Đại số và hình học của số phức tạp (dựa trên § § 17,1-17,3 của Zill) Định nghĩa 1,1 Một số phức có dạng z = (x, y), trong đó x và y là các số thực. x được gọi là phần thực của z, và y được gọi là phần tưởng tượng của z. Chúng tôi viết Re (z) = x, Im (z) = y. Ký hiệu tập hợp các số phức tạp bởi C. | Advanced Engineering Math II Math 144 Lecture Notes by Stefan Waner First printing 2003 Department of Mathematics Hofstra University 1. Algebra and Geometry of Complex Numbers based on of Zill Definition A complex number has the form z x y where x and y are real numbers. x is referred to as the real part of z and y is referred to as the imaginary part of z. We write Re z x Im z y. Denote the set of complex numbers by C . Think of the set of real numbers as a subset of C by writing the real number x as x 0 . The complex number 0 1 is called i. Examples 3 3 0 0 5 -1 -n i 0 1 . Geometric Representation of a Complex Number- in class. Definition Addition and multiplication of complex numbers and also multiplication by reals are given by x y x y x x y y x y x y xx -yy xy x y x y x y . Geometric Representation of Addition- in class. Multiplication later Examples a 3 4 3 0 4 0 7 0 7 b 3x4 3 0 4 0 12-0 0 12 0 12 c 0 y y 0 1 yi which we also write as iy . d In general z x y x 0 0 y x iy. z x iy e Also i2 0 1 0 1 -1 0 -1. i2 -1 g 4 - 3i 4 -3 . Note In view of d above from now on we shall write the complex number x y as x iy. Definitions The complex conjugate z of the complex number z x iy given by z x - iy. The magnitude Izl of z x iy is given by Izl x2 y2 . Examples and Geometric Representation of Conjugation and Magnitude - in class. Notes 1. z z x iy x-iy 2x 2Re z . Therefore Re z 2 z z 2 z - z x iỳ - x-iy 2iy 2iIm z . Therefore . . 2 .2 2 2 2 .2 2. Note that zz x iy x-iy x -i y x y lzl 3. If z 0 then z has a multiplicative inverse. Why because Im z 2i z-z zz Izl2 z z lzl2 s 7 2 1. Hence lzl lzl -1 _ z lzl2 Examples a 1 -i c 0 V2 à 1-i V2 3-4i _ 1 b 3 4i 25 1 d cos0 isin0 cos -0 isin -0 4. There is also the Triangle Inequality lz1 z2l lz1l lz2l. Proof We square both sides and compare them. Write z1 x1 iy1 and z2 x2 iy2. Then lzi z2l2 xi x2 2 yi y2 2 xi2 x 2x1x2 yi2 V 2yiy2. On the other hand lz1l lz2l 2 lz1l2 2lz1llz2l lz2l2 xi2 x yi2 x2 .
