This textbook contains the mathematics needed to study computer science in application-oriented computer science courses. The content is based on the author's many years of teaching experience.
The translation of the original German 7th edition Mathematik für Informatiker by Peter Hartmann was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.
Textbook Features
- You will always find applications to computer science in this book.
- Not only will you learn mathematical methods, you will gain insights into the ways of mathematical thinking to form a foundation for understanding computer science.
- Proofs are given when they help you learn something, not for the sake of proving.
Mathematics is initially a necessary evil for many students. The author explains in each lesson how students can apply what they have learned by giving many real world examples, and by constantly cross-referencing math and computer science. Students will see how math is not only useful, but can be interesting and sometimes fun.
The Content
- Sets, logic, number theory, algebraic structures, cryptography, vector spaces, matrices, linear equations and mappings, eigenvalues, graph theory.
- Sequences and series, continuous functions, differential and integral calculus, differential equations, numerics.
- Probability theory and statistics.
The Target Audiences
Students in all computer science-related coursework, and independent learners.
Author(s): Peter Hartmann
Edition: 1
Publisher: Springer
Year: 2023
Language: English
Pages: 600
City: Wiesbaden
Tags: Computer Science; Algebra; Analysis; Discrete Mathematics; Statistical Methods; Stochastics; Probability Theory; Cryptography
Preface
Contents
Part I Discrete Mathematics and Linear Algebra
1 Sets and Mappings
Abstract
1.1 Set Theory
Relationships between Sets
Operations with Sets
Calculation Rules for Set Operations
The Cartesian Product of Sets
1.2 Relations
Equivalence Relations
Order Relations
1.3 Mappings
The Cardinality of Sets
1.4 Comprehension Questions and Exercises
Anchor 14
2 Logic
Abstract
2.1 Propositions and Propositional Variables
Compound Propositions
Boolean Algebras
Evaluation of Propositional Formulas in a Program
2.2 Proof Principles
The Direct Proof
The Proof of Equivalence
The Proof by Contradiction
2.3 Predicate Logic (First-Order Logic)
Negation of Quantified Predicates
2.4 Logic and Testing of Programs
2.5 Comprehension Questions and Exercises
Anchor 15
3 Natural Numbers, Mathematical Induction, Recursion
Abstract
3.1 The Axioms of Natural Numbers
3.2 The Mathematical Induction
3.3 Recursive Functions
Recursions of Higher Order
Runtime Calculations for Recursive Algorithms
3.4 Comprehension Questions and Exercises
Anchor 9
4 Some Number Theory
Abstract
4.1 Combinatorics
4.2 Divisibility and Euclidean Algorithm
4.3 Modular Arithmetic
Calculating with Residue Classes
4.4 Hashing
Hash Functions
Collision Resolution
4.5 Comprehension Questions and Exercises
Anchor 11
5 Algebraic Structures
Abstract
5.1 Groups
Permutation groups
5.2 Rings
Polynomial Rings
5.3 Fields
The Field of Complex Numbers
The Field
5.4 Polynomial Division
Horner’s Method
Residue Classes in the Polynomial Ring, The Field )
Using Polynomial Division for Error Detection
5.5 Elliptic Curves
Elliptic Curves over the Field of Real Numbers
Elliptic Curves Over Finite Fields
5.6 Homomorphisms
5.7 Cryptography
Encryption with Secret Keys
Encryption with Public Keys
The RSA Algorithm
The Diffie-Hellman Algorithm
The Diffie-Hellman Algorithm with Elliptic Curves
Key Generation
Random Numbers
5.8 Comprehension Questions and Exercises
Anchor 27
6 Vector Spaces
Abstract
6.1 The Vector Spaces , and
6.2 Vector Spaces
6.3 Linear Mappings
6.4 Linear Independence
6.5 Basis and Dimension of Vector Spaces
6.6 Coordinates and Linear Mappings
6.7 Comprehension Questions and Exercises
Anchor 10
7 Matrices
Abstract
7.1 Matrices and Linear Mappings in
Composition of linear mappings
7.2 Matrices and Linear Mappings from Kn → Km
Matrix multiplication and composition of linear mappings
7.3 The Rank of a Matrix
7.4 Comprehension Questions and Exercises
Anchor 9
8 Gaussian Algorithm and Linear Equations
Abstract
8.1 The Gaussian Algorithm
8.2 Calculating the Inverse of a Matrix
8.3 Systems of Linear Equations
Geometrical Interpretation of Systems of Linear Equations
Ray Tracing, Part 1
Solution of Systems of Linear Equations Using the Gaussian Algorithm
8.4 Comprehension Questions and Exercises
Anchor 10
9 Eigenvalues, Eigenvectors and Change of Basis
Abstract
9.1 Determinants
9.2 Eigenvalues and Eigenvectors
9.3 Change of Basis
Orientation of Vector Spaces
Ray Tracing, Part 2
9.4 Comprehension Questions and Exercises
Anchor 9
10 Dot Product and Orthogonal Mappings
Abstract
10.1 Dot Product
Where Does the Mouse Click?
