复数知识点.pdf
复 数知识点1.复数的单位为i，它的平方等于 1，即1i2.复数及其相关概念：复数形如 a+bi 的数（其中Rba，）；实数当 b=0 时的复数 a+bi，即 a；虚数当0b时的复数 a+bi；纯虚数当 a=0 且0b时的复数 a+bi，即 bi.复数 a+bi 的实部与虚部 a 叫做复数的实部，b叫做虚部（注意a，b 都是实数）复数集 C全体复数的集合，一般用字母C 表示.复数是实数的充要条件：z=a+bi Rb=0（a、bR）；zRz=z；ZR22ZZ。复数是纯虚数的充要条件：z=a+bi 是纯虚数a=0 且 b0（a、bR）；z是纯虚数或 0Z+z=0；z 是纯虚数 z20。两个复数相等的定义：00babiaRdcbadbcadicbia）特别地，（其中，且.两个复数，如果不全是实数，就不能比较大小.注：若21,zz为复数，则1若021zz，则21zz.（）21,zz为复数，而不是实数 2若21zz，则021zz.（）若Ccba,，则0)()()(222accbba是cba的必要不充分条件.（当22)(iba，0)(,1)(22accb时，上式成立）2、复数加、减、乘、除法的运算法则：设),(,21Rdcbadiczbiaz，则idbcazz)()(21；ibcadbdaczz)()(21；idcadbcdcbdaczz222221。加法的几何意义：设1OZ，2OZ各与复数z1，z2对应，以1OZ，2OZ为边的平行四边形的对角线OZ就与 z1+z2对应。减法的几何意义：设1OZ，2OZ各与复数z1，z2对应，则图中向量21ZZ所对应的复数就是z2-z1。z1-z2的几何意义是分别与Z1，Z2对应的两点间的距离。3.复平面内的两点间距离公式：21zzd.其中21zz，是复平面内的两点21zz 和所对应的复数，21zzd和表示间的距离.由上可得：复平面内以0z为圆心，r为半径的圆的复数方程：）（00rrzz.曲线方程的复数形式：00zrzz表示以为圆心，r 为半径的圆的方程.21zzzz表示线段21zz的垂直平分线的方程.212121202ZZzzaaazzzz，）表示以且（为焦点，长半轴长为 a 的椭圆的方程（若212zza，此方程表示线段21ZZ，）.），（2121202zzaazzzz表示以21ZZ，为焦点，实半轴长为a 的双曲线方程（若212zza，此方程表示两条射线）.绝对值不等式：设21zz，是不等于零的复数，则212121zzzzzz.左边取等号的条件是），且（012Rzz，右边取等号的条件是），（012Rzz.212121zzzzzz.左边取等号的条件是），（012Rzz，右边取等号的条件是），（012Rzz.注：nnnAAAAAAAAAA11433221.4.共轭复数：两个复数实部相等，虚部互为相反数。即z=a+bi，则z=a-bi，（a、bR），实数的共轭复数是其本身性质zz2121zzzzazz2，i2bzz（za+bi）22|zzzz2121zzzz2121zzzz2121zzzz（02z）nnzz)(注：两个共轭复数之差是纯虚数.（）之差可能为零，此时两个复数是相等的 5.复数的乘方：)(.Nnzzzzznn对任何z，21,zzC及Nnm,有nnnnmnmnmnmzzzzzzzzz2121)(,)(,注：以上结论不能拓展到分数指数幂的形式，否则会得到荒谬的结果，如1,142ii若由11)(212142ii就会得到11的错误结论.在实数集成立的2|xx.当x为虚数时，2|xx，所以复数集内解方程不能采用两边平方法.常用的结论：1,1,143424142nnnniiiiiii)(,0321Zniiiinnnn文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2iiiiiiii11,11,2)1(2若是 1 的立方虚数根，即i2321，则.6.复数z是实数及纯虚数的充要条件：zzRz.若0z，z是纯虚数0zz.模相等且方向相同的向量，不管它的起点在哪里，都认为是相等的，而相等的向量表示同一复数.特例：零向量的方向是任意的，其模为零.注：|zz.7.复数集中解一元二次方程：在复数集内解关于x的一元二次方程)0(02acbxax时，应注意下述问题：当Rcba,时，若0，则有二不等实数根abx22,1；若=0，则有二相等实数根abx22,1；若0，则有二相等复数根aibx2|2,1（2,1x为共轭复数）.当cba,不全为实数时，不能用方程根的情况.不论cba,为何复数，都可用求根公式求根，并且韦达定理也成立.)(0,01,1,121223Znnnn文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2文档编码:CY2J2Y3G2L1 HJ7E4P6M10S5 ZG1R7P7P2P2
