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 P_n and Q_n                           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   0  3  0  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 1     1      -2    -5

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            10        36=2

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.