Complex Numbers
12 Revision Questions
Complex Numbers — Selected Questions
1. Equality of complex numbers
Find the complex number \(z=x+yi\) if \(5z+2i\overline{z}=11-4i\), where \(\overline{z}\) is the conjugate of \(z\).
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2. Addition, subtraction and multiplication
If \(z=5-3i\) and \(w=-2+4i\), express in the form \(a+bi\): \[ 3w\big(2z-\overline{z}\big). \]
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3. Conjugate and division
Express \(\displaystyle \frac{30+i}{3+5i}\) in the form \(a+bi\).
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4. Square roots
Find real \(a,b\) if \((a+bi)^2=3-4i\).
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5. Argand diagram and modulus
Let \(z=3+4i\) and \(w=1-2i\). Compute \(z+w\) and investigate whether \(|z+w|=|z|+|w|\).
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6. Interpreting an Argand diagram
The Argand diagram below shows points \(a,b,c,d,e,f\).
They represent \(z,\;2z,\;\overline{z},\;z+\overline{z},\;iz,\; (1+i)z\) .
(i) Identify the label for each of \(z,\;2z,\;\overline{z},\;iz,\;z+\overline{z},\;(1+i)z\).
(ii) Determine the acute angle z, O, (1+i)z.
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Identification rules.
- \(2z\): same argument as \(z\); modulus doubled (lies on same ray, twice as far).
- \(\overline{z}\): reflection of \(z\) in the real axis.
- \(iz\): rotation of \(z\) by \(+90^\circ\) (same modulus).
- \(z+\overline{z}\): real part is doubled, imaginary parts add to 0.
- \((1+i)z\): multiply by \(1+i=\sqrt2\,\mathrm{cis}(\pi/4)\) → scale by \(\sqrt2\) and rotate \(+45^\circ\).
Angle at \(O\). Multiplication by \(1+i\) rotates any vector by \(\arg(1+i)=45^\circ\). Thus \(\angle z, O,\,(1+i)z=\boxed{45^\circ}\) for any \(z\neq 0\).
| Complex number | Label on diagram |
|---|---|
| \(z\) | c |
| \(2z\) | e |
| \(\overline{z}\) | b |
| \(iz\) | a |
| \(z+\overline{z}\) | f |
| \((1+i)z\) | d |
7. Complex equations
If \(2+i\) is a root of \(z^2-(a-i)z+(8+bi)=0\) with \(a,b\in\mathbb{R}\), find \(a,b\) and the other root.
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8. Conjugate Roots Theorem
Suppose \(3+2i\) is a root of \(z^3+az^2+bz-52=0\) with \(a,b\in\mathbb{R}\). Find \(a,b\) and the other roots.
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9. Polar form
Express \(-2\sqrt3+2i\) in the form \(r(\cos\alpha+i\sin\alpha)\).
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10. De Moivre: Trigonometric identity
Using De Moivre’s Theorem, express \(\sin 3\theta\) as a polynomial in \(\sin\theta\).
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11. De Moivre: Large powers
Express \(\big(1-\sqrt3\,i\big)^8\) in the form \(a+bi\).
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12. Roots via De Moivre
Solve \(z^3=-64\) and give the roots in \(a+bi\) form.
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Tutorial Videos
1. Irrational numbers
Irrational numbers - surds |
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Construct root 2Using a compass and straight edge
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Construct root 3Using a compass and straight edge
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2. Complex numbers introduction
Equation with imaginary roots |
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Quadratic equation with imaginary rootsUsing the quadratic formula
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Addition and multiplication of complex numbers |
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3. Division and equity of complex numbers
Dividing by a complex number - conjugateExpressing a fraction with a real denominator
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Complex number equations"Real = real, imaginary = imaginary"
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4. Argand diagram - modulus
Plot on an Argand diagram
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5. Transformations of complex numbers
Understanding multiplication |
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Multiplication involves scaling and rotating
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Transformations involving multiplication |
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6. Conjugate roots theorem
Quadratic equation - conjugate roots |
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Showing that a conjugate is a root |
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Solving a cubic equation with two imaginary roots |
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7. Polar form of a complex number
Converting from polar to rectangular form |
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Express in polar formFinding the modulus and argument
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π, pi, 2*pi/3, √2, sqrt(5), 3*sqrt(7).
