3.2 Polarform

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{{Vald flik|[[3.2 Polär form|Teori]]}}
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{{Vald flik|[[3.2 Polär form|Theory]]}}
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{{Ej vald flik|[[3.2 Övningar|Övningar]]}}
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{{Ej vald flik|[[3.2 Övningar|Exercises]]}}
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'''Innehåll:'''
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'''Content:'''
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* Det komplexa talplanet
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* The complex plane
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* Addition och subtraktion i talplanet
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* Addition and subtraction in the complex plane
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* Belopp och argument
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* Modulus and argument
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* Polär form
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* Polar form
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* Multiplikation och division i polär form
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* Multiplication and division in polar form
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* Multiplikation med ''i'' i talplanet
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* Multiplication with ''i'' in the complex plane
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'''Lärandemål:'''
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'''Learning outcomes:'''
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Efter detta avsnitt ska du ha lärt dig att:
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After this section, you will have learnt:
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* Ha geometrisk förståelse för de komplexa talen och räkneoperationerna i talplanet.
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* Kunna omvandla komplexa tal mellan formen ''a'' + ''ib'' och polär form.
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*A geometric understanding of complex numbers and their arithmetic operations in the plane.
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* To be able to convert the complex number between the form ''a'' + ''ib'' and polar form.
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== Det komplexa talplanet ==
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== The complex plane ==
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Eftersom ett komplext tal <math>z=a+bi</math> består av en realdel <math>a</math> och en imaginärdel <math>b</math>, så kan <math>z</math> betraktas som ett ordnat talpar <math>(a,b)</math> och tolkas som en punkt i ett koordinatsystem.
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As a complex number <math>z=a+bi</math> consists of a real part and <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.
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Man bildar därför ett koordinatsystem genom att ställa en imaginär axel (en tallinje med enheten <math>i</math>) vinkelrät mot en reell axel (den reella tallinjen). Vi kan nu beskriva varje komplext tal med en punkt i detta koordinatsystem, och varje punkt beskriver ett unikt komplext tal.
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Denna geometriska tolkning av de komplexa talen kallas det ''komplexa talplanet''.
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This geometric interpretation of the complex numbers is called the ''complex plane''.
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''Anm:'' De reella talen, dvs. alla komplexa tal med imaginärdel 0, ligger alltså längs den reella axeln. Man kan därför se utvidgningen av talsystemet från <math>\mathbb{R}</math> (de reella talen) till <math>\mathbb{C}</math> (de komplexa talen) som att tillföra en ny dimension till den redan fyllda tallinjen.
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''Note:'' The real numbers, that is all complex numbers with imaginary part 0, lie along the real axis. One can therefore regards 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 .
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Addition av komplexa tal får helt naturligt en enkel tolkning i det komplexa talplanet och sker geometriskt på samma sätt som vid addition av vektorer. Subtraktion kan ses som addition av motsvarande negativa tal, dvs. <math>z-w=z+(-w)</math>.
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Addition of complex numbers has a quite natural and simple interpretation in the complex plane and is geometrically the same method as vector addition. Subtraktion can be seen as the addition of the corresponding negative numbers, that is <math>z-w=z+(-w)</math>.
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| valign="top" |<small>Geometriskt fås talet ''z''&nbsp;+&nbsp;''w'' genom att ett tänkt linjesegment från 0 till ''w'' parallellförflyttas så att startpunkten i 0 hamnar i ''z''. Då kommer linjesegmentets slutpunkt ''w'' hamna i ''z''&nbsp;+&nbsp;''w''.</small>
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| 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>
