In this module, proof by induction, the binomial theorem, complex numbers and vectors in three dimensions will be introduced, and their elegant algebraic properties will be explored in detail.  Depending on the student’s background in A-Level Mathematics, and whether or not Further Mathematics was undertaken, some or all of these topics may have been studied previously.  However, this module only assumes knowledge of A-Level Mathematics core modules, while aiming to consolidate and extend understanding of other important material which may have been studied before.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

• Understand and apply the method of proof by induction, and the binomial theorem.
• Understand and use complex numbers.
• Understand and use vectors, planes and lines in three dimensions.

27 - 50 minute lectures

5 - 50 minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study, including completion of homework exercises for tutorial classes.

Skills: 

Ability to work with complex numbers and vectors, and appreciation of their importance within mathematics.

Transferable Skills:

Ability to understand, manipulate and formulate abstract mathematical concepts.  Critical thinking and problem solving skills.  Ability to present work in a scholarly manner.

Formative assessment is by means of regular homework exercises.  Feedback to students on their performance in the homework and their progress towards learning outcomes is provided during the tutorial classes.

The in-course element of summative assessment is based on selected homework exercises.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also enables them to provide evidence of the higher levels of knowledge and understanding required for above-average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length, but mostly short.  Section B has a choice of two from three equally weighted questions.

• Mathematical induction and the Binomial Theorem
  • Proof by induction, with illustrative examples.
  • The Binomial Theorem and binomial series.

• Complex numbers
  • Definition of and motivation for complex numbers.  Addition, subtraction, multiplication and division of complex numbers.  Real and imaginary parts, conjugation, modulus and arguments.  Basic algebraic properties of complex arithmetic.  The complex plane and geometric interpretation of complex numbers.
  • Polar form and complex exponentials.  De Moivre's formula and applications.  Statement of Fundamental Theorem of Algebra.

• Vectors and coordinate geometry
  • Geometric and algebraic definitions of vectors.  Addition, subtraction, scalar multiplication, magnitudes, dot products and cross products of vectors, in geometric and algebraic forms.
  • Equations of lines and planes.  Intersections of, or distances between, points, lines and planes.

Please see Background Reading List for an indicative list.

Elementary Linear Algebra with Applications (10^th Edition), Anton, H. & Rorres, C., Wiley (2010)

Complex Numbers from A to . . . Z. Andreescu, T. & Andrica, D., Birkhäuser, (2006)