Chapter JComplex Number FieldInthischapter,wesurveythealgebraic andgeometric structureof thecomplexnumber systemWeassumevarious correspondingproperties ofreal numbers to beknown.Thepositive integernumbersystem,integernumbersystem,rationalnumbersystemandrealnumbersystemaredenoted by N,Z,Q and R, respectively.$1.1.SumsandProducts1.Definiton of Complex NumbersA complex mumber is defined as an ordered pair (x,y) of real numbers x and yIt is customaryto denoteacomplex number(x,y)by z,so that(1.1.1)z =(x,y)The real numbers xand y arecalled thereal and imaginaryparts of =,respectively,andwewriteRez=x,Imz=y(1.1.2)Two complex numbers z, =(x,yi) and z2 =(x2,y2) are equal whenever they have thesame real parts and the same imaginary parts.2. Operations of Complex NumbersThe sum z+z2 and the product zi2 of two complex numbers z=(x,yi) and22=(x2,y2）aredefinedasfollows(1.1.3)(x,y1)+(x2,y2)=(x, +X2,/ +y2),(1.1.4)(X,y)(x2,2) =(xix2 -yiy2, Jix2 +Xiy2)3. The Relationship of Real Numbers and Complex NumbersNote that the operations defined by equations (1.1.3) and (1.1.4) become the usual operations ofaddition and multiplication when restricted to the real numbers:(x,0)+(x2,0)=(x +x2,0), (x,0)(x2,0)=(xx2,0)The complex number system is, therefore, a natural extension of the real number system.4.Alternative Representation ofComplex Numbersy4Any complex number z=(x,y) can be written z=(x,y)as z =(x,0)+(O,y), and it is easy to see that(0,1)(y,0) = (0, y) . Hencei=(0,1)z =(x,0)+ (0,1)(y,0);+0x=(x,0)and, if we think of a real number x as the complexnumber (x,O),thatis,weidentifyareal numberxwith a correspondingcomplexnumber (x,O),and letFig. 1-1
Chapter Ⅰ Complex Number Field In this chapter, we survey the algebraic and geometric structure of the complex number system. We assume various corresponding properties of real numbers to be known. The positive integer number system, integer number system, rational number system and real number system are denoted by QZ,N, and R , respectively. §1.1. Sums and Products 1. Definiton of Complex Numbers A complex number is defined as an ordered pair yx ),( of real numbers x and y . It is customary to denote a complex number yx ),( by z , so that = yxz ),( . (1.1.1) The real numbers x and y are called the real and imaginary parts of , respectively; and we write z = Im,Re = yzxz . (1.1.2) Two complex numbers ),( and 111 = yxz ),( 222 = yxz are equal whenever they have the same real parts and the same imaginary parts. 2. Operations of Complex Numbers The sum and the product of two complex numbers and are defined as follows: 21 + zz 21zz ),( 111 = yxz ),( 222 = yxz ),(),(),( 2211 2121 + = + + yyxxyxyx , (1.1.3) (),)(,( , ) 2211 21212121 = − + yxxyyyxxyxyx . (1.1.4) 3. The Relationship of Real Numbers and Complex Numbers Note that the operations defined by equations (1.1.3) and (1.1.4) become the usual operations of addition and multiplication when restricted to the real numbers: )0,()0,()0,( 1 2 21 =+ + xxxx , )0,()0,)(0,( 21 21 = xxxx . The complex number system is, therefore, a natural extension of the real number system. 