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| valign="top" |<small>Subtraktionen ''z'' - ''w'' kan skrivas som ''z'' + (-''w'') och kan därför tolkas geometriskt som att ett tänkt linjesegment från 0 till -''w'' parallellförflyttas så att 0 hamnar i ''z''. Då hamnar segmentets slutpunkt -''w'' i ''z'' - ''w''.</small>
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| 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>
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<div class="exempel">
<div class="exempel">
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'''Exempel 1'''
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''' Example 1'''
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Givet <math>z=2+i</math> och <math>w=-3-i</math>. Markera <math>z</math>, <math>w</math>, <math>\overline{z}</math>, <math>\overline{z}-\overline{w}</math> och <math>z-w</math> i det komplexa talplanet.
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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.
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{| width="100%"
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| width="100%" |Vi har att
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| width="100%" |We have that
*<math>\overline{z}=2-i\,</math>,
*<math>\overline{z}=2-i\,</math>,
*<math>\overline{w}=-3+i\,</math>,
*<math>\overline{w}=-3+i\,</math>,
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Notera hur komplexkonjugerade tal är spegelsymmetriska i reella axeln.
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Note that complex conjugated numbers are mirror images in the real axis.
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</div>
</div>
<div class="exempel">
<div class="exempel">
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'''Exempel 2'''
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''' Example 2'''
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Markera i det komplexa talplanet alla tal <math>z</math> som uppfyller följande villkor:
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Indicate in the complex plane all numbers <math>z</math> which meet the following conditions:
<ol type="a">
<ol type="a">
<li><math>\mathop{\rm Re} z \ge 3\,</math>,</li>
<li><math>\mathop{\rm Re} z \ge 3\,</math>,</li>
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</ol>
</ol>
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Den första olikheten definierar området i figuren till vänster nedan och den andra olikheten området i figuren till höger nedan.
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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.
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||{{:3.2 - Figur - Området -1 mindre än Im z ≤ 2}}
||{{:3.2 - Figur - Området -1 mindre än Im z ≤ 2}}
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| valign="top" |<small>Alla tal som uppfyller Re&nbsp;''z''&nbsp;≥&nbsp;3 har en realdel som är större än eller lika med&nbsp;3. Dessa tal bildar det färgade halvplanet i figuren.</small>
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| 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>
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| valign="top" |<small>Tal som uppfyller -1&nbsp;<&nbsp;Im&nbsp;''z''&nbsp;≤&nbsp;2 har en imaginärdel som är mellan&nbsp;-1 och&nbsp;2. Dessa tal ligger därför inom det bandformade område som markerats i figuren. Den undre horisontella linjen är streckad och det betyder att punkter på den linjen inte tillhör det färgade området.</small>
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| 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>
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</div>
</div>
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== Absolutbelopp ==
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== Absolute value ==
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De reella talen går att ordna i storleksordning, dvs. vi kan avgöra om ett reellt tal är större än ett annat; ju längre till höger på den reella tallinjen desto större är talet.
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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.
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För de komplexa talen saknar man denna möjlighet. Vi kan inte utan vidare avgöra vilket tal som är störst av t.ex. <math>z=1-i</math> och <math>w=-1+i</math> . Med hjälp av begreppet ''absolutbelopp'' kan vi dock definiera ett mått på storleken av ett komplext tal.
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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.
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För ett komplext tal <math>z=a+ib</math> definieras absolutbeloppet <math>|\,z\,|</math> som <br\><br\>
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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>
<div class="regel">{{Fristående formel||<math>|\,z\,|=\sqrt{a^2+b^2}\,\mbox{.}</math>}}</div>
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Vi ser att <math>|\,z\,|</math> är ett reellt tal och att <math>|\,z\,|\ge 0</math>. För reella tal är <math>b = 0</math> och då gäller att <math>|\,z\,|=\sqrt{a^2}=|\,a\,|</math>, vilket överensstämmer med den vanliga definitionen för absolutbelopp av reella tal. Geometriskt är absolutbeloppet avståndet från talet <math>z=a+ib</math> (punkten <math>(a, b)</math>) till <math>z = 0</math> (origo), enligt Pythagoras sats.
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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 Pythagoras theorem.
