Nice Numbers

327 Pages·2016·11.722 MB·English

by  John Barnes

John Barnes
Nice Numbers
JohnBarnes
Caversham
England
ISBN978-3-319-46830-3 ISBN978-3-319-46831-0 (eBook)
DOI 10.1007/978-3-319-46831-0
LibraryofCongressControlNumber:2016957844
©SpringerInternationalPublishingSwitzerland2016
ThisbookispublishedunderthetradenameBirkhäuser
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
(www.birkhauser-science.com)
Preface
This book is based on lectures given originally at Reading University and more
recently at Oxford as part of the Continuing Education program of Oxford
University in England. In a sense it is a sequel to Gems of Geometry (now in its
second edition) and which is also based on lectures given at Reading and Oxford.
Having given the geometry lectures a few times, it was suggested that I
should develop another course based on aspects of numbers. I was extremely
busy at the time updating my book on the Ada programming language so settled
on a short course of just five lectures entitled Nice and Noughty (sic) Numbers.
However, as the time approached I was still too busy and so the course was
kindly given by Aruna Hardy. The next year, the pressure of life abated and I was
able to gave a full ten lecture course.
I have since given the course a number of times at both Reading and Oxford
and the notes eventually matured into this book. The lectures divide into two
kinds: some address the theme of simple number theoretic topics such as prime
numbers and cryptography, whereas others address the theme of daily needs and
pleasures such as keeping track of time and enjoying music.
There are ten basic lectures. The first lecture is entitled Measures. Measuring
and counting various things are fundamental human activities. In earlier times
one might have been interested in measuring the size of a field and counting
cows; these days one might be more likely to measure the size of a garden and
count money. The lecture starts by asking students what is their favourite number
and why; this not only breaks the ice but lays the foundation for many topics such
as what would be a good base for counting. That takes us into prime numbers,
divisibility, factors, and perfect numbers; the lecture concludes with a survey of
weights and measures and currencies.
The second lecture leads on from perfect numbers into Amicable numbers
and provides a good opportunity to introduce modular arithmetic. This is
followed by Probability and is more light-hearted and ends with applications to
games such as craps and poker. We then return to the numerical theme by
considering Fractions of various forms including the amazing Egyptian system
of unit fractions. This is followed by a lecture on Time including the calendar and
sunshine.
There are then two more lectures on the numerical theme covering Notations
of various forms and Prime numbers and two lectures of a more fun nature
covering Bell ringing and the evolution of Musical scales.
The final lecture looks at the topic of cryptography which has become so
important with electronic communication. And for final light relief it concludes
by looking at the gaits of animals and two popular puzzles.
There are also a number of appendices. Some provide additional material on
topics such as Pascal’s triangle and polydivisibility, whereas a few cover material
from a short course called Puzzles and Pastimes which I have also given at
Reading and Oxford. Thus the appendix on Groups draws on examples from Bell
ringing and also leads into the amazing topic of Rubik’s cube from the puzzles
course. The final appendix considers the musical keyboards designed by
Mersenne who is better known for his numbers; it concludes with a discussion of
the Tonnetz schema for revealing harmony and the strange relation between
music and the topology of the torus.
The main lectures contain some exercises (harder ones are marked with
asterisks) but answers are not provided since I anticipate giving the course again.
An important issue when writing a book is to consider who might enjoy
reading it. The mathematical background required is not hard (just a bit of simple
algebra really) and is the kind of material anyone who studied a scientific subject
to the age of perhaps 16 would have encountered. One group of potential readers
is therefore young people with a zest for knowledge (I would have been delighted
to have had such a book when I was 16). Another group as shown by students on
courses includes those of maturer years who might like to know more about
topics that they enjoyed when younger.
Students attending the courses are from various backgrounds – of all ages
and sexes. Some have little technical background but revel in activities such as
throwing dice, ringing handbells, and attempting to simulate the gaits of camels;
others have serious technical experience and enjoy perhaps a nostalgic trip
visiting some familiar topics and also meeting new ones.
I have made no attempt to avoid using mathematical notation wherever it is
appropriate (readers can skip hard bits if they are weary). I am saddened that
many popular mathematical books strive to avoid mathematics because some
publisher once said that every time an equation is added, the sales divide by two.
But I have aimed to provide various tables and illustrations to enliven the text.
I must now thank all those who have helped me in this task. First, a big thank
you to my wife, Bobby, who helped with typesetting and photography, to my
daughter Janet, who provided much background material, and to my daughter
Helen for advice on design, and to David Godwin for information on the noble
art of bell-ringing. Thanks also to the (anonymous) reviewers for their input and
to Pascal Leroy who was a great help in finding a number of errors and
suggesting many improvements and to Tucker Taft for assistance on analysing
the performance of Ackermann’s function and to Ahlan Marriott for the gift of a
corkscrew with a message in base 4.
