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{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
| style="border-bottom:1px solid #797979" width="5px" | &nbsp;
{{Vald flik|[[3.2 Polar form|Theory]]}}
{{Ej vald flik|[[3.2 Övningar|Exercises]]}}
| style="border-bottom:1px solid #797979"  width="100%"| &nbsp;
|}

{{Info|
'''Contents:'''
* The complex plane 
* Addition and subtraction in the complex plane 
*  Modulus  and argument
* Polar form 
* Multiplication and division in polar form 
* Multiplication with ''i'' in the complex plane 
}}

{{Info|
'''Learning outcomes:'''

After this section, you will have learnt: 

*A geometric understanding of complex numbers and  their arithmetic operations in the  plane. 
* To be able to convert the complex number between the form  ''a'' + ''ib'' and polar form. 
}}

== The complex plane  ==

As a complex number <math>z=a+bi</math>  consists of a real part <math>a</math> and  an imaginary part <math>b</math>,  one can consider <math>z</math> to be  an ordered pair of numbers  <math>(a,b)</math> and interpreted as a point in a coordinate system. We thus construct a coordinate system by drawing an imaginary axis ( a number axis having a unit <math>i</math>) perpendicular to a real axis (the real-number axis). We can now designate each complex number as a point in this coordinate system, and conversely each point defines a unique complex number. 


<center>{{:3.2 - Figur - Komplexa talplanet}}</center>


This geometric interpretation of the complex numbers is called the  ''complex plane''.


''Note:'' The real numbers, that is all complex numbers with imaginary part 0, lie along the real axis. One can therefore regard the extension of the number system from <math>\mathbb{R}</math> (the real numbers) to <math>\mathbb{C}</math>  (the complex numbers) to mean that one  adjoins an  extra dimension to the completely filled real-number axis . 


Addition of complex numbers has a quite natural and  simple interpretation in the complex plane and is geometrically  the same method as vector addition. Subtraction can be seen as the addition of the corresponding negative number, that is <math>z-w=z+(-w)</math>.

{| width="100%" align="center"
| width="10%" |
| width="35%" align="center" |{{:3.2 - Figur - Addition av komplexa tal}}
| width="10%" |
| width="35%" align="center" |{{:3.2 - Figur - Subtraktion av komplexa tal}}
| width="10%" |
|-
||
| valign="top" |<small> Geometrically the number ''z''&nbsp;+&nbsp;''w'' is obtained by considering a  hypothetical  line segment from  0 to ''w'' which is parallel-displaced so that its initial  point at 0 is moved to z. Then this  line segments terminal point w lands at the point ''z''&nbsp;+&nbsp;''w''.</small>
||
| valign="top" |<small>The subtraction ''z'' - ''w''  can be written as ''z'' + (-''w'') and can therefore be interpreted geometrically as a hypothetical line segment from 0 to -''w''  is parallel-displaced so that its initial  point at  0 is moved to ''z''. Then this  line segments terminal point -''w'' lands at the point ''z'' - ''w''.</small>
||
|}

<div class="exempel">
''' Example 1'''


Given <math>z=2+i</math> and <math>w=-3-i</math>. Indicate <math>z</math>, <math>w</math>, <math>\overline{z}</math>, <math>\overline{z}-\overline{w}</math> and <math>z-w</math> in the complex plane. 

{| width="100%"
| width="100%" |We have that 
*<math>\overline{z}=2-i\,</math>,
*<math>\overline{w}=-3+i\,</math>,
*<math>z-w=2+i-(-3-i)</math><br/><math>\phantom{z-w}{}=5+2i\,</math>,
*<math>\overline{z} -\overline{w} = 2-i -(-3+i)</math><br/><math>\phantom{\overline{z} -\overline{w}}{}=5-2i\quad ({}=\overline{z-w})\,</math>.
||{{:3.2 - Figur - Komplexa talplanet med z, w, z*, z - w och z* - w* markerade}}
|}

Note that  complex conjugated numbers are  mirror images  in the real axis. 
</div>

<div class="exempel">
''' Example 2'''


Indicate in the complex plane all numbers <math>z</math> which meet the following conditions: 
<ol type="a">
<li><math>\mathop{\rm Re} z \ge 3\,</math>,</li>
<li><math> -1 < \mathop{\rm Im} z \le 2\,</math>.</li>
</ol>

The first inequality defines the region in the figure on the left below, and the second inequality defines the region in the figure on the  right below. 


