1.1 The Algebra of Complex Numbers 3.Notice that and 1 retain theiridentity"properties as;that is 0+=z and 1.=z when z is complex. (a)Verify that complex subtraction is the inverse of complex addition (that is, 23 =z2-zI if and only if z3+z=z). (b)Verify that complex division,as given in the text,is the inverse of complex multiplication (that is,if0,then z3=z/if and only if=z). 4.Prove that if zz=0,then =0 or 2=0. In Problems 5-13,write the number in the form a+bi. 点0-) b)(8+i)-(5+i) a2 6.(a)(-1+i02 21 (⊙i(π-4i) 7.a)8影-】 )+ i 2+3i 四+号 8.8+20-1-) (2+)2 粉- 2+i72 10.6-0-2iJ 11.i0+1)2 12.(2+i0(-1-03-2i) 13.(3-i0}2-3i 14.Show that Re(iz)=-Im z for every complex number z. 15.Letk be an integer.Show that 4=1.1=i 7+2=-1,+3=-i 16.Use the result of Problem 15 to find (a)77 b)62 (d202 (d山4321 17.Use the result of Problem 15 to evaluate 31+6㎡3+0+1 18.Show that the complex number=-1+i satisfies the equation z2+2x+2=0
6 Complex Numbers 19.Write the complex equation 3+522=z+3i as two real equations 20.Solve each of the following equations forz. (a)iz=4-zi 子=1-1 (c)(2-i0z+8z2=0 dz2+16=0 21.The complex numberssatisfy the system ofequations (1-i0z+3z2=2-3. iz:+(1+2i)z2=1. Findz2 22.Find all solutions to the equation16=0. 23.Let z be a complex number such that Re0.Prove that Re(1/)>0. 24.Let z be a complex number such that Im z>0.Prove that Im(1/z)<0. 25.complex numbers such that+and are each negative real numbers.Prove that and z2 must be real numbers. 26.Verify that Re∑=2Re j1 and that m2=2m [The real (imaginary)part of the sum is the sum of the real (imaginary)parts.] Formulate,and then disprove,the corresponding conjectures for multiplicatior 27.Prove the binomial formula for complex numbers: a+ar=石+（)）双+…+（图侧）-*线++8， wheren is a positive integer,and the biomial coefficients are given by 28.Use the binomial formula (Prob.27)to compute(2-i)5. 9.Prove that there iso riona umberthat saisfies.[HINT Show that if p/g were a solution,where p and are integers,then 2 would have to divide both p and g.This contradicts the fact that such a ratio can always be written without common divisors】
1.2 Point Representation of Complex Numbers 30.The definition of the order relation denoted by>in the real number system is based upothe a subset(the positivereals)having the following properties (For any numbereitheror(but not both)belongs toP. (ii)If a and B belong to P.so does+B. (iii)If a and B belong toP,so doesB. When such a set P exists we write B if and only if a-B belongs toP. Prove that the compx number system does n possess a nonempty subsetPhaving properties (i).(ii),and (iii).[HINT:Argue that ncither nor-f could belong to such a set P.] Th 31.Writeac real and imaginary parts. the t(a+b)(c s fo r(rea tions (and two signe itions).Or most computers multiplication is far more tim consuming than a an algorithm for computing(a+bi)(c+d)with only three multiplications(at the cost of extra additions).[HINT:Start with (a+b)(c+d).] 1.2 Point Representation of Complex Numbers Itispresthdrsfmiliar wth the Ceyst(1.1) which establishes a one-to-one correspondence between points in the xy-plane ordered pair of real numbers.The ordered pair(-2,3),for example,corresponds to that point Pthat lies two units to theofthe y-axis and three units above theais. P-2.3) 。2+21 -4°31 Figure 1.1 Cartesian coordinate system Oncomputers this is.in fact,the method by which the tested
8 Complex Numbers The Cartesian coordinate system suggests a convenient way to represent complex numbers as points in the cy-plane;namely,to each complex number a+bi we asso- ciate that point in the xy-plane that has the coordinates (a.b).The complex number -2+3i is therefore repres ented by the point P in Fig.1.1.Also shown in Fig.1.1 are the points that represent the compubersd-4-i. When the xy-plane is used for the purpose of describing complex ntimbers it is referred to as the complex plane or z-plane.(The term Argand diagram is sometimes hfcomembersn the plane was proposednp e Argand in 1806.)Since each point on the axis represents a real number,this axis is called the realaris.Analogously,the y-axis is called the imaginary axis for it represents the pure imaginary numbers. Hereafter,we shall refer to the point that represents the complex number z as simply the point that is,the point =a+bi is the point with coordinates (a,b) Example 1 Suppose that n particles with masses..are located at the respective points .2.....n in the complex plane.Show that the center of mass of the system is the point 全=m131+m22十…+mn2烈 m1十m2+,+mn Solution. Write=x+yi,2=x+y2i,....n=+yni,and let M be the total mass.Presumably the reader will recall that the center of mass of the given system is the point with coordinates(),where M But clearly andare,respectively,the real and imaginary parts of the comple number(∑ikzk)/M=交.■ AbsoleValue.By the Pythagre tore the distance fom the point to the origin is given bya2+b2.Special notation for this distance is given in Definition 3.The absolute value or modulus of the number =a+bi is denoted by lzl and is given by kl:=Va2+b2 In particular, 0=0, 自- 3-41=√9+16=5
1.2 Point Representation of Complex Numbers 9 1z1 32 2=02*b以 4=a1+b,l Figure 1.2 Distance beiween points. The reader should note that is always a nonegative real number and that the only is th he number 0. Let zt =at +b1i and z2 =a2+b2i.Then lz1-z2l=l(a-a2)+(bt-b)il=V(a1-a2)2+(b1-b2)2, which is the distance between the points with coordinates (at.b)and(a,)(se Fig.1).Hence the distance bewee poinsandis .This of all numbers z that satisfy the equation l3-0l=r, (1) where ois a fixed complex number andr is a fixed positive realnumber.This set con sists of all points z whose distance from zo is r.Consequently Eq.(1)is the equation of a circle. Example 2 Describe the set of pointsz that satisfy the equations (a)lz+21=Iz-ll, b)lz-1川=Rez+1. Solution. (a)ApointsaisfiesE(a)id oy if i isquidistafmhe points -2 and 1.Hen e Eq.(a)is the of the perp segment joining -2 and 1;that is,Ed.(a)describes thein bisector of the line A more routine method for solving Eq.(a)is to set=iy in the equation and perform the algebra: 芝+2=2一1， +iy+=+iy-1l, (x+22+y2=x-1)2+y2 4x+4=-2x+1. x=一2 (b)The geometric interpretation of Eq.(b)is less obvious,so we proceed di- rectly with the mechanical approach and derive .ory24 which describes a parabola(see Fig.1.3)