A calculus of the absurd
1
Introduction
I
A Level mathematics (and further mathematics)
II
A Level Further Mathematics
III
General mathematics
IV
Algorithms
V
Computer science science stuff
VI
Miscellaneous
A calculus of the absurd
A calculus of the absurd
Teymour Aldridge
[email protected]
Contents
1
Introduction
I
A Level mathematics (and further mathematics)
2
Solving problems
2.1
Introduction
2.2
General things to keep in mind
2.3
Specific heuristics
2.4
Psychological tactics
3
Sequences and series
3.1
Sigma notation
3.2
"Telescoping" series
3.3
Arithmetic series
3.4
Geometric series
3.4.1
Finding the sum of a geometric series
3.4.2
Sum to infinity
3.4.3
"Hidden" geometric series.
3.5
The binomial theorem
3.5.1
Derivation
3.6
The general binomial series
4
Concrete algebra
4.1
Fractions
4.1.1
Reciprocals of fractions
4.2
The root function
4.3
Quadratics
4.3.1
quadratics introduction
4.3.2
Factorising
4.3.3
Difference of two squares
4.3.4
Comparing coefficients
4.3.5
Completing the square
4.3.6
The quadratic formula
4.4
Some useful techniques for working with polynomials (and other functions)
4.5
Surds
4.5.1
Rationalising the denominator
4.6
The factor and remainder theorems
4.6.1
An example
4.7
Transformations of functions
4.7.1
Y-axis transformations
4.7.2
X-axis transformations
4.8
Partial fractions
4.9
The modulus function
4.9.1
Graphically
4.9.2
As a function
4.9.3
Algebraically
4.9.4
Some examples
4.9.5
Proof that a value is not greater than its modulus
4.10
Parametric equations
4.11
Roots of polynomials
4.11.1
A harder example
4.11.2
Some further examples
4.12
Inequalities
4.12.1
Problems in applying functions to inequalities
4.12.2
Solutions for problems in applying functions to inequalities
4.12.3
Adding or removing terms
4.12.4
Some interesting inequality proofs
4.12.5
The Cauchy-Schwarz inequality
5
A brief digression on numbers
6
Proof
6.1
Direct proof
6.2
Proving identities
6.3
Proof by induction
6.3.1
Divisibility
6.3.2
From
k
k
to
k
+
1
k+1
6.4
Proof by contradiction
6.4.1
Proof that sqrt(2) is irrational
6.4.2
Proof that there are infinitely many primes
6.5
The pigeonhole principle
6.5.1
Proof that at least one number is greater than or equal to the mean
6.6
Existence proofs
6.7
Uniqueness proofs
7
Trigonometry
7.1
Trigonometric functions
7.2
Spangles
7.3
Trigonometric identities
7.3.1
sine and cosine identities
7.3.2
identities for tan
7.3.3
Secant, cosecant, cotangent and friends
7.3.4
Proving trigonometric identities
7.4
General solutions to trigonometric equations
7.5
homogenising trig functions
8
Exponentials and logarithms
8.1
Exponentials
8.2
Logarithms
8.3
Euler’s number
8.3.1
Definition of
e
e
9
Coordinate geometry
9.1
Distances
9.2
Circles
9.3
Vector lines
9.4
Planes
10
Differential calculus
10.1
Definition of the derivative
10.2
Derivatives of sums
10.3
Differentiating polynomials
10.4
Differentiating exponential functions
10.4.1
the derivative of e
10.5
The product rule
10.6
The chain rule
10.7
Derivatives and slopes
10.7.1
Proof that the reciprocal function is decreasing
10.8
Maxima and minima of functions
10.9
Implicit differentiation
10.9.1
Optimisation using implicit differentiation
10.10
A bunch of trigonometric limits
10.11
Derivatives of trigonometric functions
10.11.1
Derivative of sin(x)
10.11.2
Derivative of inverse trig functions
10.11.3
Derivative of arctan(x)
10.11.4
Derivative of arccos(x)
10.11.5
derivative of arcsin
10.12
Maclaurin series
10.12.1
Derivation of the sum of an infinite geometric series
10.12.2
A lower bound for the factorial function
11
Integral calculus
11.1
Introduction
11.2
Integration by parts
11.3
Integration by substitution
11.4
Integral arithmetic
11.5
Integration of trigonometric functions
11.5.1
Integral of cos(x) squared
11.5.2
Integral of sin(x)/(cos(x) + (cos(x) cubed))
11.5.3
Integral of (1-cos(x))/(1+cos(x))
11.5.4
Integral of sqrt(1-cos(x))
12
Polar coordinates
13
Differential equations
13.1
Separation of variables
13.2
Integrating Factors
13.3
Substitution
14
Complex numbers
14.1
Introduction
14.1.1
The complex conjugate
14.1.2
Dividing complex numbers
14.1.3
The modulus of a complex number
14.1.4
Properties of the complex conjugate and absolute value
14.1.5
Some example complex number problems
14.2
The Argand diagram
14.2.1
Plotting complex numbers
14.2.2
Loci on the Argand diagram
14.2.3
Polar co-ordinates and complex numbers
14.2.4
Graphical interpretation of operations on complex numbers
14.3
The trigonometric form of a complex number
14.3.1
Using the trigonometric form of a complex number
14.4
The exponential form of a complex number
14.5
Properties of the exponential form of a complex number
14.6
Helpful things to know about the exponential form
14.7
Cool stuff with trigonometry
14.7.1
Proving identities
14.7.2
Writing complex numbers in terms of the exponential function
14.7.3
Using the exponential form to show odd/evenness
14.8
The roots of unity
15
Hyperbolic functions
15.1
Definitions
15.2
Properties
15.2.1
Odd/even nature
15.2.2
Inverse functions
15.2.3
Inverse function of sinh(x)
15.3
Relationship to trig functions
15.4
Identities
II
A Level Further Mathematics
16
Linear algebra
16.1
Properties of matrices
16.1.1
Introduction
16.1.2
Adding matrices
16.1.3
Properties of matrix addition
16.1.4
Matrix multiplication
16.1.5
Properties of matrix multiplication
16.1.6
The identity matrix
16.1.7
Raising matrices to powers
17
More differential equations
17.1
Second-Order Differential Equations
17.1.1
Introduction
17.1.2
Homogenous second-order ODEs
17.1.3
Why two linearly independent solutions?