{| align="center" width="80%"
||{{:3.2 - Figur - Området Re z ≥ 3}}
| width="5%" |
||{{:3.2 - Figur - Området -1 mindre än Im z ≤ 2}}
|-
| valign="top" |<small> All the numbers that satisfy Re&nbsp;''z''&nbsp;≥&nbsp;3 have a real part that is greater than or equal to&nbsp;3. These figures form the shaded semi-plane in the figure. </small>
||
| valign="top" |<small>Numbers that satisfy -1&nbsp;<&nbsp;Im&nbsp;''z''&nbsp;≤&nbsp;2 have an imaginary part that is between&nbsp;-1 and&nbsp;2. These numbers are therefore in the ribbon-like region marked in the figure. The lower horizontal line is dotted and that means that points on that line do not belong to the coloured region. </small>
|}
</div>


== Absolute value  ==

The real numbers can be  arranged in order of magnitude, that is we can determine whether a real number is greater than another, the further to the right  on the real number line the greater the number. 

For the complex numbers this is not possibile. We cannot decide which  is the larger  of  e.g. <math>z=1-i</math> and <math>w=-1+i</math> . With the help of the concept of ''absolute value'' however, we can define a measure of the size of a complex number. 


For a complex number <math>z=a+ib</math> the  absolute value <math>|\,z\,|</math> is defined as  <br\><br\>

<div class="regel">{{Fristående formel||<math>|\,z\,|=\sqrt{a^2+b^2}\,\mbox{.}</math>}}</div>

We see that <math>|\,z\,|</math> is a real number, and that <math>|\,z\,|\ge 0</math>. For a real number  <math>b = 0</math> and then <math>|\,z\,|=\sqrt{a^2}=|\,a\,|</math>, which is consistent with the usual definition of an absolute value of a real number. Geometrically the absolute value is the distance from the number  <math>z=a+ib</math> (the point <math>(a, b)</math>) to <math>z = 0</math> (origin), according to the Pythagorean theorem. 


<center>{{:3.2 - Figur - Beloppet av z}}</center>


== Distance between complex numbers  ==

With the help of the formula for the distance between points in a coordinate system one can obtain an important and useful interpretation of the absolute value. The distance  <math>s</math> between the two complex numbers  <math>z=a+ib</math> and <math>w=c+id</math> (see fig.)  can with the help of the formula for distance  be written as

<div class="regel">{{Fristående formel||<math>s=\sqrt{(a-c)^2+(b-d)^2}\,\mbox{.}</math>}}</div>

<center>{{:3.2 - Figur - Avstånd mellan z och w}}</center>


Since <math>z-w=(a-c)+i(b-d)</math>,  one gets

<center><math>|\,z-w\,|=\sqrt{(a-c)^2+(b-d)^2}={}</math> distance between the numbers <math>z</math> and <math>w</math>.</center>


<div class="exempel"> 
''' Example 3'''


Indicate the following sets in the complex plane: 

{| width="100%"
| width="95%" |
<ol type="a">
<li><math>\,\, |\,z\,|=2</math>
<br/>
<br/>
The equation describes all numbers whose distance to the origin is 2. These numbers describe in the complex plane a circle with radius 2 and its centre at the origin. </li>
</ol>
| width="5%" |
||{{:3.2 - Figur - Cirkeln ∣z∣ = 2}}
|}

