# Mathematics X - programme subject in programmes for Specialization in General Studies (MAT2-01)

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### Mathematics X

#### Numbers theory

The aims of the training are to enable the apprentice to
• reproduce Euclid’s proof that there are infinitely many primes, use the Sieve of Eratosthenes to find prime numbers, and give an account of Fermat numbers and Mersenne numbers in the historical search for primes
• apply congruence arithmetic to analyze divisibility, solve linear congruence equations and determine whether simple diophantic equations have solutions
• give an account of practical applications of congruence arithmetic in encryption and error-correcting codes
• plan, execute and present an independent piece of research into a subject linked to the main subject area

#### Complex numbers

The aims of the training are to enable the apprentice to
• use the four basic arithmetical operations, extraction of roots, absolute values and the conjugation rules for complex numbers, with and without digital tools
• use geometric representation of complex numbers, calculate with complex numbers in trigonometrical and apply de Moivre’s formula
• find complex nth roots and solve linear and quadratic equations with complex coefficients
• give an account of and present the main elements in the history of complex numbers from the Renaissance to Caspar Wessel
• give an account of and present a self-selected subject linked to the application of complex number theory

#### Probability and statistics

The aims of the training are to enable the apprentice to
• give an account of the concepts of distribution and stochastic variables for finite probabilities, and find expectation, variance and standard deviation for a stochastic variable
• give an account of the significance of the normal distributions and calculate the probabilities linked to them
• apply the central limit theorem to work out probabilities for sums of independent stochastic variables and binomial distributions
• plan, execute and present an assignment linked to statistical applications of probability calculations in hypothesis testing and sample surveys

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