About cookies

The NCETM site uses cookies. Read more about our privacy policy

Please agree to accept our cookies. If you continue to use the site, we'll assume you're happy to accept them.

 

Personal Learning Login






Sign Up | Forgotten password?
 
Register with the NCETM

Secondary Magazine - Issue 115: Sixth Sense


Created on 07 October 2014 by ncetm_administrator
Updated on 28 October 2014 by ncetm_administrator

 

Secondary Magazine Issue 115'SIX' by Richard Gray (adapted), some rights reserved
 

Sixth Sense
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 \binom{x}{y}. 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 \binom{r\; o}{o\; r} represents an enlargement centre the origin O with a scale factor of r, and the matrix \binom{cos\Theta \; sin\Theta }{sin\Theta \; cos\Theta } 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 \binom{r\: cos\Theta \; -r\: sin\Theta }{r\: sin\Theta \; r\: cos\Theta } = \binom{r\; 0}{0\; r}\binom{cos\Theta \; -sin\Theta }{sin\Theta \; cos\Theta } = \binom{cos\Theta \; -sin\Theta }{sin\Theta \; cos\Theta }\binom{r\; 0}{0\; r} 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 \binom{r\; 0}{0\; r} and any complex number z = cos θ + i sin θ on the unit circle (see Issue 114) with the matrix \binom{cos\Theta \; -sin\Theta }{sin\Theta \; cos\Theta }. 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 \binom{r\: cos\Theta \; -r\: sin\Theta }{r\: sin\Theta \; r\: cos\Theta }. 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?

Image credit
Page header by Richard Gray (adapted), some rights reserved

 
 

 

 
 
 
 
Download the magazine as a PDF
 
Secondary Magazine Archive
 
Magazine Feed - keep informed of forthcoming issues
 
Departmental Workshops - Structured professional development activities
 
Explore the Secondary Forum
 
Contact us - share your ideas and comments 
 

 


Comment on this item  
 
Add to your NCETM favourites
Remove from your NCETM favourites
Add a note on this item
Recommend to a friend
Comment on this item
Send to printer
Request a reminder of this item
Cancel a reminder of this item

Comments

 


There are no comments for this item yet...
Only registered users may comment. Log in to comment