Introduction to Complex Numbers

complexnumbers year12 ext2

Introduction to complex numbers | NSW Curriculum Website

• ME2-12-03 uses algebraic and geometric representations of complex numbers to prove results and model and solve problems

📖 Prior Knowledge

Content	Prior knowledge	Used for
Trigonometry and Measure of Angles	- unit circle definition of trigonometry	- Euler’s formula
Working with Functions	- solving quadratic equations	- solving quadratic equations with complex solutions
Proof by Mathematical Induction	- Proof by MI	- proof of de Moivre’s theorem
Introduction to Vectors	- vectors	- complex numbers can be represented as a vector

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 ($\mathbb{C}$) 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 $a{x}^2+bx+c=0$, where $a$, $b$ and $c$ are real numbers and the discriminant $\Delta =b^2–4ac<0$
• Refer to $a$ as ‘the real part of the complex number $z=a+ib$’, denoted by $\mathrm{Re}(z)$
• Refer to $b$ as ‘the imaginary part of the complex number $z=a+ib$’, denoted by $\mathrm{Im}(z)$
• Classify numbers as belonging to the set of natural numbers ($\mathbb{N}$), integers ($\mathbb{Z}$), rational numbers ($\mathbb{Q}$), real numbers ($\mathbb{R}$) or complex numbers ($\mathbb{C}$), 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 $\mathrm{Re}(z_1)=\mathrm{Re}(z_2)$ and $\mathrm{Im}(z_1)=\mathrm{Im}(z_2)$
• 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 $\bar{z}=a-ib$ and use it to solve problems
• Define and calculate the modulus of a complex number $z=a+ib$ as $|z|=\sqrt{z\bar{z}}=\sqrt{a^2+b^2}$
• Establish relationships $\mathrm{Re}(z)=\frac{z+\bar{z}}{2}$ and $\mathrm{Im}(z)=\frac{z-\bar{z}}{2i}$ and $\frac{1}{z}=\frac{\bar{z}}{|z|}$ 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 $\arg (z)=\theta$, where $\theta$ satisfies $\sin \theta =\frac{b}{|z|}$ and $\cos \theta =\frac{a}{|z|}$, noting that the argument has multiple values that differ by multiples of $2\pi$
• Define and use the principal argument $\mathrm{Arg}(z)$ of a non-zero complex number $z$ as the unique value of the argument in the interval $(-\pi ,\pi ]$
• Define and use complex numbers in polar or modulus–argument form that expresses a complex number in terms of its modulus and argument, $z=r(\cos \theta +i\sin \theta )$, where $r$ is its modulus and $\theta$ 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: $\arg (z_1{z}_2)=\arg (z_1)+\arg (z_2)$, $\arg (\frac{z_1}{z_2})=\arg (z_1)-\arg (z_2)$ and $\arg (z^n)=n\arg (z)$ where $n$ is an integer
• Prove and use identities involving the complex conjugate of complex numbers: $\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}$, $\overline{z_1{z}_2}=\overline{z_1}\overline{z_2}$
• Prove and use the triangle inequality $|z_1+z_2|\le |z_1|+|z_2|$ for complex numbers $z_1$ and $z_2$

Solving equations with complex numbers

• Solve quadratic equations of the form $a{x}^2+bx+c=0$, 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 $\bar{z}=a-ib$ 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: $(\cos \theta +i\sin \theta {)}^n=\cos n\theta +i\sin n\theta$
• Prove that $(\cos \theta +i\sin \theta {)}^n=\cos n\theta +i\sin n\theta$ 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 $\mathrm{Re}(z)=c$ or $\mathrm{Im}(z)=k$ 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 $\arg (z)=\theta$ and $\arg (z-z_1)=\theta$, 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