10.2 Orthogonal Mappings
The orthogonal linear mappings in and
10.3 Homogeneous Coordinates
Basis Transitions in Robotics
10.4 Comprehension Questions and Exercises
Anchor 10
11 Graph Theory
Abstract
11.1 Basic Concepts of Graph Theory
Paths in Graphs
Centrality of Nodes—Who is the Most Important?
11.2 Trees
Rooted Trees
Search Trees
The Huffman Code
11.3 Traversing Graphs
Shortest Paths
11.4 Directed Graphs
11.5 Comprehension Questions and Exercises
Anchor 14
Part II Analysis
12 The Real Numbers
Abstract
12.1 The Axioms of Real Numbers
The Order Axioms
The Completeness Axiom
12.2 Topology
12.3 Comprehension Questions and Exercises
Anchor 8
13 Sequences and Series
Abstract
13.1 Sequences of Numbers
Convergent Sequences
The Big O Notation
Monotonic Sequences
13.2 Series
Convergence Tests for Series
13.3 Representation of Real Numbers in Numeral Systems
13.4 Comprehension Questions and Exercises
Anchor 11
14 Continuous Functions
Abstract
14.1 Continuity
Functions of several variables
14.2 Elementary Functions
14.3 Properties of Continuous Functions
Theorems About Continuous Functions
Logarithm and General Exponential Function
The Trigonometric Functions
Numerical Calculation of Trigonometric Functions
Radian and polar coordinates
14.4 Comprehension Questions and Exercises
Anchor 13
15 Differential Calculus
Abstract
15.1 Differentiable Functions
Differentiation Rules
Calculation of Extrema
15.2 Power Series
15.3 Taylor Series
15.4 Differential Calculus of Functions of Several Variables
Extrema
The Regression Line
15.5 Comprehension Questions and Exercises
Anchor 12
16 Integral Calculus
Abstract
16.1 The Integral of Piecewise Continuous Functions
Integration Rules
16.2 Applications of the Integral
Volumes of Shapes
The Arc of a Curve
Improper Integrals
16.3 Fourier Series
Discrete Fourier Transform
16.4 Comprehension Questions and Exercises
Anchor 12
17 Differential Equations
Abstract
17.1 What are Differential Equations?
17.2 First Order Differential Equations
Separable Differential Equations
First Order Linear Differential Equations
17.3 nth Order Linear Differential Equations
Linear Differential Equations with Constant Coefficients
Inhomogeneous Linear Differential Equations
17.4 Comprehension Questions and Exercises
Anchor 11
18 Numerical Methods
Abstract
18.1 Problems with Numerical Calculations
Real Numbers in the Computer
Propagation of Errors
Calculation Errors in Systems of Linear Equations
18.2 Nonlinear Equations
Calculation of Fixed Points
Calculation of Zeros
18.3 Splines
Cubic Splines
Parametric Splines
18.4 Numerical Integration
18.5 Numerical Solution of Differential Equations
18.6 Comprehension Questions and Exercises
Anchor 16
Part III Probability and Statistics
19 Probability Spaces
Abstract
19.1 Problems in Statistics and Probability Theory
The Election Poll
Quality Checking
Determining Estimates
Testing a Hypothesis
Random Events Over Time
Probabilistic Algorithms
Monte Carlo Methods
Distributions
19.2 The Concept of Probability
Random Events
Probability Spaces
Uniform Probability on Finite Spaces
Geometric Probabilities
19.3 Conditional Probability and Independent Events
Independent Events
19.4 Bernoulli Processes and Urn Problems
19.5 Urn Problems
19.6 Comprehension Questions and Exercises
Anchor 22
20 Random Variables
Abstract
20.1 Random Variables and Probabilty Distributions
Discrete Random Variable
Continuous Random Variables
Sets of Random Variables
Independent Random Variables
20.2 Expected Value and Variance of Random Variables
Sums and Products of Random Variables
A Short Excursion into Information Theory
20.3 Comprehension Questions and Exercises
Anchor 12
21 Important Distributions, Stochastic Processes
Abstract
21.1 Discrete Probability Distributions
The Discrete Uniform Distribution
The Binomial Distribution
The Hypergeometric Distribution
The Geometric Distribution
The Poisson Distribution
21.2 Continuous Probability Distributions, The Normal Distribution
The Continuos Uniform Distribution
The Standard Normal Distribution
The General Normal Distribution
The Exponential Distribution
The Chi-Square Distribution
21.3 Stochastic Processes
The Poisson Process
Markov Chains
Queues
21.4 Comprehension Questions and Exercises
Anchor 20
22 Statistical Methods
Abstract
22.1 Parameter Estimation
Samples
Estimators
Estimator for the Probability in a Bernoulli Process
Estimators for the Expected Value and Variance of a Random Variable
22.2 Principal Component Analysis
22.3 Confidence Intervals
22.4 Hypothesis Testing
Parameter Testing
Pearson’s Chi-Squared Test
22.5 Comprehension Questions and Exercises
Anchor 14
23 Appendix
23.1 The Greek Alphabet
23.2 The Standard Normal Distribution
Bibliography
