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.
Key Concepts
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
Complex numbers - Cartesian, Polar and Euler Form 1/1
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