Topic outline
Expressing Mathematics in English
Semester 2 Teaching:
- Semester 2 Lectures: Fridays 12-14, room 2.03
- Semester 2 Exercise classes: Wednesdays 12-14, room 3.05
Lecturer: Alex Simpson
Room 4.28 of Jadranska 21
Office hours by e-mail appointment
e-mail: Alex.Simpson@fmf.uni-lj.siTeaching assistant: Timotej Hrga
Room 5.08 of Jadranska 21
Office hours by e-mail appointment
e-mail: timotej.hrga@fmf.uni-lj.si- Semester 2 Lectures: Fridays 12-14, room 2.03
Course assessment
The requirements for passing the course are:
- Attendance: 75% attendance at lectures and at exercise classes
- Exams: either a passing mark (on aggregate) on the two midterm tests (the first in Semester 1 , the second in semester 2) or a passing mark on one of the course exams in the May/June or August/September exam period.
- Presentation: give a presentation on a mathematical topic in English in one of the lectures in the spring semester.
- Combined mark: obtain an overall pass mark on your combined mark from the exams and presentation.
Midterm tests
1st: 14. 2. 2022
2nd: 16. 5. 2022, 2.05, 16.00 - 18.00
Exams:
1st: 3. 6. 2022, 3.07, 10.00 - 12.00
2nd: 18. 8. 2022, VFP, 9.00 - 11.00
3rd: 30. 8. 2022, 3.05, 10.00 - 12.00
Old midterm exams
Below are some midterm exams from previous years.Old final exams
Lectures
Lectures are given in the classroom on the whiteboard. You are expected to take notes in the lectures.
SEMESTER 1
Lecture 1 (05. 10. 2021): Course assessment and organisation, cardinal numbers (counting numbers), ordinal numbers (position numbers), kinds of number (odd, even, square, cube, triangular, factorial, powers of 2, prime), Mersenne primes.
Lecture 2 (12. 10. 2021): Negative numbers and integers. Fractions and rational numbers. Mixed fractions and reduced fractions. Euclid's algorithm for gcd.
Lecture 3 (19. 10. 2021): The square root of 2 is irrational. Real numbers and decimal expansions. Ratios. The golden ratio. Pi as a ratio. Formulas involving pi.
Lecture 4 (26. 10. 2021): The decimal expansion of pi and the Gregory-Leibniz series. Scientific notation.
Lecture 5 (2. 11. 2021): Floating point numbers. Exponentiation and exponential functions. Logarithms. Euler's number e.
Lecture 6 (9. 11. 2021): Plane geometry. Definition of polygon. Names of polygons including special triangles and quadrilaterals. Vocabulary for angles The statement of Pythagoras' theorem.
Lecture 7 (16. 11. 2021): Congruence and similarity of 2-dimensional shapes. Proof of the Pythagorean theorem using similar triangles. 3-dimensional solids, their volumes and surface areas: cube, rectangular cuboid, cylinder, cone sphere. Polyhedra and regular polyhedra. The 5 Platonic solids.
Lecture 8 (23. 11. 2021): Cartesian coordinates. Polar coordinates. The basic trigonometric functions. Converting between cartesian and polar coordinates.
Lecture 9 (30. 11. 2021): Infinite series for sine and cosine. Motivation for complex numbers. The basic form of a complex number (real and imaginary parts). The fundamental theorem of algebra. The complex plane. The modulus and argument of a complex number. Complex conjugate. Complex numbers of modulus 1. The complex exponential and Euler's formula.
Lecture 10 (07. 12. 2021): Vectors as quantities with magnitude and direction. n-dimensional vectors. Operations on vectors: magnitude, sum, scalar multiplication, scalar (dot) product, vector (cross) product. Linear independence.
Lecture 11 (14. 12. 2021): General form of and terminology for matrices. Row vectors, column vectors and square matrices. Operations on matrices: sum, scalar multiplication, matrix multiplication. Matrix multiplication is associative but not commutative.
Lecture 12 (21. 12. 2021): Identity and inverse matrices. Characterisation of invertibility and inverse matrices in terms of determinant and adjunct. Characterisation of invertibility in terms of rank. Gaussian elimination for simultaneous linear equations. The augmented matrix, elementary row operations and row echelon form.
