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# Class Note for CSCI 124 at GW (6)

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Date Created: 02/07/15

Complex Numbers Notes for CSci 124 Poorvi L Vora September 20 2006 1 Motivation for the De nition of Complex Numbers Suppose you wished to be able to count objects You would start with the notion of one object and then add another and so on That is you would want a set of numbers that contains the number 1 and is closed with respect to addition This is the set of natural numbers N 1 2 3 Notice that N is also closed under both addition and multiplication That is m ENandy ENixy Nandmy EN However the converse is not true That is z E N and y E N does not imply that z 7 y E N Example 1 Which ofthe following result in natural numbers 5 7 3 10 7 20 7 7 l 29 7 30 5 7 3 2is anatural number 10 7 20 710 is not 77 1 6 is 29 7 30 71 is not If in addition to adding objects you wish to also take away objects from an existing set you need a set of numbers that includes N and is closed under subtraction The smallest such set is the set of integers I 72710 123 I is also said to be the closure of N under subtraction The integers allow us to add and subtract objects as well as to compare values 7 such as elevations and temperatures 7 with a predetermined zero value However just as N is not closed under subtraction I is not closed under division z E I and y E Iy 7 0 does not imply that g E I 5 25 710 Example 2 Which ofthe followmg result in integers 3 7 2 4 7 5 g g j 3 7 2 lis aninteger as are 4 7 5 71 2575 5 71 5 gis not an integer Complex N umbersVoraGWU 2 The closure of I under division except by zero is the set of rational numbers P Q E 1194761761750 The rational numbers allow us to share objects into equal parts such that each part is not necessarily a whole For example 3 apples among 2 children 11 Some properties of Q Now we see that Q satis es the following properties with respect to both addition and multiplication Let 0 represent either addition or multiplication Then 1 Closure under operation If x y E Q m o y E Q 2 Identity There is an element 5 called the identitysuch that z o e x V m For addition 5 O and for multiplication e 1 3 Inverse V x except z 0 when 0 is multiplication 3 Invm such that z o Invm e For 1 addition Invm 7x and for multiplication Invm m 4 Closure under inverse Q is closed under the inverse operation when 0 is addition and Q 0 is closed under the inverse operation when 0 is multiplication Here denotes set subtrac tion Q is however not closed with respect to square roots That is 2 y E Q does not imply that z E Q A physical implication of this is that all squares with rational values of area do not have rational values for side lengths For example the square root of 2 cannot be written in the form g for integers p and q Yet one can see that should lie between 1 the positive square root of 1 and 2 the positive square root of 4 and it physically represents the side of a square of area 2 Example 3 Which of the following equations have at least one rational solution 2 7 2 7 2 9 7 2 2 7 m 170m 7470m 7 70m 7 70 2 7 4 0 has rational solutions z 2 and z 72 2 7 0 has rational solutions z g and z f3 The other two equations do not have any rational solutions The set of numbers represented by the number line the real numbers R contains the square roots of all the natural and positive rational numbers Thus in particular it contains all solutions to equations of the form 2 7 a 0 where a E Q and a gt O R does not however contain any solutions to the equation 3 7 2 O or to 2 1 0 While the motivation for studying the roots of such equations is not as obvious as the motivation for studying say the