Complex Numbers

use the complex number system

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• develop an understanding of the classification of numbers and their associated properties, symbols and representations

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• define the number, $i$, as a root of the equation $x^2=-1$ (ACMSM067)ACMSM067

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• use the symbol $i$ to solve quadratic equations that do not have real roots

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represent and use complex numbers in Cartesian form AAM

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• use complex numbers in the form $z=a+ib$, where $a$ and $b$ are real numbers and $a$ is the real part $\mathrm{Re}(z)$ and $b$ is the imaginary part $\mathrm{Im}(z)$ of the complex number (ACMSM068, ACMSM077)ACMSM068 ACMSM077

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• identify the condition for $z_1=a+ib$ and $z_2=c+id$ to be equal

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• define and perform complex number addition, subtraction and multiplication (ACMSM070)ACMSM070

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• define, find and use complex conjugates, and denote the complex conjugate of $z$ as $\overline{z}$

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• divide one complex number by another complex number and give the result in the form $a+ib$

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• find the reciprocal and two square roots of complex numbers in the form $z=a+ib$

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represent and use complex numbers in the complex plane (ACMSM071)ACMSM071

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• use the fact that there exists a one-to-one correspondence between the complex number $z=a+ib$ and the ordered pair $(a,b)$

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• plot the point corresponding to $z=a+ib$

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represent and use complex numbers in polar or modulus-argument form, $z=r(\cos \theta +i\sin \theta )$, where $r$ is the modulus of $z$ and $\theta$ is the argument of $z$ AAM

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• define and calculate the modulus of a complex number $z=a+ib$ as $\left|z\right|=\sqrt{a^2+b^2}$

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• define and calculate the argument of a non-zero complex number $z=a+ib$ as $\arg (z)=\theta$, where $\tan \theta =\frac{b}{a}$

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• define, calculate 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 $(-\pi ,\pi ]$

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prove and use the basic identities involving modulus and argument (ACMSM080) AAMACMSM080

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• $|z_1{z}_2|=|z_1||z_2|$ and $\arg (z_1{z}_2)=\arg z_1+\arg z_2$

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• $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$ and $\arg \left(\frac{z_1}{z_2}\right)=\arg z_1-\arg z_2$, $z_2\ne 0$

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• $|z^n|={|z|}^n$ and $\arg (z^n)=n\arg z$

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• $\left|\frac{1}{z^n}\right|=\frac{1}{{|z|}^n}$ and $\arg \left(\frac{1}{z^n}\right)=-n\arg z$, $z\ne 0$

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• $\overline{z_1}+\overline{z_2}=\overline{z_1+z_2}$

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• $\overline{z_1}\overline{z_2}=\overline{z_1{z}_2}$

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• $z\overline{z}={\left|z\right|}^2$

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• $z+\overline{z}=2\mathrm{Re}(z)$

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• $z-\overline{z}=2i\mathrm{Im}(z)$

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understand Euler’s formula, $e^{ix}=\cos x+i\sin x,$ for real $x$

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represent and use complex numbers in exponential form, $z=r{e}^{i\theta }$, where $r$ is the modulus of $z$ and $\theta$ is the argument of $z$ AAM

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use Euler’s formula to link polar form and exponential form

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convert between Cartesian, polar and exponential forms of complex numbers

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find powers of complex numbers using exponential form

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use multiplication, division and powers of complex numbers in polar form and interpret these geometrically (ACMSM082) AAMACMSM082

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solve problems involving complex numbers in a variety of forms AAM

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use De Moivre’s theorem with complex numbers in both polar and exponential form AAM

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• prove De Moivre’s theorem for integral powers using proof by induction (ACMSM083)ACMSM083

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• use De Moivre’s theorem to derive trigonometric identities such as $\sin 3\theta =3{\cos }^2\theta \sin \theta -{\sin }^3\theta$

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determine the solutions of real quadratic equations

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define and determine complex conjugate solutions of real quadratic equations (ACMSM075) AAMACMSM075

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determine conjugate roots for polynomials with real coefficients (ACMSM090) AAMACMSM090

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solve problems involving real polynomials with conjugate roots

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solve quadratic equations of the form ${ax}^2+bx+c=0$, where $a,b,c$ are complex numbers AAM

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examine and use addition and subtraction of complex numbers as vectors in the complex plane (ACMSM084) AAMACMSM084

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• given the points representing $z_1$ and $z_2$, find the position of the points representing $z_1+z_2$ and $z_1-z_2$

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• describe the vector representing $z_1+z_2$ or $z_1-z_2$ as corresponding to the relevant diagonal of a parallelogram with vectors representing $z_1$ and $z_2$ as adjacent sides

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examine and use the geometric interpretation of multiplying complex numbers, including rotation and dilation in the complex plane

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recognise and use the geometrical relationship between the point representing a complex number $z=a+ib$, and the points representing $\overline{z}$, $cz$ (where $c$ is real) and $iz$

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determine and examine the $n$ th roots of unity and their location on the unit circle (ACMSM087)ACMSM087

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determine and examine the $n$ th roots of complex numbers and their location in the complex plane (ACMSM088)ACMSM088

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solve problems using $n$ th roots of complex numbers AAM

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identify subsets of the complex plane determined by relations, for example $|z-3i|\le 4$, $\frac{\pi }{4}\le \mathrm{Arg}(z)\le \frac{3\pi }{4}$, $\mathrm{Re}(z)>\mathrm{Im}(z)$ and $|z-1|=2|z-i|$ (ACMSM086)ACMSM086

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