Topic 3.1 Complex Numbers

Syllabus

Assumed knowledge (from year 11 Mathematics Specialist content)

This content will be only briefly reviewed as part of the year 12 course.

Complex numbers

2.3.7 define the imaginary number i as a root of the equation x^2=-1;

2.3.8 represent complex numbers in the rectangular form; a + bi where a and b are the real and imaginary parts;

2.3.9 determine and use complex conjugates;

2.3.10 perform complex number arithmetic: addition, subtraction, multiplication and division;

The complex plane

2.3.11 consider complex numbers as points in a plane, with real and imaginary parts, as Cartesian coordinates;

2.3.12 examine addition of complex numbers as vector addition in the complex plane;

2.3.13 develop and use the concept of complex conjugates and their location in the complex plane

Roots of equations

2.3.14 use the general solution of real quadratic equations;

2.3.15 determine complex conjugate solutions of real quadratic equations;

2.3.16 determine linear factors of real quadratic polynomials

Topic 3.1: Complex numbers (18 hours)

Cartesian forms

3.1.1 review real and imaginary parts Re(z) and Im(z) of a complex number z;

3.1.2 review Cartesian form;

3.1.3 review complex arithmetic using Cartesian forms;

Complex arithmetic using polar form

3.1.4 use the modulus |z| of a complex number z and the argument Arg(z) of a non-zero complex number z and prove basic identities involving modulus and argument

3.1.5 convert between Cartesian and polar form;

3.1.6 define and use multiplication, division, and powers of complex numbers in polar form and the geometric interpretation of these;

3.1.7 prove and use De Moivre’s theorem for integral powers

The complex plane (The Argand plane)

3.1.8 examine and use addition of complex numbers as vector addition in the complex plane;

3.1.9 examine and use multiplication as a linear transformation in the complex plane

3.1.10 identify subsets of the complex plane determined by relations such as
|z-3i|leq4,frac{pi}4leqmathrm{Arg}(z)leqfrac{3pi}4 and |z-1|=2|z-i|

Roots of complex numbers

3.1.11 determine and examine the nth roots of unity and their location on the unit circle

3.1.12 determine and examine the nth roots of complex numbers and their location in the complex plane

Factorisation of polynomials

3.1.13 prove and apply the factor theorem and the remainder theorem for polynomials

3.1.14 consider conjugate roots for polynomials with real coefficients

3.1.15 solve simple polynomial equations

Text

The textbook exercises referenced below are from Topics in Secondary Mathematics: Complex Numbers; Glen Prideaux, 2015.
complexnumbers-with-cover Support independent publishing: Buy this book on Lulu.

Lessons

1. Review of year 11 work

Textbook exercises: Chapter 1 questions 1-184, 257-280.

2. Complex Numbers in Polar form

The Prezi makes some references to course work on polar coordinates. This work is not included in the current course. All you really need to understand is that polar coordinates are represented by a radius or distance from the origin, usually denoted r and an angle measured anti-clockwise from the positive x-axis, usually denoted with the Greek letter theta (theta).
Note: The bubble on Conjugation Properties covers work not specifically mentioned in the current syllabus, but which may be seen as a straightforward application of work previously covered on the complex conjugate.
Textbook exercises: Chapter 2 questions 1-116.

3. Complex numbers and their conjugates

This is work that is not specifically mentioned in the syllabus, but covers some useful results that come from further exploration of arithmetic involving complex conjugates.
Solutions to exercises: complex-19

4. De Moivre’s Theorem

De Moire’s theorem deals with powers and roots of complex numbers.

You should be able to produce a proof of De Moivre’s theorem for positive integer powers using the principle of Mathematical Induction, and you may be asked to demonstrate this in one or more assessment tasks.
Textbook exercises: Chapter 2 questions 117-172

5. Roots and Factors of Polynomials

Recommended viewing: Numberphile: Odd Equations
Definitions are based on those in Wolfram Mathworld
Note: comments about complex roots occurring in pairs of complex conjugates are based on an assumption that we are dealing with a polynomial with real coefficients. If the coefficients are complex, that does not apply, and there can be any mix of real and single complex roots.
Note: Polynomial long division is a useful tool, but in many simpler situations it is easier to find a co-factor by multiplication rather than division. For example, if you have one linear factor of a cubic, you know the other factor must be a quadratic. Multiply a generic quadratic (ax^2+bx+c) by your known factor then equate coefficients to determine the unknowns (a, b) and (c). This is especially the case where you have complex coefficients.

Polynomial long division


The same division is probably quicker to do using the Synthetic Division algorithm rather than the long division algorithm shown here, but you’ll need to find a different resource if you want help with Synthetic Division. I find the Synthetic Division algorithm too much of a “black box” approach: it’s not at all easy to see the significance of the steps involved and why it works. I’m happy for students to use it if they understand what they’re doing, but note that if you use it in an assessment task and make an error, I will find it significantly harder to follow your working in order to be able to give you part marks.
Textbook exercises: chapter 1 questions 281-322

6. Subsets of the Complex Plane

Explore |z-z_1|=m|z-z_2| for different values of m with Desmos:

Textbook exercises: Chapter 1 questions 185-256.

Other Resources

See the Numberphile clip on the Fundamental Theorem of Algebra. Although the theorem itself is beyond the scope of this course, it’s a very nice application of some of the complex number ideas included here.

Old Prezis

These prezis were created for the former Specialist course and reflect the way that was structured.

The first is my very first Prezi and tries to pack all the content from the first semester work on complex numbers into a single Prezi. Most or all of this appears above, broken up into more manageable chunks.

3. Euler’s Formula

This is no longer included in the course.

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