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Saturday, November 30, 2013

online Encyclopedias of Mathematics

You can find a wealth of information on Mathematics of all levels at the following locations:
Also, you can find interesting (and much more than mathematical) topics at:
Finally, it will be very instructive to gain some insight into an experimental "Computational Knowledge Engine" (whatever this means to the present!)

Polynomials and equations

Here are some notes on the theory of polynomials and on various forms (polynomial, rational, irrational, eq. with absolute values) of equations.
You should test your understanding trying to workout the problems and exercises either here or there !


Straight line: notes and homework

Quadratics: notes and some homework

Here are some notes on quadratics and some homework as well.

Some background knowledge

those of you (parents or students) who are wondering what is the expected prerequisite level, for some student aiming to start an A-level or an IB HL-SL course, should take a look at the following: Background material

Thursday, November 28, 2013

Question of the week - #1

A question which is addressed to all those studying the fundamentals of differential calculus, and more specifically the definition of the derivative:

"Let a function $f(x)$ which is differentiable (and thus continuous) at a point $x=a$ of its domain. Is the derivative function $f '(x)$ necessarilly continuous at $x=a$ ?"

I am waiting for your ideas and answers.
I will post the solution next week, together with the names of those who will have communicated me correct answers (with proofs or counterexamples!) 

Indefinite Integrals - tables

A "computational" tool !! --> Indefinite integrals - tables

Mathematics Workbook - UoC

Contains 30 review exercises, indicative of the prerequisite level for someone intending to study Mathematics in a top University: Mathematics Workbook - UoC 

Notes for an introduction to Pure Mathematics (in Greek)

A general introduction to undergraduate level mathematics, focusing the viewpoint of pure mathematics (not for the faint-hearted !!)  Σημειώσεις για μια εισαγωγή στα Καθαρά Μαθηματικά


Fermat's last theorem (in Greek)

A very nice and interesting paper, with an appendix focusing on the historical development of the topic.
Unfortunately the text is in Greek only ;)    -->  Το τελευταίο θεώρημα του Fermat
(From: University of Athens, School of Mathematics)


Monday, November 25, 2013

Trigonometry I - class notes

Homework 18 (on functions)

Homework 17 (on functions)

Functions - class notes

Welcome !

   I would like to welcome everybody on my new blog.
   The primary purpose of this blog will be to post various material related to the curriculum of A-level mathematics, IB Math HL, SL and Further Math, and mathematics for grades K11-K12 or equivalent. We will post classnotes, material for deepening into the theory (notes, remarks, various proofs of propositions and theorems which are unlikely to be found on standard textbooks), solved exercises of escalating difficulty (in order to cover even the needs of the most demanding students), tests, collections of questions and exercises for practise, projects , etc.
   It is hoped that the material will be of interest to the demanding students and to instructors/teachers/ tutors offering classes at that level.

   The topics discussed here will cover a range such as:  
  • Quadratics (discriminant, roots, factorization, Vietta's forulae, graph),  
  • Straight line equation
  • elementary Number theory (Euclidean division, primes, prime factorizations, LCM, GCD), 
  • Polynomials (Euclidean division, long division, Horner's scheme, factor and remainder theorem, theorems on roots of polynomials, multiplicities and geometric interpretation),  
  • Partial fractions,  
  • arithmetic-geometric series,
  • Mathematical induction
  • Axioms of Real numbers (binary relations and their properties: reflexive, (anti)symmetric, transitive, ordering relations and equivalence relations, N, Q, R. R as an ordered field: intervals and bounds, completeness of real numbers and the "nested sphere theorem")
  • Exponentials and Logarithms (completeness of the reals and the definition of an irrational exponent, laws of logarithms, change of base, Exp and Log functions, the number e), 
  • Binomial theorem, Pascal's triangle 
  • Combinatorics (multiplication principle, inclusion-exclusion principle, permutations with or without repetition, arrangements with or without rep., combinations with or without rep., applications),    
  • Linear algebra (matrices, algebra of matrices, determinants, mxn linear systems of equations and their geometrical interpretation)
  • $2d$ and $3d$ Coordinate and geometry and Vector geometry (straight line equation in $2d$ and $3d$, equation of a plane, relative positions between two lines, two planes, a line and a plane, distance between a point and a line or a plane, Conics: circle, ellipse, parabola, hyperbola, coordinate, parametric, vector equations),  
  • Functions (graphs, "$1-1$", onto, inverse, composite,  monotonicity, linear, quadratic, polynomial, rational, irrational, exponential, logarithmic),  
  • Trigonometry ( radian measures: rad and deg, geometry of a circle, trigonometric ratios in right triangles, oriented angles, trigonometric circle, trigonometric ratios for oriented angles, elementary identities, reduction to the first quadrant, trigonometric functions sinx, cosx, tanx, cotx, secx, cosecx, graphs, trig. equations, general solution formulae, sine-cosine laws, solution of triangles, addition forumale, double angle formulae, asinx+bcosx, transformation formulae from sums to products and vice versa),  
  • Sequences (Convergent and Cauchy sequences, criteria of convergence, limits, application: e as limit of sequences),
  • Limits (abstract definitions, various computational methods),  
  • Continuity (Bolzano and Intermediate value theorem, extreme value theorem,),  
  • Differential and Integral Calculus (Differentiation, basic derivatives, products-quotients, tangents, chain rule, differentiation of the inverse function, implicit differentiation, Rolle's  and mean value theorem, local extrema, turning points, curve sketching, Integration, Indefinite integral - Antiderivative, basic integrals, integration by factors and by substitution, Definite integral, Fundamental theorem of integral Calculus, areas and volumes,  applications, mean value theorem of integral Calculus), 
  • Complex numbers (axiomatic description, Argand's diagram, operations and their geometrical interpretation, trigonometric form, exp form, De Moivre's theorem),  
  • Vectors and Vector spaces 
  • Groups, Rings, Fields,  
  • Statistics (descriptive and inferential),  
  • Probability (intuitive and axiomatic definition, sample spaces, events, mutually exclusive and independent events, Bayes' theorem, expected values, discrete and continuous distributions, gaussian, binomial, geometric, hypergeometric),  
  • Series expansions (Taylor, McLauren, ratio of convergence, criteria of convergence),  
  • Differential Equations,  
  • Arithmetic methods and approximations
  • (axiomatic) Euclidean Geometry,  
  • Mathematical Logic, .... etc
   There will also frequently appear posts related to the syllabus of the Greek Panhellenic Exams (corresponding to the three classes A' ,B' ,C' of the Greek Lyceum). These will often appear in greek (however, I will try to provide english translations for the most interesting of these).
   Finally, anything (ranging from Pure to Applied Mathematics) with sufficient interest for mathematically-oriented students/teachers/researchers may appear from time to time (under suitable headings).
   You are welcome to send me questions, problems, ideas etc related to the above material or with mathematics in general! I will try my best to respond to possible questions -as soon as possible- and to adopt best practise ideas in order to help anybody taking or giving such courses and to anyone inerested in mathematics generally.

So, keep on reading and thinking!