11–12Mathematics Extension 2 11–12 Syllabus

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Year 12

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Introduction to complex numbers

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME2-12-03
uses algebraic and geometric representations of complex numbers to prove results and model and solve problems

Introduction to complex numbers

Arithmetic of complex numbers

Define the number $i$ by $i^{2} = - 1$
Use the number $i$ to solve quadratic equations of the form $x^{2} + k = 0$ where $k$ is a positive real number
Define the complex numbers ( $C$ ) as the set of numbers of the form $a + ib$ , where $a$ and $b$ are real numbers
Use complex numbers to express the roots of quadratic equations of the form ${ax}^{2} + bx + c = 0$ , where $a$ , $b$ and $c$ are real numbers and the discriminant $∆ = b^{2} - 4 ac < 0$
Refer to $a$ as ‘the real part of the complex number $z = a + ib$ ’, denoted by $Re (z)$
Refer to $b$ as ‘the imaginary part of the complex number $z = a + ib$ ’, denoted by $Im (z)$
Classify numbers as belonging to the set of natural numbers $(N$ ), integers ( $Z$ ), rational numbers ( $Q$ ), real numbers ( $R$ ) or complex numbers ( $C$ ), each of which is an extension of the previous
Identify and use the condition for two complex numbers $z_{1}$ and $z_{2}$ to be equal, that is $z_{1} = z_{2}$ if and only if $Re (z_{1}) = Re (z_{2})$ and $Im (z_{1}) = Im (z_{2})$
Define and perform complex number addition, subtraction and multiplication, with and without digital tools
Define the complex conjugate of a complex number $z = a + ib$ as $\overset{̅}{z} = a - ib$ and use it to solve problems
Define and calculate the modulus of a complex number $z = a + ib$ as $|z| = \sqrt{z \overset{̅}{z}} = \sqrt{a^{2} + b^{2}}$
Establish relationships $Re (z) = \frac{z + \overset{̅}{z}}{2}$ and $Im (z) = \frac{z - \overset{̅}{z}}{2 i}$ and $\frac{1}{z} = \frac{\overset{̅}{z}}{|z|}$ and use them to solve problems
Divide one complex number by another non-zero complex number, with and without digital tools, and give the result in the form $a + ib$
Find the two square roots of a complex number $z = a + ib$

Geometric representation of complex numbers

Plot the complex number $z = a + ib$ as a point on the complex plane with and without graphing applications
Define and calculate the argument of a non-zero complex number $z = a + ib$ as $\arg (z) = θ$ , where $θ$ satisfies $\sin θ = \frac{b}{|z|}$ and $\cos θ = \frac{a}{|z|}$ , noting that the argument has multiple values that differ by multiples of $2 π$
Define and use the principal argument $Arg (z)$ of a non-zero complex number $z$ as the unique value of the argument in the interval $(- π, π]$
Define and use complex numbers in polar or modulus–argument form that expresses a complex number in terms of its modulus and argument, $z = r (\cos θ + i \sin θ)$ , where $r$ is its modulus and $θ$ is an argument of $z$ , and represent complex numbers in this form on the complex plane
Use multiplication, division and powers of complex numbers in polar form and interpret these geometrically
Convert between complex numbers in Cartesian form and polar form and use complex numbers in Cartesian form and polar form to solve problems
Prove and use identities involving the modulus of complex numbers: $|z_{1} z_{2}| = |z_{1}| |z_{2}|$ , $|\frac{z_{1}}{z_{2}}| = \frac{|z_{1}|}{|z_{2}|}$ and $|z^{n}| = {|z|}^{n}$ , where $n$ is an integer
Prove and use identities involving the argument of complex numbers: $a r g (z_{1} z_{2}) = a r g (z_{1}) + a r g (z_{2})$ , $a r g (\frac{z_{1}}{z_{2}}) = a r g (z_{1}) - a r g (z_{2})$ and $a r g (z^{n}) = n \arg (z)$ where $n$ is an integer
Prove and use identities involving the complex conjugate of complex numbers:
$\overset{___}{z_{1}} + \overset{___}{z_{2}} = \overset{____________}{z_{1} + z_{2}}$ , $\overset{___}{z_{1}} \overset{___}{z_{2}} = \overset{________}{z_{1} z_{2}}$
Prove and use the triangle inequality $|z_{1} + z_{2}| \leq |z_{1}| + |z_{2}|$ for complex numbers $z_{1}$ and $z_{2}$

Solving equations with complex numbers

Solve quadratic equations of the form ${ax}^{2} + bx + c = 0$ , where $a$ , $b$ and $c$ are complex numbers
Recognise that solutions to quadratic equations with real coefficients are complex conjugates of each other and use this to solve problems
Prove the complex conjugate root theorem: if the complex number $z = a + ib$ is a root of the polynomial equation $P (x) = 0$ with real coefficients, then the complex conjugate $\overset{̅}{z} = a - ib$ is also a root of $P (x) = 0$
Solve problems involving complex conjugate roots of polynomial equations with real coefficients

Powers and roots of complex numbers

Using proof by mathematical induction prove de Moivre’s theorem for positive integer powers: ${(\cos θ + i \sin θ)}^{n} = \cos n θ + i \sin n θ$
Prove that ${(\cos θ + i \sin θ)}^{n} = \cos n θ + i \sin n θ$ for negative integers $n$
Use de Moivre’s theorem to find any integer power of a given complex number
Use de Moivre’s theorem to derive trigonometric identities
Determine the $n$ ^th roots of $\pm 1$ in polar form and their location on the unit circle
Illustrate the geometrical relationship connecting the $n$ ^th roots of $\pm 1$
Determine the $n$ ^th roots of complex numbers and their location on the complex plane
Recognise that a complex number can be represented as a vector, where the magnitude and direction of the vector are determined by the modulus and argument of the complex number respectively
Examine and use addition and subtraction of complex numbers as vectors on the complex plane
Examine and use the geometric interpretation of multiplying complex numbers, including rotation and dilation on the complex plane, with and without graphing applications
Prove geometric results using complex numbers as vectors
Solve problems and prove results using the $n$ ^th roots of complex numbers

Describing lines, curves and regions

Graph vertical and horizontal lines of the form $Re (z) = c$ or $Im (z) = k$ where $c$ and $k$ are real constants
Graph the line corresponding to the equation $|z - z_{1}| = |z - z_{2}|$ , where $z_{1}$ and $z_{2}$ are complex numbers, and give a geometrical description of the line
Graph the circles corresponding to the equations $|z| = r$ and $|z - z_{1}| = r$ , where $z_{1}$ is a complex number and $r$ is a positive real number
Graph rays corresponding to the equations $\arg (z) = θ$ and $\arg (z - z_{1}) = θ$ , where $z_{1}$ is a complex number
Graph regions associated with lines, rays and circles defined using complex numbers, giving a geometrical description of any such curves or regions, and using circle geometry theorems where necessary

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11–12Mathematics Extension 2 11–12 Syllabus

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