MA183 Mathematics

Syllabus & Learning Outcomes

• 1: Elementary Number Theory

  Syllabus
  • 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
  Learning outcomes
  • You will 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. You will also be able to present a proof of Fermat's little theorem.

• 2: Matrix arithmetic

  Syllabus
  • Matrix addition, multiplication, row operations, inversion (via row operations), systems of equations, applications to resource allocation problems
  • Linear transformations, applications to cryptography and geometry
  Learning outcomes
  • You will be able to use matrices to: solve resource allocation problems; encipher messages using higher dimensional affine cryptosystems; attack higher dimensional affine cryptosystems; solve some geometric problems.

• 3: Eigenvalues and eigenvectors

  Syllabus
  • Calculation of eigenvalues, eigenvectors and matrix powers for 2x2 matrices, Hamilton-Cayley theorem (with proof for 2x2 matrices)
  • Proof by induction
  • Fibonacci sequence, Golden Ratio, applications to practical recurrence problems
  Learning outcomes
  • You will be able to 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 2x2 case.

• 4: Continuous functions

  Syllabus
  • Basic functions and their graphs.
  • Inverse functions.
  • Limits.
  • The Intermediate Value Theorem; roots of equations.
  Learning outcomes
  • You will be able to: 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. You will be able to apply the material learned to a variety of problems coming from Physics and Earth Sciences.

• 5: Rates of change and optimization

  Syllabus
  • Definition of derivative and its physical interpretation. Techniques of differentiation. Differentiability implies continuity (with proof).
  • The Mean Value Theorem; roots of equations.
  • Detecting maxima/minima, monotonicity, concavity; application to graph sketching.
  • Optimisation word problems.
  Learning outcomes
  • You will be able to: 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.

• 6: Anti-derivatives and differential equations

  Syllabus
  • Exponentials and logarithms.
  • Anti-derivatives and separable differential equations.
  • World problems involving differential equations: radioactive decay, population models.
  Learning outcomes
  • You will be able to: find the general solution of a number of basic separable differentiable equations; solve basic word problems stemming from Chemistry and Biology.

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