{| width="100%"
| width="95%" |
<ol type="a" start="2">
<li><math>\,\, |\,z-2\,|=1</math>
<br/>
<br/>
This equation is satisfied by all the numbers, whose distance from the  number 2 is equal to 1, i.e. a circle of radius 1 and with its centre at <math>z = 2</math>.</li>
</ol>
| width="5%" |
||{{:3.2 - Figur - Cirkeln ∣z - 2∣ = 1}}
|}

{| width="100%"
| width="95%" |
<ol type="a" start="3">
<li><math>\,\, |\,z+2-i\,|\le 2</math>
<br/>
<br/>
The left-hand side can be written  <math>|\,z-(-2+i)\,|</math>, which means all the numbers at a distance <math>{}\le 2</math> from the number <math>-2+i</math>, that is a circular disc a with a radius of 2 and its centre at <math>-2+i</math>.</li>
</ol>
| width="5%" |
||{{:3.2 - Figur - Cirkelskivan ∣z + 2 - i∣ ≤ 2}}
|}

{| width="100%"
| width="95%" |
<ol type="a" start="4">
<li><math>\,\, \frac{1}{2}\le |\,z-(2+3i)\,|\le 1</math>
<br/>
<br/>
The set is given by any number whose distance from <math>z=2+3i</math> is between  <math>\frac{1}{2}</math> and <math>1</math>.</li>
</ol>
| width="5%" |
||{{:3.2 - Figur - Cirkelringen 1/2 ≤ ∣z - (2 + 3i)∣ ≤ 1}}
|}

</div>

<div class="exempel"> 
''' Example 4'''


Indicate in the complex plane all numbers <math>z</math> satisfying the following conditions:


<ol type="a">
<li><math>\, \left\{ \eqalign{&|\,z-2i\,|\le 3\cr &1\le\mathop{\rm Re} z\le 2}\right.</math>
<br/>
<br/>
The first inequality gives the points on and inside a circle with  radius 3 and  center at  <math>2i</math>. The second inequality is a vertical strip of points with their real part between 1 and 2. The area satisfying both  inequalities is given by  the points which lie both within the circle and within the strip. 
</li>
<br/>
<li><math>\, |\,z+1\,|=|\,z-2\,|</math>
<br/>
<br/>
The equation can be written as <math>|\,z-(-1)\,|=|\,z-2\,|</math>. This shows then that  <math>z</math> should be at an equal distance from  <math>-1</math>  and <math>2</math>. This condition is met by all the numbers <math>z</math> that have a real part  <math>1/2</math>.
</li>
</ol>

{| align="center" width="80%"
||{{:3.2 - Figur - Området ∣z - 2i∣ ≤ 3 och 1 ≤ Re z ≤ 2}}
| width="5%" |
||{{:3.2 - Figur - Området ∣z + 1∣ = ∣z - 2∣}}
|-
||<small> The shaded region consists of the points that satisfy the inequalities |''z''&nbsp;- 2i|&nbsp;≤&nbsp;3 and 1&nbsp;≤ Re&nbsp;''z''&nbsp;≤&nbsp;2.</small>
||
||<small>The points that satisfy the equation |''z''&nbsp;+ 1|&nbsp;= |''z''&nbsp;- 2| lie on the line with real part equal to 1/2.</small>
|}

</div>


== Polar form  ==

Instead of representing a complex number <math>z=x+iy</math> by its rectangular coordinates <math>(x,y)</math> one can  use polar coordinates. This means that one represents a numbers location in the complex plane by its distance <math>r</math> to the origin, and the angle <math>\alpha</math>, made by the positive real-line axis and the line from the origin to  the number (see the figure). 


