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Elementary mathematical and computational tools for electrical engineers using Matlab - Chapter 6

Complex Numbers Kể từ khi x2 0 cho tất cả các số thực x, phương trình x2 = -1 thừa nhận không có số thực sự là một giải pháp. Để đối phó với vấn đề này, các nhà toán học trong thế kỷ 18 đã giới thiệu số lượng tưởng tượng i = -1 = j. (Vì vậy, để không nhầm lẫn biểu tượng thông thường cho một hiện tại với số lượng này, các kỹ sư điện thích việc sử dụng các biểu tượng j MATLAB chấp nhận một trong hai biểu tượng, nhưng luôn luôn cung cấp. | 6 Complex Numbers 6.1 Introduction Since x2 0 for all real numbers x the equation x2 -1 admits no real number as a solution. To deal with this problem mathematicians in the 18th century introduced the imaginary number i 4-1 j. So as not to confuse the usual symbol for a current with this quantity electrical engineers prefer the use of the j symbol. MATLAB accepts either symbol but always gives the answer with the symbol i . Expressions of the form z a jb 6.1 where a and b are real numbers called complex numbers. As illustrated in Section 6.2 this representation has properties similar to that of an ordered pair a b which is represented by a point in the 2-D plane. The real number a is called the real part of z and the real number b is called the imaginary part of z. These numbers are referred to by the symbols a Re z and b Im z . When complex numbers are represented geometrically in the x-y coordinate system the x-axis is called the real axis the y-axis is called the imaginary axis and the plane is called the complex plane. 6.2 The Basics In this section you will learn how using MATLAB you can represent a complex number in the complex plane. It also shows how the addition or subtraction of two complex numbers or the multiplication of a complex number by a real number or by j can be interpreted geometrically. 2001 by CRC Press LLC Example 6.1 Plot in the complex plane the three points P1 P2 P3 representing the complex numbers z1 1 z2 j z3 -1. Solution Enter and execute the following commands in the command window z1 1 z2 j z3 -1 plot z1 axis -2 2 -2 2 axis square hold on plot z2 o plot z3 hold off that is a complex number in the plot command is interpreted by MATLAB to mean take the real part of the complex number to be the x-coordinate and the imaginary part of the complex number to be the y-coordinate. 6.2.1 Addition Next we define addition for complex numbers. The rule can be directly deduced from analogy of addition of two vectors