Mathematics In Everyday Life

178 Pages·2019·2.233 MB·English

by  John Haigh

John Haigh
Mathematics
in Everyday Life
Second Edition
Mathematics in Everyday Life
John Haigh
Mathematics in Everyday
Life
Second Edition
123
JohnHaigh
Department ofMathematics
University of Sussex
Brighton, UK
ISBN978-3-030-33086-6 ISBN978-3-030-33087-3 (eBook)
https://doi.org/10.1007/978-3-030-33087-3
MathematicsSubjectClassification(2010): 00-01,00A05,00A99,34-01,90-01,91-01
1stedition:©SpringerInternationalPublishingSwitzerland2016
2ndedition:©SpringerNatureSwitzerlandAG2019
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Preface to the Second Edition
I thank those readers who have told me of their enjoyment and illumination from
the first edition. The main change now is the addition of a Chapter on computer
applications, for which my colleagues James Hirschfeld and Konstantin Blyuss
havebeenmosthelpful.Otherwise,thematerialhasbeenupdatedinplaces,further
ideas inthetakingofpenalties insoccer,and instock control for perishable goods
have been added. Readers seeking Solutions to the exercises should now contact
Springer directly.
Brighton, UK John Haigh
September 2019
v
Preface to the First Edition
The motivation for this book was to construct a mathematics module that
encouraged First-Year university students to appreciate the diverse ways in which
the mathematics they already knew, or were just about to learn, can impact on
everyday occurrences. The applications discussed here will give students further
practice in working with calculus, linear algebra, geometry, trigonometry and
probability,thussupportingothermodules.Thechaptersarelargelyindependentof
each other: and within most chapters, sections are often self-contained, giving
teachers freedom in selecting topics to their taste. All in all, there is likely to be
more material than would be covered in a one-semester course. Readers are
assumedtohaveabackgroundroughlyequivalenttosingle-subjectMathematicsat
A-level in the UK.
Inmanydegreecourses,studentshavetheopportunity,oreventheobligation,to
write an extended essay on a mathematical topic of their choice. They will find a
rangeofideashere,inthesetexercisesaswellastheformaltext,andreferencesto
more substantial accounts.
Mathematicshasmanyapplications:itisunrealistictoexpectonebooktodomore
than give a representative sample of them. My selection runs from the obviously
practical,suchashowtocalculatemortgagerepayments,orschedulenursestocover
hospital wards, to amusing ways in which mathematical ideas can give pointers to
tacticsinTVGameShows,orinfluencethescoringsystemsinsportingevents.Wesee
howsimpledifferentialequationscanmodelbacterialgrowth,mixingliquids,emp-
tyingbaths,evaporatingmothballsandthespreadofepidemics;whenlookingatdarts,
rouletteandhowpeopleprogressthroughhierarchicalorganisations,weusecombi-
natorics,logic,differenceequationsandcalculationswithmatricesandvectors.The
variety of voting systems in use throughout the world is inevitable, given Kenneth
Arrow’s‘ImpossibilityTheorems’—thatnosystemcanbeconstructedthatsatisfies
all among a simple list of desirable properties! Mathematics is versatile: and dis-
coveringtheunusualplaceswhereitprovidesinsightsoranswersisenjoyable.
Whenyoungchildrenfirstusetherulefordividingbyafraction—‘turnitupside
down and multiply’—they seldom understand why it works. But they quickly find
that the rule is easy to apply, and gives the right answer. Only much later are they
vii
viii PrefacetotheFirstEdition
able to appreciate a valid justification. Similarly with more advanced material:
heuristic explanations, or worked examples, can be more enlightening than an
all-encompassinggeneralapproach.Meetingandunderstandingaformalproofofa
theorem or technique the first time you use it is not essential. But it is vital that
students appreciate the necessity for formal proofs at some stage of their studies,
else they will not be confident about the conditions under which particular tech-
niquescanproperlybeapplied.Oneofthemostvital,ifunacknowledged,wordsin
mathematics is ‘if’: if certain conditions hold, then some result follows. All too
often, it is tempting to use that result without checking the ‘if’ part.
Eachchapterendswithacollectionofexercises,whichareanintegralpartofthe
book. Seeking to solve them should help students reinforce their understanding of
basic principles. Ido notindicate whether an exerciseis expected to be routine, or
quite tricky; lack of that knowledge is exactly the position mathematicians find
themselvesinwhenconfrontedwithaproblemtosolve.Iamhappytosupplyasetof
[e-mail].
(Students, please don’t pretend. The best way to learn maths is to try to solve
problemsfromscratch,withouthavinglookedatasolutionfirst.Andifhandingin
myworkinyournamewouldgainyousignificantcredit,isyourdegreequalification
worth having?)
ItisapleasuretoacknowledgethefeedbackfromthecohortsofSussexstudents
who have taken a module based on much of this material, as well as the helpful
comments from three anonymous referees, and the support from the Springer
production team.
The Appendix contains several formulae, techniques and approximations that
working mathematicians will have used so often that they are as familiar as com-
mon multiplication tables. No student will regret the time spent in committing this
list to memory.
Brighton, UK John Haigh
September 2015
Contents
1 Money. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Present Value and APR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Mortgage Repayments; Annuities . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Investing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Personal Finance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 More Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 What They Are, How They Arise . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Second-Order Equations with Constant Coefficients . . . . . . . . . . 26
2.4 Linked Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Sport and Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Lawn Tennis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Rugby . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 The Snooker Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Athletics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Darts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Tournament Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.7 Penalty Kicks in Soccer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.8 Golf: Flamboyance Versus Consistency . . . . . . . . . . . . . . . . . . . 57
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
ix
x Contents
4 Business Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1 Stock Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Linear Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Transporting Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Jobs and People . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Check Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6 Hierarchies in Large Organisations . . . . . . . . . . . . . . . . . . . . . . 77
4.7 Investing for Profits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.8 More Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1 Voting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Voting Dilemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 From the USA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 Simpson’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 False Positives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.6 Measuring Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.7 More Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 TV Game Shows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Monty Hall’s Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 The Price Is Right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4 Pointless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.5 Two Tribes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.6 The Million Pound Drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.7 Deal or No Deal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.8 The Weakest Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.9 The Colour of Money. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.10 Who Wants to Be a Millionaire? . . . . . . . . . . . . . . . . . . . . . . . . 125
6.11 Other Shows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Gambling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Lotteries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.3 Roulette . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4 The Horse Racing Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 Card Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Contents xi
7.6 Premium Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8 Computer Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.2 Pseudorandom Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3 Codes and Ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.4 Search Engines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Appendix: Useful Mathematical Facts... .... .... .... .... ..... .... 167
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 171