187777 is the number.
- Substitute x = 3 into x3 – 2x2 – 5x + 1 . (Chapter 14, Example 11)
|
x3 – 2x2 – 5x + 1
|
Osculating left to right with 3 we arrive at -5.
- Find the cycle length for 1/69 . (Chapter 6, Example 6)
Q-1 = -10 = 59 = 128 = 27 = Q37 = Q21
∴ L = 21 – -1.
∴ There are 22 digits in the cycle.
- Find the common factors of 403 and 589. (Chapter 9)
44 + 18 = 62 = 2 × 31 ∴ 31 is the only common factor.
- Factorise 6097. (Chapter 9)
6097 = 7 × 13 × 67 (using 1001, and/or 201).
- Find the remainder when 123456 is divided by 34. (Chapter 11, Example 5)
|
12 34 56
|
Osculating with -2 we find that the remainder is 2.
- Check 1234 × 4567 = 5635678 using bases 7 and 13. (Chapter 11, Example 9)
We find that 2 × 3 = 6, and that 12 × 4 = 9. Both confirm the answer.
- Convert 12435 into base ten. (Chapter 12, Example 3)
Osculating by 5 we find that 12435 = 19810.
- Factorise 2x2 – x – 6. (Chapter 13)
Since 184 = 23 × 8 ∴ 2x2 – x – 6 = (2x + 3)(x – 2).
- Divide 2x3 + 7x2 + 9x + 3 by x – 2. (Chapter 14)
Osculating with 2 from the left we get 2x2 + 11x + 31 remainder 65.
- Factorise p = 2x3 + 3x2 – 14x + 6. (Chapter 15, Example 5)
Osculating with x = ½ from the left, or x = 2 from the right, gives zero.
So, 2x3 + 3x2 – 14x + 6 = (2x – 1)(x2 + 2x – 6). (Or, using p1 = -3.)
Back Cover
An osculator is a number or variable that is applied repeatedly to another value.
It is like a key that can be applied left to right, right to left, singly or in groups.
Osculators have their own arithmetic, by which they can be combined.
This is an unusual but far-reaching concept which has many applications and which seems to be a unique.
It is used with surprising effect in factorisation (of numbers and polynomials), divisibility testing, number bases, substitutions,
recurring decimals, continued fractions etc. There are doubtless other applications that can be discovered.
Kenneth Williams has been researching and teaching Vedic Mathematics for over 50 years. He has published many articles,
books and DVDs and has been invited to many countries to give seminars and courses. His online courses are available at Math2Shine.com, including teacher training.