4. Alternative Representation of Complex Numbers Fig. 1-1 Any complex number can be written as , and it is easy to see that . Hence = yxz ),( += yxz ),0()0,( = yy ),0()0,)(1,0( xz += y )0,)(1,0()0,( ; and, if we think of a real number x as the complex number x )0,( , that is, we identify a real number x with a corresponding complex number x )0,( , and let
idenotetheimaginarynumber(O,l)(Fig.1-1)it isclearthatz=x+iy,(1.1.5)which iscalledtherectangularformof thenumberz.Thus,thecomplexnumber systemcanbewritten asC=((x,y):x,yeR)=(x+iy:x,yeR)22= zz,z”=zz,etc.,wefind thatAlso, with the convention2 = (0,1)(0,1) = (-1,0) = -1(1.1.6)Thus,the equation z? +1=0 has a root z=i in CIn view ofexpression (1.1.5), definitions (1.1.3) and (1.1.4) become(1.1.7)(x, +iyi)+(x2 +iy2)=(xi +x2)+i(yi+y2),(1.1.8)(x+iy(x+iy)=(xx-yy)+iyxz+xy)Observe that the right-hand sides of these equations can be obtained by formally manipulating theterms ontheleft replacing?2by-l when itoccurs
i denote the imaginary number )1,0( ( Fig. 1-1) it is clear that z = x + iy , (1.1.5) which is called the rectangular form of the number . Thus, the complex number system can be written as z C = ∈R = + yxiyxyxyx ∈R},:{},:),{( . Also, with the convention , etc., we find that 2 23 , == zzzzzz 1)0,1()1,0)(1,0( 2 i = −=−= . (1.1.6) Thus, the equation 01 has a root 2 z =+ = iz in C . In view of expression (1.1.5), definitions (1.1.3) and (1.1.4) become )()()()( 2211 21 21 + + + = + + + yyixxiyxiyx , (1.1.7) ())(( () ) 2211 2121 2121 + + = − + + yxxyiyyxxiyxiyx . (1.1.8) Observe that the right-hand sides of these equations can be obtained by formally manipulating the terms on the left replacing by -1 when it occurs. 2 i
Complex Number FieldIChapterIs1.2.BasicAlgebraicPropertiesVarious properties of addition and multiplication of complexnumbers are the same as forrealnumbers. We list here the more basic of these algebraic properties and verify some of them. Mostof the others are verified in the exercises.1.Commutativelaw(1.2.1)2,+22=22+212172=22712.Associative law3.Distributivelaw2(2, +z2) = 22, + 272,(1.2.3)nz=z+z+..+z and z" =zz...z4.IdentitiesTheadditiveidentity0=(0,0)and the multiplicative identity1=(10)for real numberscarry over to the entire complex number system. That is,z+0=z and z-1=z(1.2.4)foreverycomplexnumberz.Furthermore,O and1aretheonlycomplexnumberswithsuchproperties (see Exercise 9).5.Additive inverseFor eachcomplex number z =(x,y),there is an additive inverse(1.2.5)-z =(-x,-y),6.Subtraction21 -22 =z,+(-22), Vz1,32 eC.(1.2.6)So if z, =(x,y) andz,=(x2,y2),then(1.2.7)z, -z2 =(x -x2,yi - y2)=(x, -x2)+i(y1 - y2)7.Multiplicative inverseFor z=(x,y)=x+iy±, theres a numbersuch that=1,called theisZzItisfindthat theofmultiplicative inverseofZeasytomultiplicativeinversez=(x,y)=x+iy isxx-y-yZ(= + 0),+i(1.2.8)x+ y2x+y2x+yx+yFrom the discussion above, we conclude that the set C of all complex numbers becomes afield,calledthefieldofcomplexnumbers,orthecomplexnumberfield