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== Avstånd mellan komplexa tal ==
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== Distance between complex numbers ==
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Med hjälp av formeln för avstånd mellan punkter i ett koordinatsystem får man också en viktig och användbar tolkning av absolutbelopp. Avståndet <math>s</math> mellan två komplexa tal <math>z=a+ib</math> och <math>w=c+id</math> (se fig.) kan med hjälp av avståndsformeln skrivas
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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>
<div class="regel">{{Fristående formel||<math>s=\sqrt{(a-c)^2+(b-d)^2}\,\mbox{.}</math>}}</div>
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Eftersom <math>z-w=(a-c)+i(b-d)</math>, så får man att
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Since <math>z-w=(a-c)+i(b-d)</math>, one gets
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<center><math>|\,z-w\,|=\sqrt{(a-c)^2+(b-d)^2}={}</math>avståndet mellan talen <math>z</math> och <math>w</math>.</center>
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<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">
<div class="exempel">
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'''Exempel 3'''
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''' Example 3'''
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Markera följande talmängder i det komplexa talplanet:
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Indicate the following sets in the complex plane:
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<br/>
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Ekvationen beskriver alla tal vars avstånd till origo är 2. Dessa tal bildar i det komplexa talplanet en cirkel med radien 2 och medelpunkt i origo.</li>
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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>
</ol>
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<br/>
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Denna ekvation uppfylls av alla tal vars avstånd till talet 2 är 1, dvs. en cirkel med radien 1 och medelpunkt i <math>z = 2</math>.</li>
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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>
</ol>
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Vänsterledet kan skrivas <math>|\,z-(-2+i)\,|</math>, vilket innebär alla tal på avståndet <math>{}\le 2</math> från talet <math>-2+i</math>, dvs. en cirkelskiva med radien 2 och medelpunkt i <math>-2+i</math>.</li>
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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>
</ol>
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Mängden ges av alla tal vars avstånd till <math>z=2+3i</math> är mellan <math>\frac{1}{2}</math> och <math>1</math>.</li>
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The set given 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>
</ol>
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<div class="exempel">
<div class="exempel">
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'''Exempel 4'''
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''' Example 4'''
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Markera i det komplexa talplanet alla tal <math>z</math> som uppfyller villkoren
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Indicate in the complex plane all numbers <math>z</math> satisfying the following
Zeile 200: Zeile 197:
<br/>
<br/>
<br/>
<br/>
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Den första olikheten ger punkterna på och innanför cirkeln med radie 3 och medelpunkt i <math>2i</math>. Den andra olikheten ger ett vertikalt band av punkter med realdel mellan 1 och 2. Det område som uppfyller de båda olikheterna ges av de punkter som samtidigt ligger inom cirkeln och bandet.
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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>
</li>
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<br/>
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<br/>
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Ekvationen kan skrivas <math>|\,z-(-1)\,|=|\,z-2\,|</math>. Man ser då att <math>z</math> ska ligga på samma avstånd från <math>-1</math> som från <math>2</math>. Detta villkor uppfylls av alla tal <math>z</math> som har realdel <math>1/2</math>.
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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>
</li>
</ol>
</ol>
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||{{:3.2 - Figur - Området ∣z + 1∣ = ∣z - 2∣}}
||{{:3.2 - Figur - Området ∣z + 1∣ = ∣z - 2∣}}
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||<small>Det färgade området består av de punkter som uppfyller olikheterna |''z''&nbsp;- 2i|&nbsp;≤&nbsp;3 och 1&nbsp;≤ Re&nbsp;''z''&nbsp;≤&nbsp;2.</small>
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||<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>
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||<small>De punkter som uppfyller likheten |''z''&nbsp;+ 1|&nbsp;= |''z''&nbsp;- 2| ligger på linjen med realdel lika med 1/2.</small>
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||<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>
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== Polär form ==
