Mathematics
MA180: Mathematics (15 ECTS)
Mathematics MA180 is an introduction to university mathematics aimed at students studying the mathematical and physical sciences. Students should have achieved at least an OA2 or HC3 level in their Leaving Certificate. Around 66% of students will have studied higher level mathematics at Leaving Certificate. The module is a prerequisite for the Mathematics BSc and is a popular option for a wide range of degree programmes in the mathematical and physical sciences.
Taught in Semester(s) 1 and 2. Examined in Semester(s) 1 and 2.
Workload: 300 hours (96 Lecture hours, 24 Tutorial hours, 180 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- use modular arithmetic and Euler's Phi function to: detect errors in ISBNs; encipher messages using 1-dimensional affine and RSA cryptosystems; attack 1-dimensional affine cryptosystems; calculate with Chinese remainders; present a proof of Fermat's little theorem.
- use matrices to: solve resource allocation problems; encipher messages using higher dimensional affine cryptosystems; attack higher-dimensional affine cryptosystems; solve some geometric problems.
- use eigenvalues, eigenvectors and the Principle of Induction to solve practical and theoretical problems about recurrence. You will also be able to state the Hamilton-Cayley theorem and prove it in the $2\times2$ case.
- sketch the graph of a number of basic functions; calculate the limit of a function at a point or at infinity; decide whether a given function has an inverse and, if it does, calculate it; use the Intermediate Value Theorem to find roots of equations; apply the material learned to a variety of problems coming from physics and earth sciences.
- use the definition of derivative to compute the derivative of simple functions; apply different techniques of differentiation to calculate derivatives; apply the Mean Value Theorem to finding roots of equations; find maxima/minima/inflection points, and use these to sketch graphs of functions; apply differentiation techniques to solve optimisation problems arising from business and economics.
- find the general solution of a number of basic separable differential equations, solve basic problems from chemistry and biology. .
- test the validity of propositional arguments; design simple logic circuits; express statements in the language of set theory.
- express a permutation as a product of transpositions and thus determine its sign; factorize certain polynomials as products of irreducible polynomials.
- determine the determinant and eigenvectors of an $n\times n$ matrix for small values of $n$. You will be able to describe the role of eigenvectors in Google's page rank algorithm.
- distinguish between finite, countably infinite and uncountable sets of real numbers. You will be able to explain the meanings of the terms supremum and infinum, and analyze boundedness properties of given sets.; explain the concept of convergence and determine whether a given sequence of real numbers is convergent.
- explain the connection between differential and integral calculus using the Fundamental Theorem of Calculus; evaluate definite and indefinite integrals using a variety of techniques.
- sketch the level curves of a number of basic functions; calculate the partial derivatives of a function, and to give the equations of the plane tangent to a point of the graph; apply the second derivative test to classify critical points.
Indicative Content
The module covers 12 topics.
Elementary Number Theory: Modular arithmetic; Euclidean algorithm; applications to ISBNs and cryptography; Euler's Phi function; Fermat's little theorem (and its proof); application to public key cryptography; Chinese Remainder Theorem;
Matrix arithmetic: Matrix addition, multiplication, inversion; systems of equations; applications to resource allocation problems; linear transformations; applications to cryptography; cross products; applications to geometry.
Eigenvalues and eigenvectors: Calculation of eigenvalues, eigenvectors and matrix powers for 2x2 matrices; Hamilton-Cayley theorem; Fibonacci sequence; Golden Ratio; applications to practical recurrence problems.
Continuous functions: Basic functions and their graphs; inverse functions; limits; Intermediate Value Theorem; roots of equations;
Rates of change and optimization: Definition of derivative and its physical interpretation; techniques of differentiation; differentiability implies continuity (with proof); Mean Value Theorem; roots of equations.; detecting maxima/minima; monotonicity, concavity; application to graph sketching; optimisation problems.
Anti-derivatives and differential equations: Exponentials and logarithms; anti-derivatives and separable differential equations; real-world problems involving differential equations; radioactive decay; population models.
The language of mathematics: Logic; sets; relations and equivalence relations; applications to mathematical reasoning, legalistic reasoning, and design of digital computers.
Permutations and polynomial functions: functions; injective functions; surjective functions; the sign and order of a permutation; application to the 15-puzzle; polynomial division and factorisation over integers modulo a prime; application to reliable transmission of digital information.
More matrix algebra: Solving systems of linear equations; determinants and eigenvectors of nxn matrices; Google's page rank algorithm.
Properties of the Real Numbers: Bounded, unbounded, finite and infinite sets; different kinds of infinities; the completeness property of the real numbers; sequences of real numbers; convergent and divergent sequences; boundedness and monotonicity; applications to financial annuities.
Integral Calculus: Definite integrals and the Fundamental Theorem of Calculus; techniques of Integration; applications of integration to real-world problems.
Calculus of two variables: Introduction to derivatives and integrals of functions of two variables.
Module Resources
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