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Chapter 6. Complex Numbers - 6 Complex Numbers 6.1...

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© 2001 by CRC Press LLC 6 Complex Numbers 6.1 Introduction Since x 2 > 0 for all real numbers x , the equation x 2 = –1 admits no real number as a solution. To deal with this problem, mathematicians in the 18th century introduced the imaginary number (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 coordi- nate 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 com- plex number in the complex plane. It also shows how the addition (or sub- traction) of two complex numbers, or the multiplication of a complex number by a real number or by j, can be interpreted geometrically. i j = = 1 .
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© 2001 by CRC Press LLC Example 6.1 Plot in the complex plane, the three points ( P 1 , P 2 , P 3 ) representing the com- plex numbers: z 1 = 1, z 2 = j , z 3 = –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 in a plane: the x -component of the sum of two vectors is the sum of the x -components of each of the vec- tors, and similarly for the y -component. Therefore: If: z 1 = a 1 + jb 1 (6.2) and z 2 = a 2 + jb 2 (6.3) Then: z 1 + z 2 = ( a 1 + a 2 ) + j ( b 1 + b 2 ) (6.4) The addition or subtraction rules for complex numbers are geometrically translated through the parallelogram rules for the addition and subtraction of vectors. Example 6.2 Find the sum and difference of the complex numbers
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© 2001 by CRC Press LLC z 1 = 1 + 2 j and z 2 = 2 + j Solution: Grouping the real and imaginary parts separately, we obtain: z 1 + z 2 = + 3 j and z 1 z 2 = –1 + j Preparatory Exercise Pb. 6.1 Given the complex numbers z 1 , z 2 , and z 3 corresponding to the ver- tices P 1 , P 2 , and P 3 of a parallelogram, find z 4 corresponding to the fourth ver- tex P 4 . (Assume that P 4 and P 2 are opposite vertices of the parallelogram).
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