rational numbers we will Complzx szmWmGWU 3 see that these mats are 1ndeed1mpunan when we study the maihemancs renurred fur prueessrng eudru end mdeu Example Wh1h quhe fulluwmg equanuns have 3112251 une real suluuun7 12 1 o 12 74 o All Except 12 12 chmplex Numbers 1 has been shewn Lhatthe set ufcumplEX numbers a z ry1 we 72 andzz 1 o Bantams the mats ufall pulynumals wnh real ur cumplEX cue immts z 15 the malpan ufcumplex numbere z ry and ythe mummyme They ere denuted we end 1mg respeeuve1y The number zyxsxmagnary whde z 15 real when 3 o the cumplex numberrs areal number hens 72 c c c 15 represented by the plane and the cumplEX number 1 zycumespunds re the pumt z y m Lh15 p1ene see Frgure1 1n Lh15 plane the number hne a eummun representahun uf 7a cunespunds re the z ems Frgure1 The cumplEX number m 51 En the cumplEX plane 2 Opera ons on Complex Numbers All uperanuns un cumplex numbers are as Lhuse un real numbers wnh the added eundmun that r2 1 Complex N umbersVoraGWU 4 1 Equality m1 iy1 x2 iyg m1 m2 and y1 yz 2 Addition x1 iy1 x2 iyg 1 2 iy1 yg 3 Subtraction m1 iy1 7 x2 iyg x1 7 2 iy1 7 yg 4 Multiplication m1 Hg1 X x2 Hg2 xlmg Hg1mg Haw2 i2y1y2 mlmg 7 y1y2 Z1902 961242 Division is slightly more complicated A small trick helps Notice that x 2 y2 is real In fact it is the magnitude ofz ty and z 7 lg is the complex conjugate ofz iy Now 951 191 901 i241 X 952 i242 901902 241 4241902 95132 4 7 t t 7 Z 902 W2 962 W2 X 962 242 90 24 90 24 The multiplicative inverse may be similarly determined Do in class Example 5 A Determine the following 1 23t574t 2 3t727t 3 576tx372t 4 15t 1 5 r1 6 g B Determine the roots of the quadratic equation 2 2x 2 O Answers A 1 23t574t25374t77t 2 3t727t 372t177112t 3 5762 x 3722 5 x3776 x 7276 x 35 x 72 37282 4 15t 1 5 r1 7t 6 31gtlt1 rf1gtlteel1X3 11146 Bz22z30m2i W71i t Complzx mbmmcwu 5 3 Polar Representation seme upa39anuns 77e nut as easytu an mthe omenan urArgandrepresmtanun ufcumplexnumba39s 1 e usmg than 7 and ycuurdmates 171s eenee msLeaAi tn represent the cumplex number 1n arms ufxts matny nnnphnee argument n7 huge as shan 1n the dugzm The magmde 15 the antenna ufthepmnmz y hem the ungn and the anglesthe angle made by the lmejmnmg the ungn tn the pumthth the 777m 1n the annealunkwlse aneennn 11m Lhamzautude and ithe angle than 7 17245 z vCUsi 3 mm The cumplEX numbEr 1s represented 177 we See Fxgure z Fxgure z The cumplex numbEr 1n 51 1n pular farm Example 5 a and the W177 representahun nfune cumplex numba39 1 7 h and the emenen representahun nfune cumplex numba39 Whuse magmde 15 2 and angle 30 e and Lhepulzrrepresentahuns ufl 7 1 r 7 1 7 1 7 7 n Fmdthepulzrrepresentanuns n 7 e 7 7 7 Answa39s a 7 112 12 77 ewe W40 A6 radme Henee 1 7 ns Dr g Alsu denuted e h w 7ng zaneg 725774S e ma e r r r 7 Complex N umbersVoraGWU 6 j 391 if 1 d 2516251 6 2516 2el 6 31 Multiplication Recall that 951 iy1 X 952 iyz 961962 241242 iy12 951242 If 1 iy1 r15i91and 2 iyg 7625192 then rlewl X r25 m1 iy1 X 902 2142 71720030100302 7 Sim91im92 ir1r2Sim9lCos02 00301527102 T1T2008ltt91 02i71725in01 92 i9192 T1T26 That is the product of two complex numbers results in a complex number with magnitude the prod uct of the two magnitudes and angle the sum of the two angles Example 7 Using the results of Example 6 nd the polar representation of the product of the two numbers 1 i and xg 1 What is the cartesian representation Answer From Example 6 we know that 1 H ed and xi i 251 Hence 1 x 3 2 ei x371i31 32 Complex Conjugation and Inverse Theorem The inverse ofa complex number rem is e 1 Proof Let 2 rem Denote its inverse by 2 r 5119 for some as yet undetermined r and 0 which we will now determine Then 2X27111Xeigtlt0 1 Recall the result proved earlier in class 