Complex numbers: Introduction to Complex numbers
The notion of complex numbers
We have already learned about imaginary numbers, including the imaginary unit #\ii#. We can multiply such numbers with real numbers, in such a way that the union of the real and imaginary numbers satisfies the first and fourth characteristic of complex numbers. But we cannot add real numbers and imaginary numbers yet. In order to be able to satisfy all of the four characteristics, we move on to geometry. In particular, we will use the flat plane to define complex numbers.
Complex number
We start with a coordinate system with perpendicular axes in the flat plane. In this coordinate system we call the horizontal axis (the #x#-axis) the real axis and the vertical axis (the #y#-axis) the imaginary axis.
Each point in the plane is defined relative to this coordinate system by two coordinates #a# and #b#, which are real. We call the point #\rv{a,b}# a complex number. Generally speaking, the point #\rv{a,b}# will be denoted in the standard notation #a+b\cdot\ii#: \[\rv{a,b}=a+b\cdot\ii\tiny.\] In here #a# and #b# are referred to as the real and imaginary part of the complex number.
If the complex number is not used as input in an answer field, we sometimes write #a+b\cdot\ii# simply without a multiplication sign between #b# and #\ii#, hence, as #a+b\ii#. We also name #\rv{a,b}# the Cartesian coordinates of the complex number.
- The points #\rv{a,0}# on the first axis are simply denoted by #a# and
- the points #\rv{0,b}# on the second axis by #b\cdot\ii# or sometimes just #b\ii#.
In particular, the one, #1#, is equal to the point #\rv{1,0}# and the imaginary unit, #\ii#, is equal to the point #\rv{0,1}#.
For a complex number, we often use the letter #z# or #w#. The set of complex numbers is represented by #\mathbb{C}#.
When we view the flat plane as the set of complex numbers, we often refer to it as the complex plane.
Geometrically we can see how the complex numbers are an extension of the real numbers #\mathbb R#: the regular number line is identified by the real axis in the flat plane, hence, the #x#-axis. The plane in turn is identified by #\mathbb C#.
The imaginary numbers, which are of the form #r\cdot\ii# with #r# being a real number, are identified by the imaginary axis, hence, the #y#-axis, in such a way that #r\cdot\ii# with #r\gt0# is above the #x#-axis.
All other complex numbers are a sum of a real and an imaginary number: #a+b\cdot\ii# corresponds with the point #\rv{a,b}# of the plane: \[\rv{a,b}=a\cdot \rv{1,0}+b\cdot\rv{0,1}=a\cdot 1+b\cdot\ii=a+b\cdot \ii\tiny.\]
From this follows that, for all real numbers #a#, #b#, #c#, and #d#:\[ a+b\cdot\ii=c+d\cdot\ii\phantom{xx}\text{ if and only if }\phantom{xx}a=c \text{ and }b=d\tiny.\]
Now that we have established what we understand as the set #\mathbb C#, we will equip #\mathbb{C}# with addition and multiplication. As mentioned in the third and fourth characteristic, such operations must coincide with the well-known operations on #\mathbb R# (the real axis), and be in accordance with #\ii^2=-1#.
Addition and subtraction of two complex numbers
Let #z_1 =a_1 + b_1\cdot \ii# and #z_2 =a_2 +b_2\cdot \ii# be two complex numbers, in which #a_1#, #a_2#, #b_1#, #b_2# are real numbers.
- The sum or the result of addition of #z_1# and #z_2# is #z_1+z_2 = (a_1+a_2)+(b_1+b_2)\cdot \ii#.
- The difference of #z_1# and #z_2# or the result of subtraction of #z_2# from #z_1# is #z_1-z_2 = (a_1-a_2)+(b_1-b_2)\cdot \ii#.
The complex number #-z_1# stands for #0-z_1#.
Addition and subtraction of complex numbers are the coordinatewise addition and subtraction in the plane, respectively. In other words, these are the well-known vector addition and vector subtraction. For example, #(1 +\ii) + (-2 + 4 \ii) = -1 + 5 \ii#.
