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Complex Numbers & Matrices

Topic Overview
Question 3 on Paper 1 covers these two topics, which are apparently unconnected. Not that it concerns us, but the topics become interwoven at university level maths. Anyway this question is one of the more popular on the first paper, and has often been the easiest question on the paper. There have been exceptions; in 1998, Question 3(c) was particularly difficult.

Complex numbers deal with the so-called 'imaginary unit', i, which stands for the square root of -1. It might at first appear as if this quantity has nothing to do with the world we live in, not existing in the way the number 3 'exists'. However, complex numbers have many applications in engineering, physics and science in general. On our course we discuss the properties of complex numbers, see how they help us to solve equations and investigate how to use complex numbers written in polar form.

Matrices were originally introduced to simplify the maths involved in transformation geometry, although in Question 3 on Paper 1 we do not see them being used for this purpose. We concentrate on the definitions associated with matrices and how we can perform the basic operations of addition, multiplication, etc. One interesting feature of matrices is the absence of division, and the use of the inverse of a matrix to overcome this problem.

Topic Structure: Complex Numbers

The study of Senior Cycle Complex Numbers can be divided into the following sections:

1. Definitions and Basic Operations
Know the real parts and the imaginary parts
Keep the real parts and the imaginary parts separate
Multiply above and below by the conjugate of the bottom
Let the roots be a+bi and square both sides
Equate the real parts and equate the imaginary parts

2. Complex Equations
Use the quadratic formula
If x+yi is a root then so is x-yi
Use results from algebra

3. Polar Form of a Complex Number
Write as a point (real part, imaginary part)
Use the square root definition
Calculate the modulus and the argument (angle)
These can be used to simplify expressions
The proof is by induction
Write the complex number in polar form
Use De Moivres and the Binomial theorems
Write the complex number in general polar form

Topic Structure: Matrices

The study of Senior Cycle Matrices can be divided into the following sections:

1. Properties of Matrices
This terminology is frequently used

Add or subtract corresponding elements
The method of multiplication is not obvious and should be well noted
This is the matrix equivalent of the number 1

2. Inverse Matrices and Matrix Equations
This is used for the inverse, among other things
The inverse is frequently used because there is no division by a matrix
Pre-multiply or post-multiply by the inverse
You may be required to use matrix methods and not any other way
Used especially with diagonal matrices

Links
http://www.sosmath.com/complex/complex.html
This is quite a high-powered site, going very far into the theory of complex numbers. But the first section will be of interest to Leaving Cert students.

http://www.sosmath.com/matrix/matrix.html
The S.O.S. site on matrices is again aimed at students starting university, and so many of the questions refer to matrices of higher dimension than the 2 x 2 that we are used to.

http://www.ping.be/math/Pcomplex.htm
This section covers a wide area of problems about complex numbers, with many well-worked examples provided at three different levels, along with good practice material.

http://www.ping.be/math/matr.htm
More from the PING site, covering a comprehensive introduction to Matrices.

http://mathforum.org/dr.math/tocs/complex.high.html
The complex numbers page from the 'Ask Dr Math' site contains previously asked questions and their answers, many of which are relevant to our course.

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