Operations with numbers in the form a × 10k where 1 ≤ a < 10 and k is an integer.

Arithmetic sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for sums of arithmetic sequences. Applications. Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.

Geometric sequences and series. Use of the formulae for the n th term and the sum of the first n terms of the sequence. Use of sigma notation for the sums of geometric sequences. Applications.

Financial applications of geometric sequences and series: • compound interest • annual depreciation.

Laws of exponents with integer exponents. Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.

Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. The symbols and notation for equality and identity.

Laws of exponents with rational exponents. Laws of logarithms. loga xy = loga x + logay x loga = loga x − logay y m loga x = mloga x for a, x, y > 0 Change of base of a logarithm. logb x loga x = , for a, b, logba Solving exponential equations, including using logarithms.

Sum of infinite convergent geometric sequences.

The binomial theorem: expansion of (a + b) , n ∈ ℕ. Use of Pascal’s triangle and nCr.

Counting principles, including permutations and combinations. Extension of the binomial theorem to fractional and negative indices, ie (a + b) , n ∈ ℚ.

Partial fractions.

Complex numbers: the number i, where i2 = − 1. Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument. The complex plane.

Modulus–argument (polar) form: z = r(cosθ + isinθ) = rcisθ Euler form: z = reiθ Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.

Complex conjugate roots of quadratic and polynomial equations with real coefficients. De Moivre’s theorem and its extension to rational exponents. Powers and roots of complex numbers.

Proof by mathematical induction. Proof by contradiction. Use of a counterexample to show that a statement is not always true.

Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, infinite number of solutions or no solution.