Homework (27th-31st December): Instead of a lecture and exercise class in the week of 27th-31st December, please watch and understand (both the English and the mathematics in) the two short videos below.
Lecture 13 (04. 01. 2022): Transformations between m-dimensional and n-dimensional spaces. Example of a 2-dimensional transformation: rotation by pi/2. Definition of linear transformation. Example: rotation by an arbitrary angle theta. Calculating the rotation matrix using basis vectors. Composition of linear transformations. The composite transformation is calculated by matrix multiplication.
Lecture 14 (11. 01. 2022): Countability and uncountability in the English language (count nouns and non-count nouns). The notion of equal cardinality. The notion of countable set. Countable infinite sets have the same cardinality. The sets of natural numbers, rational numbers and real numbers are all countable. Cantor's theorem: the set of real numbers is uncountable.
SEMESTER 2
Lecture 15 (16. 02. 2022): Finite and infinite sequences. Defining sequences explicitly and by recursion. Arithmetic progressions. Geometric progressions.
Lecture 16 (18. 02. 2022): The notion of limit of an infinite sequence. A sequence converges if and only if it is a Cauchy sequence. A convergence test: any sequence that is monotonically increasing and bounded converges. Application: convergence of the Basel-problem sequence (the sum of the reciprocals of square numbers).
Lecture 17 (04. 03. 2022): Series and series convergence. Examples of convergent sequences. The Euler zeta function. The harmonic series is not convergent. Harmonic numbers (and the pill jar puzzle). Convergence test for alternating series. Convergence test for absolutely convergent series.
Lecture 18 (11. 03. 2022): Three notions of limit: (1) of a sequence (recap); (2) of a function at infinity; (3) of a function at a real number. Expressing quantifiers in English. Expressing questions and assumptions. Formulating negations in English.
Lecture 19 (18. 03. 2022): Terminology associated with functions. Intervals of real numbers. The definition of continuity. The importance of continuity in mathematics.
Lecture 20 (25. 03. 2022): Brownian motion trajectories as examples of functions that are continuous but not smooth. Differentiability is the mathematics of smoothness. Intuition: derivatives = slope of tangent line. Subtle points with this intuition. Precise mathematical definition of derivative. Laws of differentiation for: monomials, sums, scalar multiples, products (the "product rule") and quotients (the "quotient rule").
Lecture 21 (01. 04. 2022): Power series for sin x and cos x revisited. The chain rule. Worked example of a differentiable function with a non-continuous derivative. Continuously differentiable functions. The hierarchy of n-times continuously differentiable functions.
Lecture 22 (08. 04. 2022): Comparing analysis and calculus. The Newton-Raphson method for solving the root finding problem. Increasing and decreasing (differentiable) functions characterised by their derivatives. Critical points and local and global extrema.
Lecture 23 (15. 04. 2022): The axiomatic method in mathematics.: axioms (postulates), definitions, theorems, propositions, lemmas and corollaries. Axioms for the real numbers. The reals form an Abelian group under addition. The nonzero reals form an Abelian group under multiplication. Adding distributivity: the real numbers form a field. Adding order: the real numbers form an ordered field. Adding completeness: the real numbers are a complete ordered field.
Lecture 24 (22. 04. 2022): Motivating integration. Definite integrals calculate area. The fundamental theorem of calculus. Antiderivatives and indefinite integrals. Laws of integration. Proof that the integral of the reciprocal function is a logarithm function.
Lecture 25 (29. 04. 2022): Simultaneous live and Zoom lecture. Samples and population in statistics. Desired properties of a sample. Different kinds of data. Summary statistics. Averages: mean, median and mode. Measures of spread: standard deviation and variance. Normal distributions.
Lecture 26 (06. 05. 2022): Instructions for presentations and marking criteria. The geometric mean. Convexity. Proof of the inequality of arithmetic and geometric means using the concavity of the logarithm function.
(13. 05. 2022): Student presentations 1
(18. 05. 2022): Student presentations 2
(20. 05. 2022): Student presentations 3
(23. 05. 2022): Student presentations 4
Exercise classes
In class 14 we went through a past midterm instead of an exercise sheet.