Complex Numbers - The Basics

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
    • Cartesian, \(z=a+bi\)
    • Polar, \(z=r(cos\theta+isin\theta)=rcis\theta\)
    • Euler, \(z=re^{i\theta}\)

Summary

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Test Yourself

Here is a quiz about Cartesian Form, Polar Form and Euler Form of a complex number


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Here is a quiz about the properties of of the modulus and arugment of a complex number


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Exam-style Questions

Question 1

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

 

Question 2

It is given that \(z_1=2+3i\) and \(z_2=4+ai\)

Find a if \(Im(z_1z_2^*)=0\)

Hint

Full Solution

Question 3

Let \(z=a+bi\)

Find a and b if \(z^2=\left|z\right|^2-4\)

Hint

Full Solution

 

Question 4

Show that \((1-\frac{1}{3} e^{2iθ})(1-\frac{1}{3} e^{-2iθ})=\frac{1}{9}(10-6\cos 2θ)\)

Hint

Full Solution

 

Question 5

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

 

Question 6

a) Given that \(z=a+bi\), show that \(z+\frac{1}{z}=2cos\theta\)

b) Hence, show that \(z^n+\frac{1}{z^n}=2cosn\theta\)

c) Use the Binomial expansion to expand \((z+{1\over z})^5\)

d) Hence, show that \(cos^5 θ={1 \over 16}cos⁡5θ+{5 \over 16} cos3θ+{5\over 8} cosθ\)

Hint

Full Solution

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