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== Polar form ==
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I stället för att ange ett komplext tal <math>z=x+iy</math> i dess rektangulära koordinater <math>(x,y)</math> kan man använda polära koordinater. Detta innebär att man anger talets läge i det komplexa talplanet genom dess avstånd, <math>r</math>, till origo, samt den vinkel, <math>\alpha</math>, som bildas mellan den positiva realaxeln och sträckan från origo till talet (se figuren).
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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).
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Eftersom <math>\,\cos\alpha = x/r\,</math> och <math>\,\sin\alpha = y/r\,</math> så är <math>\,x = r\cos\alpha\,</math> och <math>\,y= r\sin\alpha</math>. Talet <math>z=x+iy</math> kan därför skrivas som
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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>
<div class="regel">{{Fristående formel||<math>z=r\cos\alpha + i\,r\sin\alpha = r(\cos\alpha + i\,\sin\alpha)\,\mbox{,}</math>}}</div>
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vilket kallas den ''polära formen'' av ett komplext tal <math>z</math>. Vinkeln <math>\alpha</math> kallas ''argumentet'' för <math>z</math> och skrivs
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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>
<div class="regel">{{Fristående formel||<math>\alpha=\arg\,z\,\mbox{.}</math>}}</div>
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Vinkeln <math>\alpha</math> kan t.ex. bestämmas genom att lösa ekvationen <math>\tan\alpha=y/x</math>. Denna ekvation har dock flera lösningar, varför man måste se till att man väljer den lösning <math>\alpha</math> som gör att <math>z= r(\cos\alpha + i\sin\alpha)</math> hamnar i rätt kvadrant.
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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.
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Argumentet till ett komplext tal är inte heller unikt bestämt eftersom vinklar som skiljer sig åt med <math>2\pi</math> anger samma riktning i det komplexa talplanet. Normalt brukar man dock ange argumentet som en vinkel mellan 0 och <math>2\pi</math> eller mellan <math>-\pi</math> och <math>\pi</math>.
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The argument for a complex number is not uniquely determined because angles that differ by <math>2\pi</math> aindicate 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>.
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Det reella talet <math>r</math>, avståndet till origo, känner vi redan som beloppet av <math>z</math>,
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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="regel">{{Fristående formel||<math>r=\sqrt{x^2+y^2}=|\,z\,|\,\mbox{.}</math>}}</div>
<div class="exempel">
<div class="exempel">
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'''Exempel 5'''
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''' Example 5'''
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Skriv följande komplexa tal i polär form:
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Write the following complex numbers in polar form:
<ol type="a">
<ol type="a">
<li><math>\,\,-3</math>
<li><math>\,\,-3</math>
<br/>
<br/>
<br/>
<br/>
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Vi har att <math>|\,-3\,|=3</math> och <math>\arg (-3)=\pi</math>, vilket betyder att <math>\ -3=3(\cos\pi+i\,\sin\pi)</math>.
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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>
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<br/>
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Vi har att <math>|\,i\,|=1</math> och <math>\arg i = \pi/2</math> så i polär form är <math>\ i=\cos(\pi/2)+i\,\sin(\pi/2)\,</math>.
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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>
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Formeln för beloppet av ett komplext tal ger att <math>|\,1-i\,|=\sqrt{1^2+(-1)^2}=\sqrt{2}</math>. Det komplexa talet ligger i den fjärde kvadranten och bildar vinkeln <math>\pi/4</math> med den positiva reella axeln, vilket ger att <math>\arg (1-i)=2\pi-\pi/4=7\pi/4</math>. Alltså är <math>\ 1-i=\sqrt{2}\,\bigl(\cos(7\pi/4)+i\sin(7\pi/4)\,\bigr)</math>.
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The formula for a the absolut 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 angl <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>
Zeile 275: Zeile 272:
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Beloppet är enklast att räkna ut
+
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>}}
{{Fristående formel||<math>|\,2\sqrt{3}+2i\,|=\sqrt{(2\sqrt{3}\,)^2+2^2}=\sqrt{16}=4\,\mbox{.}</math>}}
-
Om vi kallar argumentet för <math>\alpha</math> så uppfyller det sambandet
+
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>}}
{{Fristående formel||<math>\tan\alpha=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}}</math>}}
-
och eftersom talet ligger i den första kvadranten (positiv real- och imaginärdel) så är <math>\alpha=\pi/6</math> och vi har att
+
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>}}
{{Fristående formel||<math>2\sqrt{3}+2i=4\bigl(\cos\frac{\pi}{6}+i\,\sin\frac{\pi}{6}\bigr)\,\mbox{.}</math>}}
</li>
