Plaiting The Strands
Complex numbers and matrices are two of the key ideas in Further Pure Mathematics, both at AS and A2. They have deep connections to each other, but are often taught separately with only fleeting opportunities for students to think about them in a joined-up way. The card matching activity available with this article allows a number of strands of mathematics to be plaited together, and also paves the way for the future so that students get into the habit of looking for new connections all the time, and especially when they revisit these two topics.
Any complex number z = x + iy can be represented by the point (x, y) on a 2-D co-ordinate diagram, usually referred to as an Argand diagram in this context. Immediately we have a way of visualising real, imaginary and complex numbers on the same representation, and we can start to see direct links between complex numbers and the co-ordinate geometry of points, lines and circles studied first at GCSE then in AS Mathematics. Alternatively, any complex number z = x + iy can be represented by the vector . Treated as a position vector relative to the origin O, this gives another way of describing the point representing z on the Argand diagram: the complex number z at the “tip” of the position vector z. However, treated as a free vector, we also have a way of visualising addition of the complex numbers z and w: we start the free vector representing z on the Argand diagram at the “tip” of the position vector representing w. Adding z to w can then be thought of as a translating w by z on the Argand diagram, which links back to work in GCSE Mathematics, and connects yet again arithmetic and geometry.
2 x 2 matrices can be used to represent (amongst other transformations) enlargements and rotations in the 2-D plane. The matrix represents an enlargement centre the origin O with a scale factor of r, and the matrix represents a rotation centre the origin O through θ° anticlockwise. This extends GCSE Mathematics work into AS Further Mathematics: students love that they can “do” transformations without drawing anything! The matrix represents the combined transformation of the enlargement followed by the rotation, or the other way round. This gives a good opportunity to consider the commutativity of matrix multiplication and transformation composition: this pair combines commutatively (students should explain – geometrically – why) but is not generally true when multiplying 2 x 2 matrices (and students should suggest pairs of transformations that they can see won’t combine commutatively).
Finally, we can associate any real number r with the matrix and any complex number z = cos θ + i sin θ on the unit circle (see Issue 114) with the matrix . Then we can think of multiplication by a real number r as an enlargement (centre O, scale factor r) on the Argand diagram and multiplication by cos θ + i sin θ as a rotation (anticlockwise through θ° about O) on the Argand diagram. More generally, any complex number z = x + iy = r(cos θ + i sin θ) where r = |z| and θ = arg z can be associated with the matrix . Then we can think of multiplication by a complex number as a spiral symmetry, namely the combined transformation described above: an enlargement (centre O, scale factor r) followed by a rotation (anticlockwise through θ° about O) on the Argand diagram, or vice versa.
This card matching activity establishes links between complex numbers, coordinates, vectors, transformations and matrices, and will get students looking for further connections between different strands of mathematics as they move forward in their studies. Could your students add cards in the future to these to increase the connections (having studied loci in the Argand diagram, for example, or eigenvalues and eigenvectors), or make up similar sets of cards to display sets of connections in other areas of the syllabus they’re studying?
Page header by Richard Gray (adapted), some rights reserved