Chapter Complex Number Field Ⅰ 1 §1.2. Basic Algebraic Properties Various properties of addition and multiplication of complex numbers are the same as for real numbers. We list here the more basic of these algebraic properties and verify some of them. Most of the others are verified in the exercises. 1. Commutative law 12211221+ = + , = zzzzzzzz (1.2.1) 2. Associative law 3. Distributive law 21 21 + )( = + zzzzzzz , (1.2.3) 6474 484 L n +++= zzznz and 876 L n n = zzzz . 4. Identities The additive identity = )0,0(0 and the multiplicative identity = )0,1(1 for real numbers carry over to the entire complex number system. That is, + 0 = zz and ⋅1 = zz (1.2.4) for every complex number z . Furthermore, 0 and 1 are the only complex numbers with such properties (see Exercise 9). 5. Additive inverse For each complex number = yxz ),( , there is an additive inverse − = − −yxz ),( , (1.2.5) 6. Subtraction − = + − 2121 ∀ ,),( zzzzzz 21 ∈C . (1.2.6) So if ),( and , then 111 = yxz ),( 222 = yxz )()(),( 212121 21 21 =− − − = − + − yyixxyyxxzz . (1.2.7) 7. Multiplicative inverse For ),( == + iyxyxz ≠ 0 , there is a number such that , called the multiplicative inverse of . It is easy to find that the multiplicative inverse of is −1 z 1 1 = − zz z ),( +== iyxyxz , )0( 2222 2222 1 ≠ + − + + =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + = − z yx y i yx x yx y yx x z . (1.2.8) From the discussion above, we conclude that the set of all complex numbers becomes a field, called the field of complex numbers, or the complex number field. C
$1.3.FurtherPropertiesInthissectionwementionanumberofotheralgebraicpropertiesofadditionandmultiplicationof complexnumbersthatfollowfromtheonesalreadydescribedinSec.1.2.Becausesuchproperties continue to be anticipated, the reader can easily pass to Sec.l.4 without seriousdisruption.1.Expunctive lawIf z=,=0,then either z,=0 or z2=0; or possibly both z and =2 equal zero.Another way to state this result is that if two complex numbers z and z2 are nonzero, then sois their product z,-2.2.DivisionDivisionbyanonzerocomplexnumber isdefinedasfollows:=222(z, ±0)(1.3.1)Z2If z,=(x,)= x, +iy) and z, =(x2,y2)=x, +iy2,then= +yy2+iy-xy2(=2±0)(1.3.2)x,+ysx2+y2Z2Although expression (1.3.2) is not easy to remember, it can be obtained by writing (see Exercise4)三_ (x +iy)(xz-iy2)(1.3.3)22(x2 +iy2)(x2 -iy2)3.Useful identities=2(=2 ± 0)(1.3.4)22(2, 0),(1.3.5)2222(2/22)(z2)=(2-7")(2232) =1 (22 ±0)()=(2,2) =27(z ± 0,z2 ± 0)(1.3.6)2,222122Z122(z3 ± 0,=4 ± 0),(1.3.7)23-423Z4Example.Computations such as thefollowing are now justified:5+i115+i1112-3i1+i)=5-i5+i(5-i)(5+i)(2 -3i)(1+i)5+i_5-5.1i26262626264.Binomial formulaIf z, and z, are any two complex numbers, then(n)n-(z +z,)">(n = 1,2,...)(Binomial Formula)(1.3.8)22Kk=o(k)n!(k = 0,1,2,..,n) and where it is agreed that Ol=1wherek!(n-k)!