</li>
Zeile 289: Zeile 286:
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== Multiplikation och division i polär form ==
+
== Multiplication and division of polar forms ==
-
Den stora fördelen med att ha komplexa tal skrivna i polär form är att multiplikation och division då blir väldigt enkla att utföra. För godtyckliga komplexa tal <math>z=|\,z\,|\,(\cos\alpha+i\sin\alpha)</math> och <math>w=|\,w\,|\,(\cos\beta+i\sin\beta)</math> kan man genom de trigonometriska additionsformlerna visa att
+
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">
<div class="regel">
Zeile 297: Zeile 294:
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Vid multiplikation av komplexa tal ''multipliceras'' alltså beloppen, medan argumenten ''adderas''. Vid division av komplexa tal ''divideras'' beloppen och argumenten ''subtraheras''. Detta kan kortfattat skrivas:
+
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">
<div class="regel">
-
{{Fristående formel||<math>|\,z\cdot w\,|=|\,z\,|\cdot |\,w\,|\quad \mbox{och}\quad \arg(z\cdot w)=\arg\,z + \arg\,w\,\mbox{,}</math>}}
+
{{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{ och}\quad \arg\Bigl(\frac{z}{w}\Bigr)=\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>
</div>
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I det komplexa talplanet innebär alltså en multiplikation av <math>z</math> med <math>w</math> att <math>z</math> förlängs med faktorn <math>|\,w\,|</math> och roteras moturs med vinkeln <math>\arg\,w</math>.
+
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>.
Zeile 315: Zeile 312:
<div class="exempel">
<div class="exempel">
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'''Exempel 6'''
+
''' Example 6'''
-
Beräkna följande uttryck och genom att skriva om på polär form:
+
Simplify the following expressions by writing them in polar form:
<ol type="a">
<ol type="a">
<li><math>\Bigl(\frac{1}{\sqrt2} -\frac{i}{\sqrt2}\Bigr) \Big/
<li><math>\Bigl(\frac{1}{\sqrt2} -\frac{i}{\sqrt2}\Bigr) \Big/
Zeile 324: Zeile 321:
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<br/>
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Vi skriver täljaren och nämnaren i polär form
+
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>}}
{{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>}}
-
och då följer att
+
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>}}
{{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>
</li>
Zeile 333: Zeile 330:
<br/>
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Faktorerna i uttrycket skriver vi i polär form
+
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>}}
{{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>}}
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Genom att utföra multiplikationen i polär form får vi att
+
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>}}
{{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>
</li>
Zeile 342: Zeile 339:
<div class="exempel">
<div class="exempel">
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'''Exempel 7'''
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''' Example 7'''
<ol type="a">
<ol type="a">
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<li>Beräkna <math>iz</math> och <math>\frac{z}{i}</math> om <math>\ z=2\Bigl(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\Bigr)</math>. Svara på polär form.
+
<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.
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Eftersom <math>\ i=1\cdot \left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)\ </math> så är
+
Since <math>\ i=1\cdot \left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)\ </math> så är
{{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>}}
{{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>
</li>
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<li> Beräkna <math>iz</math> och <math>\frac{z}{i}</math> om <math>\ z=3\left(\cos\frac{7\pi}{4}+i\sin\frac{7\pi}{4}\right)\,</math>. Svara på polär form.
+
<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.
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Använder vi den polära formen av <math>i</math> så fås att
+
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>}}
{{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>
</li>
</ol>
</ol>
-
Vi ser här att multiplikation med ''i'' innebär en rotation <math>\pi/2</math> moturs, medan division med ''i'' medför en rotation <math>\pi/2</math> medurs.
+
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"
{| width="80%" align="center"
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||{{:3.2 - Figur - Komplexa talplanet med z, iz och z/i är markerade, där arg z = 7π/4}}
||{{:3.2 - Figur - Komplexa talplanet med z, iz och z/i är markerade, där arg z = 7π/4}}
|-
|-
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||<small>De komplexa talen ''z'', ''iz'' och ''z''/''i'' när |''z''|&nbsp;=&nbsp;2 och arg&nbsp;''z''&nbsp;= π/6.</small>
+
||<small>Complex numbers ''z'', ''iz'' and ''z''/''i'' when |''z''|&nbsp;=&nbsp;2 and arg&nbsp;''z''&nbsp;= π/6.</small>
||
||
-
||<small>De komplexa talen ''z'', ''iz'' och ''z''/''i'' när |''z''|&nbsp;=&nbsp;3 och arg&nbsp;''z''&nbsp;= 7π/4.</small>
+
||<small>Complex numbers ''z'', ''iz'' and ''z''/''i'' when |''z''|&nbsp;=&nbsp;3 and arg&nbsp;''z''&nbsp;= 7π/4.</small>
|}
|}
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Version vom 18:37, 23. Jul. 2008