Thanks also to colleagues in Continuing Education at Oxford and especially
to Aruna Hardy for digging me out of a hole by giving the first short numbers
course and to Julian Gallop and Iryna Schlackow for some inspiring examples of
probability and numerical puzzles.
I am grateful to many students on my courses for their feedback and
encouragement. I must especially thank: Felix Lam for introducing me to the life
of John William Colenso who wrote fascinating books on Arithmetic and became
Bishop of Natal; Susan Vaughan for help with the presentation of the strange
material on music; Rita Sawrey-Woodwards for taking the tricky photographs of
measuring devices and other artifacts; and Felicity Wood for introducing me to
the Maria Theresa thaler. Susan Vaughan also introduced me to Geoff Chew at
Royal Holloway who kindly brought me up to date on the world of music theory.
I am also grateful to the authors of the many books that I have read and
enjoyed. I cannot mention them all here but I must mention a few. Two old
favourites are An Introduction to Probability Theory by Feller and On the
Sensations of Tone by Helmholtz. Regarding the ancient world, Mathematics in
the Time of the Pharaohs by Richard Gillings is excellent. For more traditional
number theory topics, I am grateful to my daughter for lending me her copies of
two books entitled Elementary Number Theory by Kenneth Rosen and David
Burton. Another wonderful book is Makers of Mathematics by Stuart
Hollingdale.
For general fun, The Penguin Dictionary of Curious and Interesting
Numbers by David Wells is vital and for puzzles the classic Amusements in
Mathematics by Henry Dudeney is hard to beat. And I must thank David
Singmaster for giving me a copy of his intriguing little book on the Rubik cube.
Probably the first book to trigger my interest in numbers was Arithmetic by
D AYoung. This was aimed at accountants in the old days when everything was
worked out with a pencil (my father was an accountant). But among basic stuff
were little advanced bits in small print which dabbled in odd topics such as
finding a cube root, continued fractions, and recurring decimal fractions. I am
sure I spent hours browsing through this when still in short trousers!
Finally, and most important of all, I must thank Dorothy Mazlum, Sabrina
Hoecklin, and all others at Springer who made this book actually happen.
I hope that all those who read or browse through it will find something to
enjoy. I enjoyed writing it and learnt a great deal in the process.
John Barnes
Caversham
England
September 2016
Contents
1 Measures 1
Favourite numbers, Prime numbers, Factors,
Weights and measures, Musical notes, Currencies,
Further reading, Exercises.
2 Amicable Numbers 25
Perfect numbers, Modular arithmetic, Mersenne the
monk, Amicable numbers, Amicable multiplets,
Sociable cycles, Fermat numbers, Fibonacci
numbers, Further reading, Exercises.
3 Probability 43
Heads or tails, Distributions, Shake, rattle, and roll,
The normal distribution, An abnormal distribution,
Tossing for π, Double or quits, Amedical problem,
Which box has the prize?, Craps and poker, Further
reading, Exercises.
4 Fractions 69
Real numbers, Vulgar fractions, Egyptian fractions,
Dividing loaves, The table of 2/n, The method of
false position, Decimal fractions, Roots, Continued
fractions, The eye of Horus, Further reading,
Exercises.
5 Time 97
Basic rhythms, The Roman calendar, The Gregorian
calendar, Start of the year and quarters, The week,
Time of day, Sunshine, Further reading, Exercises.
6 Notations 115
Types of notation, Roman numbers, Babylonian
system, Place systems and bases, Divisibility,
Fractions and bases, Fermat’s Little Theorem,
Further reading, Exercises.
7 Bells 137
Rounds and plain hunting, Plain Bob Minimus,
Plain Bob Doubles, Stedman, Grandsire, Groups,
Further reading, Exercises.
8 Primes 161
Greatest Common Divisor, Prime factors, Fermat’s
method, Eratosthenes revisited, Complex numbers,
Complex primes, Polynomials, Further reading,
Exercises.
9 Music 179
Frequency and vibrations, The Pythagorean scale,
Cents, Just intonation, Minor scales, Meantone
temperament, Equal temperament, Other
arrangements, Dividing the octave, Further reading.
10 Finale 205
The RSAalgorithm, Linear congruences, Euler’s
function, Encryption and decryption, Code blocks,
Animal gaits, Towers and rings, Further reading.
A Ackermann 227
Recursion and iteration, Further reading.
B Pascal’s Triangle 231
Basic properties, Fibonacci numbers, Squares and
pyramids, Further reading.
C Stochastics 239
War games, Queuing, Further reading.
D Polydivisibility 245
The problem, Other bases.
E Groups 249
Basics, Subgroups, Generators, Cosets,
Permutations, Polyhedral groups, Direct products,
The quaternion group, Further reading.
