Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Given that \(z=1+2i\) and \(w=2+i\), calculate
a) \(z+w \)
b) \(zw\)
c) \(\frac{z}{w}\)
d) \(z^{*}\)
e) \(\frac{z^{*}}{w^{*}}\)
Solutions
Get Ready
Questions
Given the complex number \(z=a+bi\), determine each of these in the form \(x+yi\):
a) \(z+z^{*}\)
b) \(z-z^{*}\)
c) \(z\times z^{*}\)
d) \(\left(z^{*}\right)^{*}\)
e) \(\left(z^{*}\right)^{2}\)
f) \(\left(z^{2}\right)^{*}\)
Solutions
Notes
From the Get Ready we can see the following properties of conjugates.
For any complex number \(z=a+bi\):
\(\left(z^{*}\right)^{*}=z\)
\(z+z^{*}=2a\) which is real
\(z-z^{*}=2bi\) which is purely imaginary
\(z\times z^{*}=a^{2}+b^{2}\) which is real and known as the Sum of Two Squares
\(\left(z^{*}\right)^{n}=\left(z^{n}\right)^{*}\)
Below we shall prove some further properties of conjugates of two complex numbers.
Your Turns
Your Turn
Prove that $$\left(z_{1}+z_{2}\right)^{*}=z_{1}^{*}+z_{2}^{*}$$
Your Turn
Prove that $$\left(z_{1}-z_{2}\right)^{*}=z_{1}^{*}-z_{2}^{*}$$
Your Turn
Prove that $$\left(z_{1}\times z_{2}\right)^{*}=z_{1}^{*}\times z_{2}^{*}$$
Your Turn
Prove that $$\left(\frac{z_{1}}{ z_{2}}\right)^{*}=\frac{z_{1}^{*}}{z_{2}^{*}}$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.