715191 X 725192 T1T26ilt9192 Applying this result to the LHS of 1 we get 7 39 39 I z X z 1 r519 X 7519 rrel99 And the above is equal to the RHS of1 hence Mei99 1 X eiXO Complex N umbersVoraGWU 7 Equating magnitudes and angles of the complex numbers on the LHS and the RHS we get 1 TT 1 a r 7 7 and 00 00 70 This completes the proof Theorem IfX rem show that X re w Example 8 What is the polar representation of 2V3 2 Hence what is the polar representation of its inverse Its compleX conjugate What are their cartesian representations Answer r v 22 x 3 4 V16 4 2501710 i and point in the rst quadrant hence t9 g The polar representation of 2V3 2 is 451 The polar representation of its inverse is 57 and that of its conjugate 45 Their cartesian representations are g 7 g and 2V3 2 respectively 33 Powers Now that we can multiply two compleX numbers given their polar representations we can multiply a number with itself that is we can determine the square of a compleX number it is simply another compleX number whose magnitude is the square of the original magnitude and whose angle is twice the angle of the original Similarly we can compute any power of a compleX number by induction i9n Theorem re Tneme for n gt 0 n a natural number Proof by induction Base Case The proposition is trivially true for n0 Suppose the statement is true for all values of n g k 7 1 We now prove that this implies it is true for n k The proposition is true for n k 7 1 Hence Tei9k71 Tk7leik719 Teieyg Tk7leik719 X rem Tk saw where the last equality follows from the result that rlewl x 7625192 T1T26ilt9192 Hence the proposition is true for n k Hence the proposition is true for all nite natural numbers n This completes the proof Example 9 What is the value of 1 i4 1 ix36 Complex N umbersVoraGWU 8 Answer 1 Qelg polar representation 44 a 1 NW 24ei a 1ixi 251 HmQG 265i0 34 71 roots ofrew Let 2 r ele be an nth root of re Then 2 rewYL rem 7 r and 710 0 m27r 7 7 and m2lm07177n71 n 71 Example 10 What are the fth roots of complex number 1 ixg Answer 4 391 4 25 1 1 2V Qel 3 polar representatzon 1 39 7r 27f 2 2351E1 m 1 39 7r 1 397 quot 1 137r 1 394 quot 1 392 quot 2E51 72E51E Qggl QEElE QEElT 7 35 71 roots of unity the number 1 We can determine the nth roots of 1 as we would the nth roots of any other complex number Let 2 r ewl be an nth root of 1 Then 2 rewYL 1 1ei0 OJ 1 and 710 m27r 7 1 Complex N umbersVoraGWU 9 2 and ml m 01n71 n 7 0 27139 47139 n 7127r 7 7 n 7 n 739quot n Example 11 What are the eighth roots of unity Answer 28 1 1 X ei0 polar representation 2 1Xeixm1xeixm 37w 57r 37r 77r i1 11 l 17e47e27e47e e4 e2 e4 Note that among the eighth roots of unity are 17 i 717 7i 4 Regions 0n the Argand Diagram One can use the Argand diagram to denote values of 2 that satisfy given equalities and inequalities See Figures 3 and 4 Figure 3 should be clear The four inequalities in Figure 4 may be worked through as follows 1 lt Re4z lt 3 Suppose z z iy 1 lt Re4z 1lt Re4m 42y 7 1 lt 4x HMOJWWW gt 1 lt 7 z 4 1 lt Magnitude22 lt 2 Suppose z re 1 lt Magnitudezz 1 lt Magnitude ze e 7 1 lt r2 71lt3 town to 0 lt Argumen zs lt ml Complex N umbersVoraGWU Suppose z re 10 0ltArgument23 lt g a 0 lt Argument geige lt g 0lt30m27r lt g 77mg lt0 lt g7m2gm07172 0 lt 0 lt gm 0 27139 47139 7139 7139 0137777 0 7737 1 3 3 lt 2 2m 47139 27139 7139 0127 0 lt 7775gm2 Magnitudez 3 7 3239 1 Suppose z z iy Magnitudez 3 7 3i 1 Magnitudez 3 y 7 1 m32y732 121 is a circle with center at 737 2 Complzx mbuxVmGWU 1 1 ma gt 3 Magmaqu lt 2 7AA 0 lt Argumenzm lt M Rea lt 72 gure 3 Sums mequalmes Complzx mbuxVmGWU 1 z 1lt Magmaqu lt 2 radm D lt mummy lt W Mammaqu 1 gure 4 Sums mequalmes and equalmes

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