\(\displaystyle z = 5\!\left(\cos\!\big(\tan^{-1}\!\tfrac{4}{3}\big)+ i\sin\!\big(\tan^{-1}\!\tfrac{4}{3}\big)\right)\)
\(\displaystyle z=\sqrt2(\cos\tfrac{\pi}{4}+i\sin\tfrac{\pi}{4})\)
\(\displaystyle z=5\!\left(\cos\!\big(\pi-\tan^{-1}\!\tfrac{4}{3}\big)+i\sin\!\big(\pi-\tan^{-1}\!\tfrac{4}{3}\big)\right)\)
\(\displaystyle z=\sqrt2(\cos\tfrac{3\pi}{4}+i\sin\tfrac{3\pi}{4})\)
\(\displaystyle z=5\!\left(\cos\!\big(-\pi+\tan^{-1}\!\tfrac{3}{4}\big)+i\sin\!\big(-\pi+\tan^{-1}\!\tfrac{3}{4}\big)\right)\)
\(\displaystyle z=5(\cos\tfrac{\pi}{2}+i\sin\tfrac{\pi}{2})\)
\(\displaystyle z=6(\cos\pi+i\sin\pi)\)
\(\displaystyle z=4(\cos(-\tfrac{\pi}{3})+i\sin(-\tfrac{\pi}{3}))\)
\(\displaystyle z=2(\cos\tfrac{\pi}{6}+i\sin\tfrac{\pi}{6})\)
\(\displaystyle z=2(\cos\tfrac{5\pi}{6}+i\sin\tfrac{5\pi}{6})\)
\(\displaystyle z=2(\cos(-\tfrac{2\pi}{3})+i\sin(-\tfrac{2\pi}{3}))\)
\(\displaystyle z=4\sqrt2(\cos(-\tfrac{\pi}{4})+i\sin(-\tfrac{\pi}{4}))\)
\(\displaystyle z=2(\cos(-\tfrac{\pi}{2})+i\sin(-\tfrac{\pi}{2}))\)
\(\displaystyle z=7(\cos0+i\sin0)\)
\(\displaystyle z=10(\cos\tfrac{2\pi}{3}+i\sin\tfrac{2\pi}{3})\)
\(\displaystyle z=2\sqrt2(\cos\tfrac{\pi}{4}+i\sin\tfrac{\pi}{4})\)
\(\displaystyle z=3\sqrt2(\cos(-\tfrac{3\pi}{4})+i\sin(-\tfrac{3\pi}{4}))\)
Polar already: \(\displaystyle z = 6(\cos\tfrac{\pi}{6}+i\sin\tfrac{\pi}{6})\).
\(\displaystyle z=2(\cos(-\tfrac{\pi}{3})+i\sin(-\tfrac{\pi}{3}))\)
\(\displaystyle z=4(\cos\pi+i\sin\pi)\)
8. Products and quotients in polar form
Multiplying and dividing in polar formMultiplication: Multiply the module and add the arguments
Division: Divide the module and subtract the arguments
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9. De Moivre's Theorem
Raising a number to a power - expanding |
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Express in polar form and expand |
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10. Applications of de Moivre's Theorem
Raising a number to a power |
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Proving trigonometric identities using de Moivre's Theorem |
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Finding roots |
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Solving Complex Equations:
Real parts = Real parts Imaginary parts = Imaginary parts |
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Transformations in the Complex Plane
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Complex conjugate roots
Solving quadratic and cubic equations with imaginary roots. |
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Writing a complex number in Polar Form
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Multiplying and dividing in Polar Form and using deMoivre's Theorem to expand a complex number to a power.
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Using deMoivre's Theorem to find roots of a Complex Equation.
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Using deMoivre's Theorem to prove Trigonometric identities. This question also requires use of the Binomial Theorem.
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The Syllabus
Abraham de Moivre (1667–1754)
Abraham de Moivre was a French mathematician best known for De Moivre’s Theorem, which links complex numbers and trigonometry through powers of \(\cos\theta + i\sin\theta\). Forced to leave France, he lived most of his life in London and became associated with Newton and Halley. His work in probability and complex numbers remains foundational in modern mathematics.
De Moivre’s Theorem
Induction proof for positive integers \(n\)Theorem. For \(n \ge 1\) and real \(\theta\),
\[ (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta). \]
Base case \(n = 1\). We have \((\cos\theta + i\sin\theta)^1 = \cos\theta + i\sin\theta = \cos(1\theta) + i\sin(1\theta)\), so the result holds for \(n=1\).
Induction hypothesis. Assume for some integer \(k \ge 1\) that
\[ (\cos\theta + i\sin\theta)^k = \cos(k\theta) + i\sin(k\theta). \tag{1} \]
Induction step. Then
\[ (\cos\theta + i\sin\theta)^{k+1} = (\cos\theta + i\sin\theta)(\cos(k\theta) + i\sin(k\theta)). \]
Expanding and using \(i^2=-1\),
\[ \begin{aligned} (\cos\theta + i\sin\theta)^{k+1} &= \cos\theta\cos(k\theta) - \sin\theta\sin(k\theta) \\ &\quad {}+ i\bigl(\cos\theta\sin(k\theta) + \sin\theta\cos(k\theta)\bigr). \end{aligned} \]
By the angle–addition formulae, \(\cos(a+b)=\cos a\cos b-\sin a\sin b\) and \(\sin(a+b)=\sin a\cos b+\cos a\sin b\), with \(a=\theta\), \(b=k\theta\), this is
\[ \cos((k+1)\theta) + i\sin((k+1)\theta). \]
Conclusion. The theorem holds for \(n=1\), and if it holds for \(n=k\) it holds for \(n=k+1\). By induction, De Moivre’s theorem is true for all integers \(n \ge 1\).
