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MATH111: Numbers and Relations
Niels Laustsen
n.laustsen@lancaster.ac.uk
Department of Mathematics and Statistics
Lancaster University
Michaelmas 2016
Contents:
1
Course introduction
1.1
Practical information
Course format
Lecture notes
Past exam papers
Further reading
1.2
Course overview
1.3
Preliminaries
2
Logic
2.1
Introduction to logic
2.2
The parity model
2.3
The logic model
2.4
Quantifiers
3
Mathematical proofs
3.1
Motivation
3.2
Direct proof
3.3
Proof by contraposition
3.4
Proof by contradiction
3.5
Counterexamples
3.6
Terminology
4
Number theory
4.1
Divisibility
4.2
HCFs and linear combinations
4.3
The Euclidean algorithm
4.4
Applications of Bézout’s Theorem
4.5
Lowest common multiples
4.6
Prime numbers
4.7
Applications of prime factorization
5
Congruences
5.1
Introduction to congruences
5.2
Solving linear congruences
5.3
The Chinese Remainder Theorem
6
Equivalence relations and their applications
6.1
Introduction to relations
6.2
Equivalence relations
6.3
Combining congruence classes
6.4
Constructing number systems
7
Polynomials
7.1
Polynomial arithmetic
7.2
HCFs and the Euclidean algorithm revisited
A
Proof of the Fundamental Theorem of Arithmetic
B
Exercises
Fun quiz
Week 1
Workshop exercises
Further exercises
Tutor-assessed exercises
Online-assessed exercises
Bonus exercises
Week 2
Workshop exercises
Further exercises
Tutor-assessed exercises
Online-assessed exercises
Bonus exercise
Week 3
Workshop exercises
Further exercises
Tutor-assessed exercises
Online-assessed exercises
Bonus exercises
Week 4
Workshop exercises
Further exercises
Tutor-assessed exercises
Online-assessed exercises
Bonus exercise
Week 5
Workshop exercises
C
Self-Explanation
C.1
How to Self-Explain
C.2
Example Self-Explanations
C.3
Self-Explanation Compared with Other Comments
C.4
Practice Proof 1
C.5
Practice Proof 2
Remember…
D
Problem-solving worksheets
Week 2: machines
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Week 3: divisibility
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Week 4: games
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
E
Solutions
E.1
Workshop exercises from week 1
E.2
Workshop exercises from week 2
E.3
Workshop exercises from week 3
E.4
Workshop exercises from week 4
E.5
Online-assessed exercises from week 1
E.6
Online-assessed exercises from week 2
E.7
Online-assessed exercises from week 3
E.8
Tutor-assessed exercises from week 1
E.9
Tutor-assessed exercises from week 2
E.10
Tutor-assessed exercises from week 3
F
End of module info
G
The Greek alphabet