Short summmary:
I. INTRODUCTION
1. Maths within science and all human activity (this is a small talk).
* Maths is a science and an art;
* We don't know why the universe behaves according to rules possible
to describe mathematically but we enjoy it;
* Examples: multiplication of numbers and modern cryptography;
gravitation and quadratic function;
* Pure maths --> applications --> Physics --> Engineering
--> Art
2. How do we do maths? Elements of logic.
* Mathematics starts with axioms and derives true statements
according to the rules of logic.
* A mathematical statement is a sentence that is either true or false
but not both. A proof of a statement is a sequence of sentences
convincing everybody that the statement is true; "everybody" is
a concept of an ideal being who perfectly understand mathematics;
a good practical approximation is a large group of competent
mathematicians.
* the truth tables for conjunction, disjunction, negation,
implication, if and only if; many examples.
* quantifiers:
- there exists
- for all
Examples emphasizing the difference between the two quantifiers.
3. Proofs.
* what is a proof;
* many examples of proofs.
- for all n\in N: n is odd ==> n^2 is odd; [direct proof]
- \sqrt{2} is irrational; [contradiction]
- for all n \in N: 6^n - 1 is divisible by 5; [induction]
- for all n \in N: \sum_{i=1}^n i = n(n+1)/2;
- for all n \in N: \sum_{i=1}^n(2i-1) = n^2;
- (a+b)^n =...
Links:
mathematics,
logic,
truth table,
proof.
II. COORDINATES AND MAPS
1. Coordinates and many dimensions.
* examples: line, plane, space; more dimensions.
2. Examples of maps and functions.
* definition of a function;
* constant, linear, quadratic etc.
3. Image and preimage (inverse image).
4. Description of geometric figures as preimages and
as solutions of equations.
Links:
Image and preimage,
injectivity,
surjectivity,
coordinate system,
n-dimensional space.
III. POLYNOMIAL EQUATIONS
1. A linear and quadratic equation
* how to solve; real and not real solutions.
2. Equations of higher degrees.
3. Long division of polynomials.
* practical examples
4. Polynomial equations with integer coefficients.
* the theorem about rational solutions.
Links:
Galois theory,
Gauss,
rational root theorem,
Abel's theorem,
Newton's method;
check your
calculations here.
IV. COMPLEX NUMBERS I
1. Numbers: natural, integers, rational, real
2. The square root of a negative number
3. Complex numbers and algebraic operations.
4. The fundamental theorem of algebra
5. Triangle inequality
6. Polar form; argument, modulus; multiplication; conjugation
Links:
Complex number,
other curious numbers,
triangle inequality,
polar form.
V. COMPLEX NUMBERS II
1. Proof of De Moivre's Theorem
2. Exponent notation
3. Roots of unity and other complex numbers.
4. Cauchy's theorem
Links:
De
Moivre's theorem,
Euler's formula,
roots of unity,
VI. MORE COMPLEX NUMBERS
1. Trigonometric identities
2. Calculating exact values of trigonometric functions
Links:
Trigonometric
identities,
VII. COMPLEX FUNCTIONS
1. Simple polynomial maps
2. The inverse
3. Iterations
4. Mandelbrot set
VIII. SYSTEMS OF LINEAR EQUATIONS
1. Gaussian elimination
2. The set of solutions
3. Definition of a vector space
4. Other motivation and examples
IX. LINEAR MAPS
1. Matrices
2. Composition, multiplication
3. Determinant, trace
X. EIGENVALUES AND EIGENVECTORS
1. Characteristic polynomial
2. Examples