<center>{{:3.2 - Figur - Polär form av z}}</center>


Since <math>\,\cos\alpha = x/r\,</math> and <math>\,\sin\alpha = y/r\,</math> then <math>\,x = r\cos\alpha\,</math> and <math>\,y= r\sin\alpha</math>. The number <math>z=x+iy</math> can be written as 

<div class="regel">{{Fristående formel||<math>z=r\cos\alpha + i\,r\sin\alpha = r(\cos\alpha + i\,\sin\alpha)\,\mbox{,}</math>}}</div>

which is called the ''polar form'' of a complex  number  <math>z</math>. The angle <math>\alpha</math>  is called the ''argument'' of <math>z</math> and is written

<div class="regel">{{Fristående formel||<math>\alpha=\arg\,z\,\mbox{.}</math>}}</div>

The angle <math>\alpha</math>, for example, can  be determined by solving the equation  <math>\tan\alpha=y/x</math>. This equation, however, has a number of solutions, so we must ensure that we choose the solution  <math>\alpha</math> that allows <math>z= r(\cos\alpha + i\sin\alpha)</math> to  end up in the correct quadrant. 

The argument for a complex number is not uniquely determined because  angles that  differ by <math>2\pi</math> indicate the same direction in the complex plane. Normally, one uses for the argument the angle between 0 and  <math>2\pi</math> or between <math>-\pi</math> and <math>\pi</math>.


The real number <math>r</math>, the distance to the origin   as we have already seen, is the absolute value of <math>z</math>,

<div class="regel">{{Fristående formel||<math>r=\sqrt{x^2+y^2}=|\,z\,|\,\mbox{.}</math>}}</div>

<div class="exempel">
''' Example 5'''


Write the following complex numbers in polar form: 
<ol type="a">
<li><math>\,\,-3</math>
<br/>
<br/>
We have that <math>|\,-3\,|=3</math> and <math>\arg (-3)=\pi</math>, which means that  <math>\ -3=3(\cos\pi+i\,\sin\pi)</math>.
</li>

<li><math>\,i</math>
<br/>
<br/>
We have that <math>|\,i\,|=1</math> and <math>\arg i = \pi/2</math> which   in polar form is <math>\ i=\cos(\pi/2)+i\,\sin(\pi/2)\,</math>.
</li>

<li><math>\,1-i</math>
<br/>
<br/>
The formula for a the absolute value of a complex number gives <math>|\,1-i\,|=\sqrt{1^2+(-1)^2}=\sqrt{2}</math>. The complex number lies in the fourth quadrant and has an angle <math>\pi/4</math> with the positive real axis, which gives <math>\arg (1-i)=2\pi-\pi/4=7\pi/4</math>. Thus <math>\ 1-i=\sqrt{2}\,\bigl(\cos(7\pi/4)+i\sin(7\pi/4)\,\bigr)</math>.
</li>

<li><math>\,2\sqrt{3}+2i</math>
<br/>
<br/>
The absolute value is the easiest to calculate
{{Fristående formel||<math>|\,2\sqrt{3}+2i\,|=\sqrt{(2\sqrt{3}\,)^2+2^2}=\sqrt{16}=4\,\mbox{.}</math>}}
If we call the argument <math>\alpha</math> then it satisfies the relationship 
{{Fristående formel||<math>\tan\alpha=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}}</math>}}
and since the number is in the first quadrant (positive real and imaginary parts) one gets <math>\alpha=\pi/6</math> and we have  that
{{Fristående formel||<math>2\sqrt{3}+2i=4\bigl(\cos\frac{\pi}{6}+i\,\sin\frac{\pi}{6}\bigr)\,\mbox{.}</math>}}
</li>
</ol>

<center>{{:3.2 - Figur - Komplexa talplanet med -3, i, 1 - i och 2√3 + 2i markerade}}</center>

</div>


== Multiplication and division of polar forms ==

The big advantage of having the complex numbers written in polar form is that multiplication and division then becomes very easy to perform. For arbitrary complex numbers <math>z=|\,z\,|\,(\cos\alpha+i\sin\alpha)</math> and <math>w=|\,w\,|\,(\cos\beta+i\sin\beta)</math>,  it can be shown using the trigonometric formulas for addition  that 

<div class="regel">
{{Fristående formel||<math>\begin{align*}z\cdot w&=|\,z\,|\,|\,w\,|\,\bigl(\cos(\alpha+\beta)+i\,\sin(\alpha+\beta)\bigr)\,\mbox{,}\\[4pt] \frac{z}{w}&=\frac{|z|}{|w|}\bigl(\cos(\alpha-\beta)+i\,\sin(\alpha-\beta)\bigr)\,\mbox{.}\end{align*}</math>}}
</div>

