#ComplexNumbers #Year12 #Ext2 >[!info]- [Introduction to complex numbers | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-2-11-12-2024/content/n12/fa0be3067b) >- 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 $ax^2+ bx+ c =0$, 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: $|z_1z_2|=|z_1||z_2|$, $|\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}$ and $|z^n|=|z|^n$, where $n$ is an integer - Prove and use identities involving the argument of complex numbers: $\arg(z_1z_2)=\arg(z_1)+\arg(z_2)$, $\arg(\frac{z_1}{z_2})=\arg(z_1)-\arg(z_2)$ and $\arg(z^n)=n\arg(z)$ where $n$ is an integer - Prove and use identities involving the complex conjugate of complex numbers: $\overline{z_1+z_2}=\bar{z_1}+\bar{z_2}$, $\overline{z_1z_2}=\bar{z_1}\bar{z_2}$ - 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 $ax^2+bx+c=0$, 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 $\bar{z}=a-ib$ 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 $\pm1$ in polar form and their location on the unit circle - Illustrate the geometrical relationship connecting the $n$th roots of $\pm1$ - 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