Exploring mathematics. Problem-solving and proof

308 Pages·2018·2.851 MB·English

by  Grieser D

Springer Undergraduate Mathematics Series
Daniel Grieser
Exploring
Mathematics
Problem-Solving and Proof
Springer Undergraduate Mathematics Series
Advisory Board
M. A. J. Chaplain, University of St. Andrews
A. MacIntyre, Queen Mary University of London
S. Scott, King’s College London
N. Snashall, University of Leicester
E. Süli, University of Oxford
M. R. Tehranchi, University of Cambridge
J. F. Toland, University of Bath
More information about this series at http://www.springer.com/series/3423
Daniel Grieser
Exploring Mathematics
Problem-Solving and Proof
123
DanielGrieser
Institut für Mathematik
Carl vonOssietzkyUniversität Oldenburg
Oldenburg
Germany
ISSN 1615-2085 ISSN 2197-4144 (electronic)
SpringerUndergraduate MathematicsSeries
ISBN978-3-319-90319-4 ISBN978-3-319-90321-7 (eBook)
https://doi.org/10.1007/978-3-319-90321-7
LibraryofCongressControlNumber:2018939472
MathematicsSubjectClassification(2010): 00-01,00A07,00A09,97D50
TranslationfromtheGermanlanguageedition:MathematischesProblemlösenundBeweisenbyDaniel
Grieser,©SpringerFachmedienWiesbaden2013.AllRightsReserved.
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For Ricarda and Leonard
Preface to the English edition
The classical curriculum for university mathematics education has
emphasized calculation and – on a higher level – proofs and system-
atic development of mathematical theories. In recent years there has
beenagrowinginterestinsupplementingthiswithaproblem-solving
approach. It has become clear that this is not only fun but also very
useful as preparation for understanding the mathematical theories,
these tremendous advances that mathematics has made over the cen-
turies. These new developments have occurred in Britain, the United
States, Germany and certainly other countries as well.
In 2011, the University of Oldenburg introduced a new course
into the curriculum: ‘Mathematical problem-solving and proving’
(Mathematisches Problemlo¨sen und Beweisen). It was aimed at first-
yearstudentswhohavejustfinishedGymnasium(roughlyequivalent
to A-levels in England and high school plus epsilon in the US) and
are starting a degree in mathematics or mathematics education. This
course has been very successful, both in bridging the gap between
high school and university, and in providing a fun introduction to the
higher mathematics taught at university. Similar courses have sprung
up at other universities in Germany, and in Britain and the US, and at
otherplacespeoplearediscussingthepossibilityofintroducingthem,
or at least introducing elements of such an approach into existing
courses.
This book arose from my teaching the course twice in Oldenburg,
and since many colleagues in English-speaking countries have ex-
pressed a great interest in it I decided to translate it into English.
There are many other excellent books on problem-solving. What
sets this book apart is that it starts at a very elementary level, but
then quite explicitly tries to be a bridge to higher mathematics: the
emphasis is on your discovery of mathematics by solving problems,
but along the way concepts, notation and terminology of higher
mathematics (e.g. sets and mappings) are introduced. In addition,
most chapters have a section with additional material (titled Going
7
8 PrefacetotheEnglishedition
further) which provides a preview of where similar ideas are used in
advanced mathematical disciplines.
Contents
Introduction 13
1 First explorations 23
1.1 Cutting up a log . . . . . . . . . . . . . . . . . . . . . . . 23
1.2 A problem with zeroes . . . . . . . . . . . . . . . . . . . 24
1.3 A problem about lines in the plane . . . . . . . . . . . . 28
1.4 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Recursion – a fundamental idea 41
2.1 Recursion in counting problems . . . . . . . . . . . . . 41
2.2 The number of subsets . . . . . . . . . . . . . . . . . . . 44
2.3 Tilings with dominoes . . . . . . . . . . . . . . . . . . . 50
2.4 Solving the Fibonacci recurrence relation. . . . . . . . 54
2.5 Triangulations . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Mathematical induction 71
3.1 The induction principle . . . . . . . . . . . . . . . . . . 71
3.2 Colourings . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 Graphs 83
4.1 The Euler formula for plane graphs . . . . . . . . . . . 83
4.2 Counting in two ways for graphs . . . . . . . . . . . . . 90
4.3 Handshakes and graphs . . . . . . . . . . . . . . . . . . 94
4.4 Five points in the plane, all joined by edges . . . . . . . 95
9
10 Contents
4.5 Going further: Euler’s formula for polyhedra, topol-
ogy and the four colour problem . . . . . . . . . . . . . 98
4.6 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Counting 107
5.1 The basic principles of counting . . . . . . . . . . . . . 107
5.2 Counting using bijections . . . . . . . . . . . . . . . . . 117
5.3 Counting in two ways . . . . . . . . . . . . . . . . . . . 123
5.4 Going further: double sums, integrals and infinities . . 128
5.5 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 General problem solving strategies 139
6.1 General problem solving strategies . . . . . . . . . . . . 139
6.2 The diagonal of a cuboid . . . . . . . . . . . . . . . . . . 143
6.3 The problem of trapezoidal numbers . . . . . . . . . . . 145
6.4 Going further: sum representations of integers . . . . . 153
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7 Logic and proofs 157
7.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8 Elementary number theory 181
8.1 Divisibility, prime numbers and remainders . . . . . . 181
8.2 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . 186
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9 The pigeonhole principle 195
9.1 The pigeonhole principle, first examples. . . . . . . . . 195
9.2 Remainders as pigeonholes . . . . . . . . . . . . . . . . 199
9.3 An exploration: approximation by fractions . . . . . . . 201
9.4 Order in chaos: the pigeonhole principle in
graph theory . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.5 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214