This page will allow you to become confident in the basic principles of complex numbers. It is important to understand, and be able to use, the three different forms of a complex number: Cartesian, Polar and Euler. You will learn about the Argand diagram, how useful the complex conjugate is and also how to make good use of the properties of the modulus and argument of a complex number.
On this page, you should learn to
- Understand that the number \(i\), where \(i^2=-1\)
- Use the notation for
- real part, \(Re(z)\)
- imaginary part, \(Im(z)\)
- conjugate, \(z^*\)
- modulus, \(|z|\)
- argument, \(arg(z)\)
- Use the Argand diagram to represent the complex plane
- Use and convert between the three forms of a complex number
- Cartesian, \(z=a+bi\)
- Polar, \(z=r(cos\theta+isin\theta)=rcis\theta\)
- Euler, \(z=re^{i\theta}\)
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Here is a quiz about Cartesian Form, Polar Form and Euler Form of a complex number
START QUIZ!
Which of the following is the correct Cartesian Form for the complex number, z
z = \(2(cos\frac {2 \pi }{ 3 } +isin\frac {2 \pi }{ 3 } )\) is the correct answer in Polar Form
It must be written in the z = a + ib
Which of the following is the correct Polar Form for the complex number, z
|z| = \(\sqrt{(-3)^2+(-3)^2}=3\sqrt{2}\)
Arg(z) = \(-\frac{3\pi}{4}\)
Which of the following is the correct Euler Form for the complex number, z
Arg(z) = \(arctan(\frac{-2}{2\sqrt{3}})\)
|z| = \(\sqrt{(-2)^2+(2\sqrt{3})^2}\)
Which of the following Argand diagrams represents the complex number, z = \(5e^{i(-\frac{\pi}{2})}\)
A | B |
C | D |
Arg(z) = \(-\frac{\pi}{2}\)
Which of the following is true about the complex number z = \(-4\sqrt{3}-4i\)?
Find all correct answers
Remember that Arg(z) is the principal argument, \(-\pi
z* = \(-4\sqrt{3}+4i\)
Which of the following is true about the complex number z
\(z=2{ e }^{ -\frac { 3\pi }{ 4 } }\)
Find all of the correct answers
z* is the complex conjugate of z
Arg(z) represents the principal argument, therefore -\(\pi\)\(<\) Arg(z) \(\leqslant\) \(\pi\) , but as arg(z) is not the principal argument arg(z) =\(\frac{5\pi}{4}\)
z is a complex number such that
Arg(z) = \(-\frac{3\pi}{4}\)
|z| = \(5\sqrt{2}\)
If z = a + ib , work out a and b
a =
b =
z is a complex number such that
|z| = \(4\sqrt{3}\)
Re(z) = -6
0 < Arg(z) < \(\pi\)
If Arg(z) = \(\frac{a}{b}\pi\) , find a and b
a =
b =
\(arccos(\frac{-6}{4\sqrt{3}})=\frac{5}{6}\pi\)
z is a complex number such that
|z| = 10
Im(z) = -5
\(-\frac{\pi}{2}<Arg(z)<\frac{\pi}{2}\)
If Arg(z*) = \(\frac{a}{b}\pi\), find a and b
a =
b =
z is a complex number such that iz = -3 -4i
If z = a + ib , find a and b
a =
b =
Multiplying a complex number by i represents a 90° anticlockwise rotation.
Hence, to find z, we need to perform a 90° clockwise rotation on the complex number -3 -4i
Here is a quiz about the properties of of the modulus and arugment of a complex number
START QUIZ!