When multiplying  complex numbers, the absolute values ''are multiplied'', while the arguments ''are  added''. For  division of complex numbers, absolute values ''are divided''  and the arguments ''subtracted''.  This can be summarised as:

<div class="regel">
{{Fristående formel||<math>|\,z\cdot w\,|=|\,z\,|\cdot |\,w\,|\quad \mbox{and}\quad  \arg(z\cdot w)=\arg\,z + \arg\,w\,\mbox{,}</math>}}
{{Fristående formel||<math>\Bigl|\,\frac{z}{w}\,\Bigr|=\frac{|\,z\,|}{|\,w\,|}\quad\quad\quad\; \mbox{ and}\quad  \arg\Bigl(\frac{z}{w}\Bigr)=\arg \,z - \arg\,w\,\mbox{.}</math>}}
</div>

In the complex plane this means  that multiplication of <math>z</math> with <math>w</math> causes <math>z</math> to be stretched by a factor  <math>|\,w\,|</math> and rotated counterclockwise by an angle <math>\arg\,w</math>.


{| width="80%" align="center"
||{{:3.2 - Figur - Komplexa tal z och w med argument α och β}}
| width="5%" |
||{{:3.2 - Figur - Komplexa produkten zw med argument α + β}} 
|}


<div class="exempel">
''' Example 6'''


Simplify the following expressions  by writing them in polar form: 
<ol type="a">
  <li><math>\Bigl(\frac{1}{\sqrt2} -\frac{i}{\sqrt2}\Bigr) \Big/
    \Bigl( -\frac{1}{\sqrt2} +\frac{i}{\sqrt2}\Bigr)</math>
<br/>
<br/>
We write the numerator and denominator in polar form 
{{Fristående formel||<math>\begin{align*}\frac{1}{\sqrt2} -\frac{i}{\sqrt2} &= 1\cdot\Bigl(\cos\frac{7\pi}{4}+i\,\sin\frac{7\pi}{4}\Bigr)\\[4pt] -\frac{1}{\sqrt2} +\frac{i}{\sqrt2} &= 1\cdot\Bigl(\cos\frac{3\pi}{4}+i\,\sin\frac{3\pi}{4}\Bigr)\end{align*}</math>}}
and it follows that 
{{Fristående formel||<math>\begin{align*}&\Bigl(\frac{1}{\sqrt2} -\frac{i}{\sqrt2}\Bigr) \Big/ \Bigl(-\frac{1}{\sqrt2} +\frac{i}{\sqrt2}\Bigr) = \smash{\frac{\cos\dfrac{7\pi}{4}+i\,\sin\dfrac{7\pi}{4}\vphantom{\Biggl(}}{\cos\dfrac{3\pi}{4}+i\,\sin\dfrac{3\pi}{4}\vphantom{\Biggl)}}}\\[16pt] &\qquad\quad{}= \cos\Bigl(\frac{7\pi}{4}-\frac{3\pi}{4}\Bigl)+i\,\sin\Bigl(\frac{7\pi}{4}-\frac{3\pi}{4}\Bigr)= \cos\pi+i\,\sin\pi=-1\,\mbox{.}\end{align*}</math>}}
</li>
<br/>
<li> <math>(-2-2i)(1+i)</math>
<br/>
<br/>
The factors in the expression are written in polar form
{{Fristående formel||<math>\begin{align*}-2-2i&=\sqrt8\Bigl(\cos\frac{5\pi}{4}+i\,\sin\frac{5\pi}{4}\Bigr)\,\mbox{,}\\[4pt] 1+i&=\sqrt2\Bigl(\cos\frac{\pi}{4}+i\,\sin\frac{\pi}{4}\Bigr)\,\mbox{.}\end{align*}</math>}}
Multiplication in polar form, gives 
{{Fristående formel||<math>\begin{align*}(-2-2i)(1+i)&=\sqrt8 \cdot \sqrt2\,\Bigl(\cos\Bigl(\frac{5\pi}{4}+\frac{\pi}{4}\Bigr)+i\,\sin\Bigl(\frac{5\pi}{4}+\frac{\pi}{4}\Bigr)\Bigr)\\[4pt] &=4\Bigl(\cos\frac{3\pi}{2}+i\,\sin\frac{3\pi}{2} \Bigr)=-4i\,\mbox{.}\end{align*}</math>}}
</li>
</ol>
</div>