§1.3. Further Properties In this section, we mention a number of other algebraic properties of addition and multiplication of complex numbers that follow from the ones already described in Sec.1.2. Because such properties continue to be anticipated, the reader can easily pass to Sec.1.4 without serious disruption. 1. Expunctive law If 0 , then either or zz 21 = 0 z1 = 0 z2 = ; or possibly both and equal zero. Another way to state this result is that if two complex numbers and are nonzero, then so is their product . 1 z 2 z 1 z 2 z 21 zz 2. Division Division by a nonzero complex number is defined as follows: )0( 2 1 21 2 1 = ≠ − zzz z z (1.3.1) If and 11111 ),( +== iyxyxz 22222 = ),( = + iyxyxz , then )0( 2 2 2 2 2 2121 2 2 2 2 2121 2 1 ≠ + − + + + = z yx yxxy i yx yyxx z z . (1.3.2) Although expression (1.3.2) is not easy to remember, it can be obtained by writing (see Exercise 4) ))(( ))(( 2222 2211 2 1 iyxiyx iyxiyx z z −+ −+ = . (1.3.3) 3. Useful identities )0( 1 2 1 2 2 = ≠ − zz z . (1.3.4) )0( 1 2 2 1 2 1 ≠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = z z z z z . (1.3.5) )0(1))(())(( 2 1 22 1 11 1 2 1 121 = ≠= −− − − zzzzzzzzz . )0,0( 11 )( 1 1 2 21 1 2 1 1 1 21 21 ≠≠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ === −−− zz zz zzzz zz . (1.3.6) )0,0( 3 4 4 2 3 1 43 21 ≠≠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = zz z z z z zz zz . (1.3.7) Example. Computations such as the following are now justified: . 26 1 26 5 2626 5 26 5 )5)(5( 5 5 5 5 1 )1)(32( 1 1 1 32 1 i ii ii i i i iiiii +=+= + = +− + = + + ⋅ − = +− ⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − 4. Binomial formula If and are any two complex numbers, then 1 z 2 z ∑= − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =+ n k n kkn nzz k n zz 0 21 21 )( K),2,1( (Binomial Formula) (1.3.8) where ),2,1,0( )!(! ! k n knk n k n = K − =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ and where it is agreed that = 1!0
$1.4.ModuliIt is natural to associateany nonzero complexnumber z=x+iywiththe directed line segmentor vector,from the origintothe point (x,y)thatrepresents z(Sec.1.1)in the complexplaneIn fact, we often refer to zas the point zor the vector z,In Fig.1-2,the numberz = x+ iy and - 2+i are displayed graphically as both points and radius vectors.J4(-2,1)人2+ix+iy(x,y)xo-2Fig. 1-2Accordingtothedefinition of thesumof two complexnumbersz,=x,+iy,andz2 = x, +iy2, z, + z2 may be obtained vectorially as shown in Fig. 1-3.The differencez1-=2 = z, +(-z2) corresponds to the sum of the vectors z and -z2(Fig. 1-4).yAy4(x2y)(xy)ZX8x02,-2Fig. 1-4Fig. 1-31.ModulusThe modulus, or absolute value, of a complex number z = x+iy is defined as thenonnegative real number x? +y?and is denoted by I=l; that is,[==/x? +y2(1.4.1)Geometrically,thenumber=isthedistancebetweenthepoint (x,y)andtheorigin,orthelength of the vector representing z: It reduces to the usual absolute value in the real numbersystemwheny=o.2.DistanceofcomplexnumbersThe distance between two points =, = x, +iy, and ≥, = x, +iy2 is defined by
§1.4. Moduli It is natural to associate any nonzero complex number z = x + iy with the directed line segment, or vector, from the origin to the point yx ),( that represents z (Sec. 1.1) in the complex plane. In fact, we often refer to z as the point z or the vector z . In Fig. 1-2, the number z x += iy and − 2 + i are displayed graphically as both points and radius vectors. Fig. 1-2 According to the definition of the sum of two complex numbers and , 1 1 1 z = x + iy 2 2 2 z = x + iy 1 2 z + z may be obtained vectorially as shown in Fig. 1-3. The difference ( ) 1 2 1 2 z − z = z + −z corresponds to the sum of the vectors and (Fig. 1-4). 1 z 2 − z 1. Modulus The modulus, or absolute value, of a complex number z = x + iy is defined as the nonnegative real number 2 2 x + y and is denoted by | z |; that is, 2 2 | z |= x + y . (1.4.1) Geometrically, the number is the distance between the point and the origin, or the length of the vector representing | z | (x, y) z . It reduces to the usual absolute value in the real number system when y = 0 . 2.Distance of complex numbers The distance between two points 1 1 1 z = x + iy and 2 2 2 z = x + iy is defined by