 
  1. REDIRECT Template:Gewählter Tab
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Content:

  • 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

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 \displaystyle z=a+bi consists of a real part and \displaystyle a and an imaginary part \displaystyle b, one can consider \displaystyle z to be an ordered pair of numbers \displaystyle (a,b) 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 \displaystyle i) 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.


3.2 - Figur - Komplexa talplanet


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 regards the extension of the number system from \displaystyle \mathbb{R} (the real numbers) to \displaystyle \mathbb{C} (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. Subtraktion can be seen as the addition of the corresponding negative numbers, that is \displaystyle z-w=z+(-w).

3.2 - Figur - Addition av komplexa tal 3.2 - Figur - Subtraktion av komplexa tal
Geometrically the number z + 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 + w. 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.

Example 1


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

We have that
  • \displaystyle \overline{z}=2-i\,,
  • \displaystyle \overline{w}=-3+i\,,
  • \displaystyle z-w=2+i-(-3-i)
    \displaystyle \phantom{z-w}{}=5+2i\,,
  • \displaystyle \overline{z} -\overline{w} = 2-i -(-3+i)
    \displaystyle \phantom{\overline{z} -\overline{w}}{}=5-2i\quad ({}=\overline{z-w})\,.
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.

Example 2


Indicate in the complex plane all numbers \displaystyle z which meet the following conditions:

  1. \displaystyle \mathop{\rm Re} z \ge 3\,,
  2. \displaystyle -1 < \mathop{\rm Im} z \le 2\,.

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.


3.2 - Figur - Området Re z ≥ 3 3.2 - Figur - Området -1 mindre än Im z ≤ 2
All the numbers that satisfy Re z ≥ 3 have a real part that is greater than or equal to 3. These figures form the shaded semi-plane in the figure. Numbers that satisfy -1 < Im z ≤ 2 have an imaginary part that is between -1 and 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.


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. \displaystyle z=1-i and \displaystyle w=-1+i . 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 \displaystyle z=a+ib the absolute value \displaystyle |\,z\,| is defined as

  1. REDIRECT Template:Abgesetzte Formel

We see that \displaystyle |\,z\,| is a real number, and that \displaystyle |\,z\,|\ge 0. For a real number \displaystyle b = 0 and then \displaystyle |\,z\,|=\sqrt{a^2}=|\,a\,|, 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 \displaystyle z=a+ib (the point \displaystyle (a, b)) to \displaystyle z = 0 (origin), according to Pythagoras theorem.


3.2 - Figur - Beloppet av z


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 \displaystyle s between the two complex numbers \displaystyle z=a+ib and \displaystyle w=c+id (see fig.) can with the help of the Formula for distance be written as

  1. REDIRECT Template:Abgesetzte Formel
3.2 - Figur - Avstånd mellan z och w


Since \displaystyle z-w=(a-c)+i(b-d), one gets

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


Example 3


Indicate the following sets in the complex plane:

  1. \displaystyle \,\, |\,z\,|=2

    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.
3.2 - Figur - Cirkeln ∣z∣ = 2
  1. \displaystyle \,\, |\,z-2\,|=1

    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 \displaystyle z = 2.
 3.2 - Figur - Cirkeln ∣z - 2∣ = 1
  1. \displaystyle \,\, |\,z+2-i\,|\le 2

    The left-hand side can be written \displaystyle |\,z-(-2+i)\,|, which means all the numbers at a distance \displaystyle {}\le 2 from the number \displaystyle -2+i, that is a circular disc a with a radius of 2 and its centre at \displaystyle -2+i.
 3.2 - Figur - Cirkelskivan ∣z + 2 - i∣ ≤ 2
  1. \displaystyle \,\, \frac{1}{2}\le |\,z-(2+3i)\,|\le 1

    The set given is given by any number whose distance from \displaystyle z=2+3i is between \displaystyle \frac{1}{2} and \displaystyle 1.
 3.2 - Figur - Cirkelringen 1/2 ≤ ∣z - (2 + 3i)∣ ≤ 1

Example 4


Indicate in the complex plane all numbers \displaystyle z satisfying the following


  1. \displaystyle \, \left\{ \eqalign{&|\,z-2i\,|\le 3\cr &1\le\mathop{\rm Re} z\le 2}\right.

    The first inequality gives the points on and inside a circle with radius 3 and center at \displaystyle 2i. 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.


  2. \displaystyle \, |\,z+1\,|=|\,z-2\,|

    The equation can be written as \displaystyle |\,z-(-1)\,|=|\,z-2\,|. This shows then that \displaystyle z should be at an equal distance from \displaystyle -1 and \displaystyle 2. This condition is met by all the numbers \displaystyle z that have a real part \displaystyle 1/2.
3.2 - Figur - Området ∣z - 2i∣ ≤ 3 och 1 ≤ Re z ≤ 2 3.2 - Figur - Området ∣z + 1∣ = ∣z - 2∣
The shaded region consists of the points that satisfy the inequalities |z - 2i| ≤ 3 and 1 ≤ Re z ≤ 2. The points that satisfy the equation |z + 1| = |z - 2| lie on the line with real part equal to 1/2.


