NSW Syllabuses

# Mathematics Extension 2 Stage 6 - Year 12 Extension 2 - Complex Numbers MEX-N1 Introduction to Complex Numbers

## Outcomes

#### A student:

• MEX12-1

understands and uses different representations of numbers and functions to model, prove results and find solutions to problems in a variety of contexts

• MEX12-4

uses the relationship between algebraic and geometric representations of complex numbers and complex number techniques to prove results, model and solve problems

• MEX12-7

applies various mathematical techniques and concepts to model and solve structured, unstructured and multi-step problems

• MEX12-8

communicates and justifies abstract ideas and relationships using appropriate language, notation and logical argument

## Subtopic Focus

The principal focus of this subtopic is the development of the concept of complex numbers, their associated notations, different representations of complex numbers and complex number operations in order to solve problems.

Students develop a suite of tools to represent and operate with complex numbers in a range of contexts.

## Content

• N1.1: Arithmetic of complex numbers
• Students:
• develop an understanding of the classification of numbers and their associated properties, symbols and representations
• discuss the solution of quadratics and identify situations where there are no real solutions
• define the imaginary number $$i$$, where $$i^2 = -1$$ (ACMSM067)
• use the symbol $$i$$ to solve quadratic equations that do not have real roots
• represent and use complex numbers in Cartesian form (standard or rectangular form) (ACMSM078)
• use complex numbers in the form $$z = a+ib$$, where $$a$$ is the real part $$Re(z)$$ and $$b$$ is the imaginary part $$Im(z)$$ of the complex number (ACMSM068, ACMSM077)
• identify the condition for $$z_1 = a+ib$$ and $$z_2=c+id$$ to be equal
• define, find and use complex conjugates, and denote the complex conjugate of $$z$$ as $$\bar z$$ (ACMSM069)
• define and perform complex number addition, subtraction and multiplication (ACMSM070, ACMSM079)
• divide one complex number by another complex number and give the result in the form $$a+ib$$ (ACMSM070)
• find the reciprocal of complex numbers in the form $$z=a+ib$$
• note that there are always two square roots of a non-zero complex number
• find the square roots of a complex number $$z=a+ib$$
• N1.2: Geometric representation of a complex number
• Students:
• use the fact that there exists a one-to-one correspondence between the complex number $$z=a+ib$$ and the ordered pair $$(a,b)$$
• plot the point corresponding to $$z=a+ib$$ on an Argand diagram (ACMSM071)
• represent and use complex numbers in polar or modulus-argument form, $$z=r\left(\cos\theta + i\sin\theta\right)$$, where $$r$$ is the modulus of $$z$$ and $$\theta$$ is the argument of $$z$$
• define and calculate the modulus of a complex number $$z=a+ib$$ as $$r=\left|z\right| = \sqrt{a^2 + b^2}$$ (ACMSM072)
• define and calculate the argument of a complex number $$z=a+ib$$ as $$\arg(z) = \theta$$ where $$\tan\theta = \frac{b}{a}$$ (ACMSM072)
• define, calculate and use the principal argument $$Arg(z)$$ of a non-zero complex number $$z$$ as the unique value of the argument in the range $$(-\pi,\pi]$$
• prove and use the basic identities involving modulus and argument:
$$\left|z_1z_2\right| = \left|z_1\right| \ \left|z_2\right|$$ and $$\arg(z_1z_2) = \arg z_1 + \arg z_2$$
$$\left\lvert \frac{z_1}{z_2}\right\rvert = \frac{\left|z_1\right|}{\left|z_2\right|}$$ and $$\arg\left(\frac{z_1}{z_2}\right) = \arg z_1-\arg z_2$$
$$\left|z^n\right| = \left|z\right|^n$$ and $$\arg\left(z^n\right) = n\arg z$$
$$\left\lvert\frac{1}{z^n}\right\rvert = \frac{1}{\lvert z\rvert^n}$$ and $$\arg\left(\frac{1}{z^n}\right) = -n\arg z$$
$$\overline{z_1} + \overline{z_2} = \overline{z_1+z_2}$$
$$\overline{z_1}\,\overline{z_2} = \overline{z_1z_2}$$
$$z\overline{z} = \left|z\right|^2$$
$$z + \overline{z} = 2 Re(z)$$ (ACMSM080)
• N1.3: Other representations of complex numbers
• Students:
• establish Euler's formula, $$e^{ix} = \cos x + i\sin x$$, for real $$x$$, by considering the Maclaurin series expansions of $$e^x$$, $$\cos x$$ and $$\sin x$$
• understand a Maclaurin polynomial can be used to find a series approximation to a function
• derive and use the first few terms of the Maclaurin series of $$e^x$$, $$\cos ⁡x$$ and $$\sin⁡ x$$ and know the interval of validity for those series
• represent and use complex numbers in exponential form, $$z=re^{i\theta}$$, where $$r$$ is the modulus of $$z$$ and $$\theta$$ is the argument of $$z$$
• use Euler’s formula to link polar form and exponential form
• find powers of complex numbers using exponential form
• convert complex numbers between Cartesian, polar, exponential and geometrical forms (ACMSM081)
• use multiplication, division and powers of complex numbers in polar and geometric forms (ACMSM082)
• solve problems involving complex numbers in a variety of forms