Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Simplify
a) \(\sqrt{32} \)
b) \(\sqrt{108}\)
c) \(\frac{1}{\sqrt{2}}\)
d) \(\frac{1+\sqrt{3}}{2+\sqrt{3}}\)
Solutions
Get Ready
Questions
Solve
a) \(\frac{x}{5}=3\)
b) \(\frac{5}{x}=3\)
c) \(x+3=5\)
d) \(x+5=3\)
e) \(x^{2}=4\)
f) \(x^{2}=2\)
g) \(x^{2}-5=3\)
h) \(x^{2}+5=3\)
Solutions
Notes
We define a new number$$i=\sqrt{-1}$$Whilst we call it imaginary, it is a number we have constructed to solve a particular type of equation.
Originally this number was introduced by Girolamo Cardano in his book Ars Magna (1545) as a way to reach a real solution for cubic equations. It was a step on the way to reach a solution that made more sense, but he did not consider \(\sqrt{-1}\) to be a number in the way we do today.
But we can treat \(i\) like any other surd, and the laws of surds apply equally to imaginary numbers.
Examples and Your Turns
Example
Write \(\sqrt{-4}\) in terms of \(i\).
-
$$\sqrt{-4}=\sqrt{4}\sqrt{-1}=2i$$
Your Turn
Write each of these in terms of \(i\):
a) \(\sqrt{-16}\)
b) \(\sqrt{-81}\)
c) \(\sqrt{-\frac{9}{4}}\)
d) \(\sqrt{-7}\)
e) \(\sqrt{-12}\)
-
a) \(\sqrt{-16}=4i\)
b) \(\sqrt{-81}=9i\)
c) \(\sqrt{-\frac{9}{4}}=\frac{3}{2}i\)
d) \(\sqrt{-7}=\sqrt{7}i\)
e) \(\sqrt{-12}=2\sqrt{3}i\)
Notes
A complex number is a number that can be written in the form \(a+bi\) where \(a,b\in\mathbb{R}\).
The set of all complex numbers is denoted with the symbol \(\mathbb{C}\).
We usually use the letter \(z\) to represent a complex number, much as convention is to use \(x\) for real numbers and \(n\) for natural numbers.
If \(z=3+2i\) then we define:
the real part of \(z\) as \(Re(z)=3\)
the imaginary part of \(z\) as \(Im(z)=2\)
A complex number \(z=a+bi\) where \(a=0\) is known as purely imaginary, and one where \(b=0\) is known as a real number.
The conjugate of the complex number \(z=a+bi\) is$$z^{*}=a-bi$$where the real part does not change but he sign of the imaginary part swaps.
Examples and Your Turns
Example
For the complex number \(z=3+2i\), state:
a) the real part
b) the imaginary part
c) the conjugate
-
a) \(Re\left(z\right)=3\)
b) \(Im\left(z\right)=2\)
c) \(z^{*}=3-2i\)
Your Turn
For the complex number \(z=5i-4\), state:
a) the real part
b) the imaginary part
c) the conjugate
-
a) \(Re\left(z\right)=-4\)
b) \(Im\left(z\right)=5\)
c) \(z^{*}=-5i-4\)
Your Turn
For the complex number \(z=-\frac{2}{3}+\sqrt{3}i\), state:
a) the real part
b) the imaginary part
c) the conjugate
-
a) \(Re\left(z\right)=-\frac{2}{3}\)
b) \(Im\left(z\right)=\sqrt{3}\)
c) \(z^{*}=-\frac{2}{3}-\sqrt{3}i\)
Your Turn
For the complex number \(z=\frac{\sqrt[3]{11}i-\sqrt{11}}{\pi}\), state:
a) the real part
b) the imaginary part
c) the conjugate
-
a) \(Re\left(z\right)=-\frac{\sqrt{11}}{\pi}\)
b) \(Im\left(z\right)=\frac{\sqrt[3]{11}}{\pi}\)
c) \(z^{*}=\frac{-\sqrt[3]{11}i-\sqrt{11}}{\pi}\)
Notes
Complex numbers are numbers, and so we should be able to perform numerical calculations with them.
We can do this following the standard rules of algebra, such as collecting like terms and expanding brackets.
Complex numbers form a closed set. That means that, under all operations, the answer is another complex number.
