Chapter 6. Complex Numbers

# Chapter 6. Complex Numbers - 6 Complex Numbers 6 .1...

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0-8493-????-?/00/\$0.00+\$.50 © 2000 by CRC Press LLC © 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. ij =−= 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 deFne 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 + 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 ±ind the sum and difference of the complex numbers
© 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, Fnd 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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## This note was uploaded on 10/19/2010 for the course ENGINEERIN ELEC121 taught by Professor Tang during the Spring '10 term at University of Liverpool.

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Chapter 6. Complex Numbers - 6 Complex Numbers 6 .1...

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The provided text is an excerpt from a chapter on complex numbers, introducing the concept, notation, and basic operations such as addition and subtraction. It includes an example and a preparatory exercise.
multiple_questions

the text contains multiple mathematical questions or problems, including:
1. an example to plot points representing complex numbers in the complex plane.
2. an example to find the sum and difference of two complex numbers.
3. a preparatory exercise to find the fourth vertex of a parallelogram given three vertices.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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Chapter 6. Complex Numbers

# Chapter 6. Complex Numbers - 6 Complex Numbers 6 .1...

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0-8493-????-?/00/\$0.00+\$.50 © 2000 by CRC Press LLC © 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. ij =−= 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 deFne 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 + 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 ±ind the sum and difference of the complex numbers
© 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, Fnd 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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## This note was uploaded on 10/19/2010 for the course ENGINEERIN ELEC121 taught by Professor Tang during the Spring '10 term at University of Liverpool.

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Chapter 6. Complex Numbers - 6 Complex Numbers 6 .1...

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The provided text is an excerpt from a chapter on complex numbers, introducing the concept, notation, and basic operations such as addition and subtraction. It includes an example and a preparatory exercise.

TYPE: MULTIPLE_QUESTIONS

The text contains multiple mathematical questions or problems, including:
1. An example to plot points representing complex numbers in the complex plane.
2. An example to find the sum and difference of two complex numbers.
3. A preparatory exercise to find the fourth vertex of a parallelogram given three vertices.