Polar form

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


3.2 - Figur - Polär form av z


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

  1. REDIRECT Template:Abgesetzte Formel

which is called the polar form of a complex number \displaystyle z. The angle \displaystyle \alpha is called the argument of \displaystyle z and is written

  1. REDIRECT Template:Abgesetzte Formel

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

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


The real number \displaystyle r, the distance to the origin as we have already seen, is the absolute value of \displaystyle z,

  1. REDIRECT Template:Abgesetzte Formel

Example 5


Write the following complex numbers in polar form:

  1. \displaystyle \,\,-3

    We have that\displaystyle |\,-3\,|=3 and \displaystyle \arg (-3)=\pi, which means that \displaystyle \ -3=3(\cos\pi+i\,\sin\pi).
  2. \displaystyle \,i

    We have that \displaystyle |\,i\,|=1 and \displaystyle \arg i = \pi/2 which in polar form is \displaystyle \ i=\cos(\pi/2)+i\,\sin(\pi/2)\,.
  3. \displaystyle \,1-i

    The formula for a the absolut value of a complex number gives \displaystyle |\,1-i\,|=\sqrt{1^2+(-1)^2}=\sqrt{2}. The complex number lies in the fourth quadrant and has an angl \displaystyle \pi/4 with the positive real axis, which gives \displaystyle \arg (1-i)=2\pi-\pi/4=7\pi/4. Thus \displaystyle \ 1-i=\sqrt{2}\,\bigl(\cos(7\pi/4)+i\sin(7\pi/4)\,\bigr).
  4. \displaystyle \,2\sqrt{3}+2i

    The absolute value is the easiest to calculate
    1. REDIRECT Template:Abgesetzte Formel
    If we call the argument \displaystyle \alpha then it satisfies the relationship
    1. REDIRECT Template:Abgesetzte Formel
    and since the number is in the first quadrant (positive real and imaginary parts) one gets \displaystyle \alpha=\pi/6 and we have that
    1. REDIRECT Template:Abgesetzte Formel
3.2 - Figur - Komplexa talplanet med -3, i, 1 - i och 2√3 + 2i markerade


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 \displaystyle z=|\,z\,|\,(\cos\alpha+i\sin\alpha) and \displaystyle w=|\,w\,|\,(\cos\beta+i\sin\beta), it can be shown using the trigonometric formulas for addition that

  1. REDIRECT Template:Abgesetzte Formel

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:

  1. REDIRECT Template:Abgesetzte Formel
  2. REDIRECT Template:Abgesetzte Formel

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


3.2 - Figur - Komplexa tal z och w med argument α och β 3.2 - Figur - Komplexa produkten zw med argument α + β


Example 6


Simplify the following expressions by writing them in polar form:

  1. \displaystyle \Bigl(\frac{1}{\sqrt2} -\frac{i}{\sqrt2}\Bigr) \Big/ \Bigl( -\frac{1}{\sqrt2} +\frac{i}{\sqrt2}\Bigr)

    We write the numerator and denominator in polar form
    1. REDIRECT Template:Abgesetzte Formel
    and it follows that
    1. REDIRECT Template:Abgesetzte Formel


  2. \displaystyle (-2-2i)(1+i)

    The factors in the expression are written in polar form
    1. REDIRECT Template:Abgesetzte Formel
    Multiplication in polar form, gives
    1. REDIRECT Template:Abgesetzte Formel

Example 7


  1. Simplify \displaystyle iz and \displaystyle \frac{z}{i} if \displaystyle \ z=2\Bigl(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\Bigr). Answer in polar form.

    Since \displaystyle \ i=1\cdot \left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)\ så är
    1. REDIRECT Template:Abgesetzte Formel


  2. Simplify \displaystyle iz and \displaystyle \frac{z}{i} if \displaystyle \ z=3\left(\cos\frac{7\pi}{4}+i\sin\frac{7\pi}{4}\right)\,. Answer in polar form.

    Rewriting \displaystyle i in polar form gives
    1. REDIRECT Template:Abgesetzte Formel

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

3.2 - Figur - Komplexa talplanet med z, iz och z/i är markerade, där arg z = π/6 3.2 - Figur - Komplexa talplanet med z, iz och z/i är markerade, där arg z = 7π/4
Complex numbers z, iz and z/i when |z| = 2 and arg z = π/6. Complex numbers z, iz and z/i when |z| = 3 and arg z = 7π/4.
Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/3.2_Polarform“