17.2
Systems of differential equations
18
Combinatorics
18.1
Basic counting principles
18.1.1
Finding the number of permutations
18.1.2
From English to maths
18.2
Arranging people in a circle
18.3
Listing all permutations of a string
III
General mathematics
19
Linear algebra, take 2
19.1
Systems of linear equations
19.1.1
Gaussian elimination
19.1.2
Elementary matrices
19.1.3
LU decomposition
19.1.4
Performing LU decomposition
19.2
Vector spaces
19.2.1
The vector space axioms
19.2.2
Linear independence
19.2.3
Bases
19.2.4
Subspaces
19.3
Linear transformations
19.3.1
Introduction to linear transformations
19.3.2
The matrix representation of a linear transformation
19.3.3
Gaussian elimination strikes back (linear independence, span and Gaussian elimination)
19.4
Determinants
19.4.1
Combinatorial approach
19.4.2
Using matrices
19.5
Spectral theory
19.5.1
Eigenvalues and eigenvectors
19.6
Inner product spaces
19.6.1
Inner products
19.6.2
Norms
19.6.3
Orthogonality
19.6.4
Some useful properties orthogonal vectors possess.
19.6.5
Orthonormal bases
19.6.6
Gram-Schmidt orthonormalisation
19.6.7
Orthogonal complement
19.6.8
Orthogonal projection
19.6.9
The method of least squares
20
Set theory
20.1
Basic set theory notions
20.1.1
Some example set theory proofs
20.2
Russell’s paradox
20.3
Relations
21
More interesting combinatorics
21.1
The principle of inclusion-exclusion
22
Number theory
22.1
Division
22.1.1
Introduction to division
22.1.2
A divides b
22.1.3
How large are divisors?
22.1.4
Bezout’s lemma
22.2
Modular arithmetic
22.2.1
Core definitions
22.2.2
The properties which make modular arithmetic
22.2.3
Fermat’s little theorem
22.2.4
The Chinese remainder theorem (and related problems)
22.3
Diophantine equations
22.4
Continued fractions
23
High-level logic
23.1
Introduction
23.2
Truth tables
23.3
Some useful operators
23.3.1
Logical and
23.3.2
Logical or
23.3.3
The material conditional
23.3.4
The material bi-conditional
23.4
Law of the excluded middle
24
Lower-level logic
24.1
Proof systems
24.2
Logics
24.2.1
Foundational things
24.2.2
Definition of propositional logic
24.2.3
Definition of predicate logic
24.2.4
Logical calculi
25
Graph theory
25.1
Finding shortest paths
25.2
Hamiltonian paths
25.2.1
The Hamiltonian path problem
25.2.2
Dirac’s theorem
25.3
Matchings in graphs
25.3.1
Bipartite matchings
26
Real analysis
26.1
Sequences
26.1.1
Convergence of sequences
26.1.2
A sequence can only converge to one value
27
Abstract algebra
27.1
Groups
27.1.1
Lagrange’s theorem
28
Probability
28.1
Discrete random variables
28.1.1
The linearity of expectation
28.1.2
Variance of a discrete random variable
28.2
The binomial distribution
28.3
The geometric distribution
28.4
The normal distribution
28.5
Continuous probability distributions
28.6
Transformations of continuous random variables
28.7
Hypothesis testing
29
Statistics
29.1
p-values
29.2
Testing for a median using the binomial distribution
29.3
Wilcoxon Matched-Pairs
29.4
Wilcoxon Signed-Rank
IV
Algorithms
30
Algorithms
30.1
Max-flow
31
Numerical methods
31.1
QR decomposition
31.1.1
Definition and some uses
31.1.2
Householder transformations
31.1.3
Givens rotations
31.2
Fourier analysis
31.2.1
Convolution
31.2.2
Periodic convolution
31.2.3
The Fourier series
31.2.4
There…
31.2.5
…and back again
31.2.6
The convolution theorem
31.2.7
“Geht es besser?” (The Fast Fourier Transform)
31.2.8
Filtering with Fourier
31.2.9
Circulant matrices
31.2.10
Proof of the convolution theorem using circulant matrices
32
Randomized algorithms
32.1
Introduction
32.2
Weighted uniform sampling
32.3
Target shooting
32.4
A sampling lemma
V
Computer science science stuff
33
Introduction
34
Pipelining
34.1
Latency
34.2
Throughput
VI
Miscellaneous
35
Resources
36
Assorted problems
36.0.1
Cones problem
36.0.2
Divisibility of abc base 10
36.0.3
Rationalising