F Rubik 265
Basics, Restoring a cube, Summary of restoration,
The cube group, Cosets, Finally, Further reading.
G Differences 287
Rings, Towers, Fibonacci numbers, Hats, Further
reading.
H Chinese Remainders 295
Linear congruences, Simultaneous equations,
Pirates, Eggs, Squares.
J Mersenne 303
Mersenne’s 31-note keyboard, Aschema, The 12-
note keyboards, An 18-note keyboard, A26-note
keyboard, The 31-note keyboard, Intervals, Abetter
keyboard, Modulation, Tonnetz revisited, Further
reading.
Bibliography 325
Index 327
1 Measures
THISFIRSTLECTURElooks at some simple properties of positive whole numbers
such as whether they are prime and what factors they have.
The number of factors of a number is important in determining whether it is
good as a basis for measuring and counting. Accordingly, we also look at some
historical weights and measures and currencies as well as at musical intervals
which are another source of curious measures.
Favourite numbers
MANY PEOPLE have favourite numbers. Some are said to have mystical
significance, some might be based on birthdays, and of course we all need
easy to remember pin numbers.
A recent survey of some twenty (a score) of numerate folk produced the
following list of favourite numbers. Names have been changed for anonymity.
0 Three people chose zero. Julia chose it because it filled a gap in number
theory. Jim chose it to be different (but it seems he was unsuccessful because
it was one of the most popular numbers!) and also because it differs most
from all the others and because it is useless outside mathematics (hmm) and
most importantly it is beautiful. Jane chose zero because it shows she has no
defects in her software and research.
1/3 Trevor chose this because calculators just can’t quite get it right.
1 My school number said Richard. Also chosen for the same reason by Bob.
Angela liked it for consistency.
√2 Liked by Cyril simply because he likes the square root sign.
2 The first choice of John Abecause two heads are better than one and it takes
two to tango (which is fun) whereas three is a crowd. Note that tango is Latin
for I touch. See also 12. Also chosen by John B because it’s the smallest
prime and the only even prime; moreover, for a computer scientist 2 makes
life easier.
e Simon and William chose e becomes it pops up/crops up in odd/interesting
places. See also 5040.
π Liked by Yorick and Dod because it underpins the order of the universe and
goes around.
4 An old astrology book said it was my lucky number and I can hit it on a
dartboard said Trevor.
2 Nice Numbers
7 Five people chose 7. Reasons were: it’s magic (Joseph), don’t know (Adrian)
and, I seemed to have success when picking this number as a child in games
of chance (John C – an early big time gambler clearly). Big enough to be
interesting but small enough to remember says Daniel. Hmm, I wonder if
Daniel has trouble with his pin numbers. Finally, Bertie hadn’t really thought
about it but it was a useful small prime and related to his favourite colour,
Red. (I wonder what the relation is.)
12 Another choice of John Abecause it has more factors than 10. (I didn’t know
that we were allowed more than one choice.) We should all use base 12 he
says. I agree and there is a base 12 society but I think it has a slim chance
now.
13 Asignificant date for Derek. And not proven to be unlucky said Robert.
17 I’m not sure why I like it said Stewart.
22 This is guaranteed to come up whenever I play roulette said Robin.
23 Reason unknown said Peter.
27 A good age to be (it was a long time ago for me). But quite recent for the
owner I think. Ladies should never reveal their age but Alice has given us a
lower bound.
34 Is the favourite Fibonacci number of Rustin.
509 Ackermann(3, 6) – a good compiler test for Clive.
777 Looks nicer than 999 and not the number of the Beast (666) said Douglas.
998 Engine capacity of Malcolm’s first car.
1729 Alfred chose this for the same reason as Ramanujan. It is the smallest
number that can be expressed as the sum of two cubes in two different ways:
123+ 13= 103+ 93. This relates to a story about a taxi number. The taxi was
taken by Hardy when visiting Ramanujan who was ill. Also chosen by
Martin who thought it might be prime; but it is 19×91 which is rather nice.
1961 Agood number for palindromes (invert the page) and Gerald won a bottle
of whisky in a raffle with it.
5040 This was Simon’s second choice (after e). 5040 is of course 7! which is
the number of changes for a classical peal of bells.
77385107 Turn it upside down and it is my name says ... – well do it on your
calculator and see who said that.
137438691328 This is the seventh perfect number. The choice of Tony, just to
be awkward/different he says. And he succeeded in being different unlike
Jim who chose zero.
It is interesting that 7 was the most popular. It seems that if you ask people to
choose a number between 1 and 10 then most people choose 7. Indeed, seven has
an air of mysticism about it. There are seven days in the week, seven deadly sins,
seven wonders of the ancient world, seven Vestal Virgins, seven muses, and so on.