<div class="exempel">
''' Example 7'''

 
<ol type="a">
<li> Simplify <math>iz</math> and <math>\frac{z}{i}</math> if  <math>\ z=2\Bigl(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\Bigr)</math>. Answer in polar form. 
<br/>
<br/>
Since <math>\ i=1\cdot \left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)\ </math> it follows that
{{Fristående formel||<math>\begin{align*} iz &= 2\Bigl(\cos\Bigl(\frac{\pi}{6}+\frac{\pi}{2}\Bigr)+i\,\sin\Bigl(\frac{\pi}{6}+\frac{\pi}{2}\Bigr)\,\Bigr)= 2\Bigl(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\Bigr)\,\mbox{,}\\[4pt] \frac{z}{i} &= 2\Bigl(\cos\Bigl(\frac{\pi}{6}-\frac{\pi}{2}\Bigr)+i\,\sin\Bigl(\frac{\pi}{6}-\frac{\pi}{2}\Bigr)\,\Bigr) = 2\Bigl(\cos\frac{-\pi}{3}+i\,\sin\frac{-\pi}{3}\Bigr)\,\mbox{.}\end{align*}</math>}}
</li>
<br/>
<li> Simplify <math>iz</math> and <math>\frac{z}{i}</math> if <math>\ z=3\left(\cos\frac{7\pi}{4}+i\sin\frac{7\pi}{4}\right)\,</math>.  Answer in polar form. 
<br/>
<br/>
Rewriting <math>i</math> in polar form gives
{{Fristående formel||<math>\begin{align*} iz &= 3\Bigl(\cos\Bigl(\frac{7\pi}{4}+\frac{\pi}{2}\Bigr)+i\,\sin\Bigl(\frac{7\pi}{4}+\frac{\pi}{2}\Bigr)\,\Bigr) = 3\Bigl(\cos\frac{9\pi}{4}+i\sin\frac{9\pi}{4}\Bigr)\\[4pt] &= 3\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)\,\mbox{,}\\[6pt] \frac{z}{i} &= 2\Bigl(\cos\Bigl(\frac{7\pi}{4}-\frac{\pi}{2}\Bigr)+i\,\sin\Bigl(\frac{7\pi}{4}-\frac{\pi}{2}\Bigr)\,\Bigr)= 2\Bigl(\cos\frac{5\pi}{4}+i\,\sin\frac{5\pi}{4}\Bigr)\,\mbox{.}\end{align*}</math>}}
</li>
</ol>

We  see that multiplying by ''i'' leads to  a counter-clockwise rotation  <math>\pi/2</math>,  while  division with ''i'' results in a clockwise rotation <math>\pi/2</math>.

{| width="80%" align="center"
||{{:3.2 - Figur - Komplexa talplanet med z, iz och z/i är markerade, där arg z = π/6}}
| width="5%" |
||{{:3.2 - Figur - Komplexa talplanet med z, iz och z/i är markerade, där arg z = 7π/4}}
|-
||<small>Complex numbers  ''z'', ''iz'' and ''z''/''i'' when |''z''|&nbsp;=&nbsp;2 and arg&nbsp;''z''&nbsp;= π/6.</small>
||
||<small>Complex numbers ''z'', ''iz'' and ''z''/''i'' when |''z''|&nbsp;=&nbsp;3 and arg&nbsp;''z''&nbsp;= 7π/4.</small>
|}

</div>
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