Six Gems for AS Further Pure Mathematics Let Maths take you Further…

General philosophy – What makes a AS Further Pure gem? For a gem to be precious it should motivate students’ interest aid students’ understanding reinforce connections within mathematics

Gem 1: Complex Numbers and coordinate geometry How do loci on the Argand diagram link to the coordinate geometry that students are studying in AS Core units? Circles Lines

Gem 2: Matrices and Google Trigonometry How Google uses matrices to do page ranking Using composition of linear maps and matrix multiplication to prove the addition formulae for sine and cosine

Gem 3: Proof by induction and pictures Tiling a chessboard

Gem 4: The Mandelbrot Set

The Mandelbrot Set The Mandelbrot Set is the most famous example of a fractal. The word "fractal" has two related meanings. In everyday life, it means a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as "infinitely complex." In mathematics a fractal is a geometric object that satisfies a specific technical condition, namely having a Hausdorff dimension greater than its topological dimension (yikes…)

The Mandelbrot Set You are now going to consider the iteration Let’s work out what happens if you start with the value 0, for various different values of c. First of all let’s take c = 1. This gives the sequence 0, 1, 2, 5, 26,….. Now c = –1. This gives the sequence 0, -1, 0, -1, 0 ,-1,….

The Mandelbrot Set What happens if c = 0 ? What happens if c = i ? What about c = 2, -2, 0.5, 1 + i, 2i, 1 – i ? A spreadsheet program can help us with this. The Mandelbrot set consists of those numbers c for which the sequence starting with 0 and calculated above (with ) does not tend to infinity.

Gem 5: Roots and coefficient of polymonials How does the graph of a cubic relate to the graph of one with related roots? Connections to AS Core. What is special about the points where a straight line crosses a cubic?

Gem 6: Fun with series Why does the sum of the first n cubes equal the square of the sum of the first n natural numbers. A surprising result, the sum of the harmonic series.