z is a complex number
|z| = \(\sqrt{3}\)
Work out |z6|
|z6| =
|z6| = \(|z|^6\)
z and w are complex numbers
|z| = \(\sqrt{2}\) and |w| = \(3\sqrt{2}\)
Work out |zw|
|zw| =
|zw| = |z|\(\times\)|w| = \(\sqrt{2}\times3\sqrt{2}=6\)
z and w are complex numbers
arg(z) = 80° and arg(w) = 120°
Work out Arg(zw) in degrees
Arg(zw) =
arg(zw) = arg(z)+arg(w) = 80° + 120° = 200°
Remember that the principal argument for -180°\(<\) Arg(zw) \(\leqslant\) 180°
Arg(zw) = -160°
z is a complex number
arg(z) = \(\frac{\pi}{3}\)
If arg(zn) = \(-\frac{\pi}{3}\) and n>0, what is the smallest possible value of n
n =
\(-\frac{\pi}{3}=\frac{5\pi}{3}\)
arg(zn) = 5arg(z)
z is a complex number
arg(z) = \(\frac{5\pi}{8}\)
If Re(zn) = 0 and n>0, find the smallest value of n
n =
arg(z²) = \(\frac{10\pi}{8}\)
arg(z3) = \(\frac{15\pi}{8}\)
arg(z4) = \(\frac{20\pi}{8}\) = \(\frac{5\pi}{2}\)
Which of the following is true about the complex numbers z and w
z = 3 + 3i and w = \(4(cos\frac { \pi }{ 3 } +isin\frac { \pi }{ 3 } )\)
Find all of the correct answers
arg(z) = \(\frac{\pi}{4}\)
arg(w) = \(\frac{\pi}{3}\)
arg(zw) = \(\frac{\pi}{4}+\frac{\pi}{3}\)
|z| = \(3\sqrt{2}\)
|w| = 4
|zw| = \(3\sqrt{2}\times4=12\sqrt{2}\)
z and w are complex numbers
z = \(2{ e }^{ i\frac { \pi }{ 2 } }\) and w = -1 - i
Arg(zw) = \(\frac{a}{b}\pi\)
Work out a and b
a =
b =
arg(z) = \(\frac{\pi}{2}\)
arg(w) = \(-\frac{3\pi}{4}\)
arg(zw) = \(\frac{\pi}{2}+(-\frac{3\pi}{4})=-\frac{\pi}{4}\)
z and w are complex numbers
z = -i and arg(w) = \(\frac{\pi}{3}\)
Arg(\(\frac{z}{w*}\)) = \(\frac{a}{b}\pi\)
Work out a and b
a =
b =
arg(z) = \(-\frac{\pi}{2}\)
arg(w*) = \(-\frac{\pi}{3}\)
arg(\(\frac{z}{w*}\)) = \(-\frac{\pi}{2}-(-\frac{\pi}{3})=-\frac{\pi}{6}\)
z is a complex number
Im(z5) = 0
Which of the following is definitely FALSE?
If Im(z5) = 0, then Arg(z5) = 0° or 180°
arg(z5) = ..., -360°, -180°, 0°,180°, 360°, ...
arg(z) = ..., -72°, -36°, 0°,36°, 72, ...
arg(z*) = ..., 72°, 36°, 0°,-36°, -72, ...
z and w are complex numbers
z = 2(cos20° + i sin20°)
w = 3ei(-30°)
\(Im(\frac { { z }^{ 3 } }{ { w }^{ n } } )=0\)
Given that n is positive, find the smallest possible value of n
n =
arg(w) = -30°
\(arg(\frac { { z }^{ 3 } }{ { w }^{ n } } )\) = 60°-(-30n°)
If \(Im(\frac { { z }^{ 3 } }{ { w }^{ n } } )=0\), then \(arg(\frac { { z }^{ 3 } }{ { w }^{ n } } )=\)..., -360°, -180°, 0°,180°, 360°, ...
If n=1, 60+30n = 90
If n=2, 60+30n = 120
If n=3, 60+30n = 150
If n=4, 60+30n = 180
Find the possible values of a, given that \(|\frac{z_1}{z_2}|=2\) and \(z_1=a+2i\) and \(z_2=1-2i\)
Hint
Full Solution
It is given that \(z_1=2+3i\) and \(z_2=4+ai\)
Find a if \(Im(z_1z_2^*)=0\)
Hint
Full Solution
Show that \((1-\frac{1}{3} e^{2iθ})(1-\frac{1}{3} e^{-2iθ})=\frac{1}{9}(10-6\cos 2θ)\)
Hint
Full Solution
Given that z is a complex number and \(\frac{3z-4}{5}=\frac{p-2i}{3-i}\), where \(p \in\mathbb{R} \)
a) Express z in the form \(a+bi\), \(a,b\in\mathbb{R}\)
Given that \(arg(z)=-\frac{\pi}{2}\)
b) Find the value of p
Hint
Full Solution
How much of Complex Numbers - The Basics have you understood?
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