This is not the case for other sets of numbers. For example:
The natural numbers are not closed under subtraction, as \(5-7=-2\notin \mathbb{N}\)
The integers are not closed under division, as \(5\div 3=\frac{5}{3}\notin \mathbb{Z}\)
The rationals are not closed under square rooting as \(\sqrt{\frac{1}{2}}=\frac{\sqrt{2}}{2}\notin\mathbb{Q}\)
The reals are not closed under square rooting as \(\sqrt{-1}=i\notin\mathbb{R}\)
Examples and Your Turns
Example
If \(z=3+i\) and \(w=5-2i\), calculate:
a) \(z+w\)
b) \(z-w\)
c) \(zw\)
d) \(\frac{w}{2}\)
-
a) $$\begin{align}z+w&=\left(3+i\right)+\left(5-2i\right)\\&=8-i\end{align}$$
b) $$\begin{align}z-w&=\left(3+i\right)-\left(5-2i\right)\\&=-2+3i\end{align}$$
c) $$\begin{align}zw&=\left(3+i\right)\left(5-2i\right)\\&=15+5i-6i-2i^{2}\\&=15-i-2\left(-1\right)=17-i\end{align}$$
d) $$\begin{align}\frac{w}{2}&=\frac{5-2i}{2}\\&=\frac{5}{2}-i\end{align}$$
Your Turn
If \(z=3+2i\) and \(w=4-i\), calculate:
a) \(z+w\)
b) \(z-w\)
c) \(zw\)
d) \(\frac{w}{2}\)
-
a) $$\begin{align}z+w&=\left(3+2i\right)+\left(4-i\right)\\&=7+i\end{align}$$
b) $$\begin{align}z-w&=\left(3+2i\right)-\left(4-i\right)\\&=-1+3i\end{align}$$
c) $$\begin{align}zw&=\left(3+2i\right)\left(4-i\right)\\&=12+8i-3i-2i^{2}\\&=12+5i-2\left(-1\right)=14+5i\end{align}$$
d) $$\begin{align}\frac{w}{2}&=\frac{4-i}{2}\\&=2-\frac{1}{2}i\end{align}$$
Notes
To perform a divsion by a complex number, we use an analogous method to rationalising the denominator for surds.
We multiply the fraction by \(1=\frac{z^{*}}{z^{*}}\) so the value is not changed but the form is.
Examples and Your Turns
Example
Write \(\frac{3+2i}{5-i}\) in the form \(x+yi\).
-
$$\begin{align}\frac{3+2i}{5-i}&=\frac{3+2i}{5-i}\times \frac{5+i}{5+i}\\&=\frac{15+3i+10i+2i^{2}}{25-i^{2}}\\&=\frac{13+13i}{26}\\&=\frac{1}{2}+\frac{1}{2}i\end{align}$$
Your Turn
Write \(\frac{4-i}{2+i}\) in the form \(x+yi\).
-
$$\begin{align}\frac{4-i}{2+i}&=\frac{4-i}{2+i}\times \frac{2-i}{2-i}\\&=\frac{8+4i-2i-i^{2}}{4-i^{2}}\\&=\frac{9+2i}{5}\\&=\frac{9}{5}+\frac{2}{5}i\end{align}$$
Your Turn
Given that \(z_{1}=5+5i\), \(z_{2}=1+2i\) and \(z_{3}=3-2i\), calculate:$$\frac{z_{1}}{z_{2}}$$
Your Turn
at \(z_{1}=5+5i\), \(z_{2}=1+2i\) and \(z_{3}=3-2i\), calculate:$$\frac{z_{1}^{2}}{z_{2}\times z_{3}^{*}}$$
Notes
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, $$z_{1}=z_{2} \iff Re\left(z_{1}\right)=Re\left(z_{2}\right)\text{ and }Im\left(z_{1}\right)=Im\left(z_{2}\right)$$We can use this to solve equations involving complex numbers, by equating the real and imaginary parts.
Examples and Your Turns
Example
Solve the equation for \(z\in\mathbb{C}\)$$2z+8=4i$$
Your Turn
Solve the equation for \(z\in\mathbb{C}\)$$5z-3+2i=2z-1$$
Your Turn
Solve the equation for \(z\in\mathbb{C}\)$$z\left(2-i\right)=-i$$
Your Turn
Find the values of the real numbers \(x\) and \(y\) such that$$\left(x+2i\right)\left(1-i\right)=5+yi$$
Example
Solve \(z^{2}=i\).
Your Turn
Solve \(z^{2}=\frac{9}{4}+3i\)
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.