Principles of counting

Key knowledge
• the concept of one-to-one correspondence of sets and its application to consideration of countability
• the pigeon-hole principle as a problem-solving technique
• techniques of counting such as permutations and combinations and the inclusion–exclusion principle
• identities involving Pascal’s triangle.

Key skills
• use one-to-one correspondence to demonstrate the countability of certain subsets of R
• solve problems which involve techniques of counting
• use deductive reasoning to solve problems involving counting techniques, the pigeon-hole principle and Pascal’s triangle.

Number systems and recursion

Key knowledge
• the representation of natural, integer, rational and irrational real numbers in various structures and contexts including arithmetic and geometric sequences and series
• the concepts of identity, inverse, conjugate and limit
• operations on number, order properties, and algorithms for computation in a variety of contexts
• the representation of complex numbers and the conventions for arithmetic of complex numbers in cartesian form.

Key skills
• define and represent number in various structures and contexts such as integer, rational, real and complex number systems, ordered sets of numbers such as sequences and series
• identify and determine special forms such as identity, inverse, conjugate, and limit value
• perform exact and approximate computations and apply algorithms in various structures and contexts, including sequences and series, and interpret results
• apply deductive reasoning, including mathematical induction, and use appropriate language in the construction of mathematical arguments and proofs involving number and algebra.