Note that #-z_1=-a_1+(-b_1)\cdot\ii=-a_1-b_1\cdot\ii# and #z_1-z_1=0#.
The product of two complex numbers
We define the product or the result of multiplication of two complex numbers #z_1 = a_1 + b_1 \cdot\ii# and #z_2 =a_2 +b_2\cdot \ii# by
\[
z_1\cdot z_2 =(a_1\cdot a_2-b_1\cdot b_2)+(a_1\cdot b_2+a_2\cdot b_1)\cdot \ii\tiny.
\]
This seems a complicated formula, but it is actually easy to remember. Expand the product #(a_1+b_1\cdot\ii)\cdot (a_2+b_2\cdot\ii)#, using the rules of calculation for real numbers and the additional property #\ii^2=-1#; this results in the formula of the right-hand member. For example:
\[ \begin{array}{rclcl}
(1+\ii)\cdot (-2 + 4 \ii) &=& -2 + 4\ii + (-2)\cdot\ii + \ii \cdot(4\ii)&&\phantom{}\color{blue}{\text{expanding the brackets}}\\&=& -2 + 2\ii + 4\cdot(\ii)^2&&\phantom{}\color{blue}{\text{terms with }\ii\text{ taken together}}\\&=& -2 + 2\ii - 4&&\phantom{}\color{blue}{\ii^2=-1}\\&=&-6 + 2 \ii&&\phantom{}\color{blue}{\text{simplified}}\end {array}\]
The notation "#\cdot#", here used for the complex multiplication, can generate confusion with the dot product vectors of length two. The two give different results. In particular, the value of the dot product is always a real number. In this chapter we will therefore avoid the notation "#\cdot #" for the dot product as much as possible.
With this we achieved our goal with regard to addition and multiplication:
The algebraic structure of the complex numbers
For the addition and multiplication of complex numbers (and their interaction) the same rules of calculation apply as for the real numbers, with the additional feature #\ii^2=-1#.
Regarding the rules of calculation for addition and subtraction, think of the following three rules:
- The addition of two complex numbers is commutative: for all complex numbers #z# and #w# we have #z+w=w+z#.
- The addition is also associative: for all complex numbers #z_1, z_2, z_3# we have #(z_1+z_2)+z_3 = z_1 + (z_2+z_3)#.
- For any complex number #z# we have #0+z=z# and #z+(-z) = 0#.
By expansion, these rules are easy to verify, although sometimes it will take some time. The latter, for example, can be verified as follows:
\[ \begin{array}{rclcl}z+(-z)&=&a+b\cdot\ii+\left(-a+(-b)\cdot\ii\right)&&\phantom{xyzxyz}\color{blue}{\text{definition }-z}\\&=&\left(a+(-a)\right)+\left(b+(-b)\right)\cdot\ii&&\phantom{xyzxyz}\color{blue}{\text{added}}\\&=&0+0\cdot\ii&&\phantom{xyzxyz}\color{blue}{\text{simplified }}\\&=&0&&\end {array}\]
The fact that #\ii^2=-1# follows from the following calculation: \[ \begin{array}{rclcl}\ii^2&=&\ii\cdot \ii&\phantom{q}&\color{blue}{\text{definition square}}\\&=&(0+1\cdot\ii)\cdot(0+1\cdot\ii)&\phantom{q}&\color{blue}{\ii\text{ written in standard form}}\\&=&(0\cdot 0-1\cdot 1)+(0\cdot 1+1\cdot 0)\cdot\ii&\phantom{q}&\color{blue}{\text{definition complex multiplication}}\\&=&-1&\phantom{q}&\color{blue}{\text{simplified}}\end {array}\]
Besides commutativity #w\cdot z=z\cdot w# and associativity #\left((z_1\cdot z_2)\cdot z_3 = z_1 \cdot (z_2\cdot z_3)\right)# of the multiplication, the so called distributivity of the multiplication over the sum is an example of a rule which applies to all complex numbers #z_1, z_2, z_3# \[z_1 \cdot (z_2 + z_3) = z_1\cdot z_2 + z_1\cdot z_3\tiny.\]
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