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  # Math144Notes - Advanced Engineering Math II Math 144...
  
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  Advanced Engineering Math II Math 144 Lecture Notes by Stefan Waner (First printing: 2003) Department of Mathematics, Hofstra University
  
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  2 1. Algebra and Geometry of Complex Numbers (based on §§17.1–17.3 of Zill) Definition 1.1 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 I . Think of the set of real numbers as a subset of C I by writing the real number x as (x, 0) . The complex number (0, 1) is called i. Examples 3 = (3, 0), (0, 5), (-1, - π ), i = (0, 1). Geometric Representation of a Complex Number- in class. Definition 1.2 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 1.3 (a) 3+4 = (3, 0)+(4, 0) = (7, 0) = 7 (b) 3 4 = (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, i 2 = (0, 1)(0, 1) = (-1, 0) = -1. i 2 = -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 1.4 The complex conjugate , z , of the complex number z = x+iy given by z = x - iy. The magnitude , |z| of z = x+iy is given by |z| = x 2 +y 2 . Examples and Geometric Representation of Conjugation and Magnitude - in class. Notes 1. z + z = (x+iy) + (x-iy) = 2x = 2 Re (z) . Therefore, Re (z) = 1 2 (z+z)
  3 z - z = (x+iy) - (x-iy) = 2iy = 2i Im (z) . Therefore, Im (z) = 1 2i (z-z) 2. Note that zz = (x+iy)(x-iy) = x 2 -i 2 y 2 = x 2 +y 2 = |z| 2 zz = |z| 2 3 . If z 0 , then z has a multiplicative inverse. Why? because: z · z |z| 2 = zz |z| 2 = |z| 2 |z| 2 = 1. Hence, z -1 = z |z| 2 Examples (a) 1 i = -i (b) 1 3+4i = 3-4i 25 (c) 1 1 2 (1+i) = 1 2 (1-i) (d) 1 cos ø + i sin ø = cos (-ø) + i sin (-ø) 4. There is also the Triangle Inequality: |z 1 + z 2 | |z 1 | + |z 2 |. Proof We square both sides and compare them. Write z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 . Then |z 1 + z 2 | 2 = (x 1 +x 2 ) 2 + (y 1 +y 2 ) 2 = x 1 2 + x 2 2 + 2x 1 x 2 + y 1 2 + y 2 2 + 2y 1 y 2 . On the other hand, (|z 1 | + |z 2 |) 2 = |z 1 | 2 + 2|z 1 ||z 2 | + |z 2 | 2 = x 1 2 + x 2 2 + y 1 2 + y 2 2 + 2|z 1 ||z 2 |. Subtracting, (|z 1 | + |z 2 |) 2 - |z 1 + z 2 | 2 = 2|z 1 ||z 2 | - 2(x 1 x 2 + y 1 y 2 ) = 2 [ |(x 1 ,y 1 )||(x 2 ,y 2 )| - (x 1 ,y 1 ).(x 2 ,y 2 ) ] (in vector form) = 2 [ |(x 1 ,y 1 )||(x 2 ,y 2 )| - |(x 1 ,y 1 )||(x 2 ,y 2 )| cos ] = 2 |(x 1 ,y 1 )||(x 2 ,y 2 )| (1 - cos ) 0, giving the result. Note The triangle inequality can also be seen by drawing a picture of z 1 + z 2 . 5. We now consider the polar form of these things: If z = x+iy, we can write x = r cos ø and y = r sin ø, getting z = r cos ø + ir sin ø, so This is called the polar form of z . It is important to draw pictures in order to feel comfortable with the polar representation. Here r is the magnitude of z, r = |z| , and ø is called the argument of z , denoted arg (z). To calculate ø , we can use the fact that tan ø = y/x. Thus ø is not arctan (y/x) as claimed in the book, but by: since the arctan function takes values between - π /2 and π /2. The principal value of arg (z) is the unique choice of ø such that - π < ø π . We write this as Arg (z) - π < Arg (z) ≤ π - π ≤ A
  
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  4 Examples (a) Express z = 1+i in polar form, using the principal value (b) Same for 3 + 3 3 i (c) 6 = 6( cos 0 + i sin 0) 6.
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explanation: The provided text is an excerpt from lecture notes on Advanced Engineering Math II, specifically covering the topic of complex numbers. The document defines complex numbers, their algebraic and geometric representations, and operations such as addition, multiplication, and conjugation. It also introduces the concept of the magnitude of a complex number and the triangle inequality. The text includes several examples and notes to illustrate these concepts.
  
  The document does not contain a specific question or a question-and-answer pair. It is primarily a set of definitions, examples, and notes.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  # Math144Notes - Advanced Engineering Math II Math 144...
  
  This preview shows pages 1–5. Sign up to view the full content.
  
  Advanced Engineering Math II Math 144 Lecture Notes by Stefan Waner (First printing: 2003) Department of Mathematics, Hofstra University
  
  This preview has intentionally blurred sections. Sign up to view the full version.
  
  View Full Document
  2 1. Algebra and Geometry of Complex Numbers (based on §§17.1–17.3 of Zill) Definition 1.1 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 I . Think of the set of real numbers as a subset of C I by writing the real number x as (x, 0) . The complex number (0, 1) is called i. Examples 3 = (3, 0), (0, 5), (-1, - π ), i = (0, 1). Geometric Representation of a Complex Number- in class. Definition 1.2 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 1.3 (a) 3+4 = (3, 0)+(4, 0) = (7, 0) = 7 (b) 3 4 = (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, i 2 = (0, 1)(0, 1) = (-1, 0) = -1. i 2 = -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 1.4 The complex conjugate , z , of the complex number z = x+iy given by z = x - iy. The magnitude , |z| of z = x+iy is given by |z| = x 2 +y 2 . Examples and Geometric Representation of Conjugation and Magnitude - in class. Notes 1. z + z = (x+iy) + (x-iy) = 2x = 2 Re (z) . Therefore, Re (z) = 1 2 (z+z)
  3 z - z = (x+iy) - (x-iy) = 2iy = 2i Im (z) . Therefore, Im (z) = 1 2i (z-z) 2. Note that zz = (x+iy)(x-iy) = x 2 -i 2 y 2 = x 2 +y 2 = |z| 2 zz = |z| 2 3 . If z 0 , then z has a multiplicative inverse. Why? because: z · z |z| 2 = zz |z| 2 = |z| 2 |z| 2 = 1. Hence, z -1 = z |z| 2 Examples (a) 1 i = -i (b) 1 3+4i = 3-4i 25 (c) 1 1 2 (1+i) = 1 2 (1-i) (d) 1 cos ø + i sin ø = cos (-ø) + i sin (-ø) 4. There is also the Triangle Inequality: |z 1 + z 2 | |z 1 | + |z 2 |. Proof We square both sides and compare them. Write z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 . Then |z 1 + z 2 | 2 = (x 1 +x 2 ) 2 + (y 1 +y 2 ) 2 = x 1 2 + x 2 2 + 2x 1 x 2 + y 1 2 + y 2 2 + 2y 1 y 2 . On the other hand, (|z 1 | + |z 2 |) 2 = |z 1 | 2 + 2|z 1 ||z 2 | + |z 2 | 2 = x 1 2 + x 2 2 + y 1 2 + y 2 2 + 2|z 1 ||z 2 |. Subtracting, (|z 1 | + |z 2 |) 2 - |z 1 + z 2 | 2 = 2|z 1 ||z 2 | - 2(x 1 x 2 + y 1 y 2 ) = 2 [ |(x 1 ,y 1 )||(x 2 ,y 2 )| - (x 1 ,y 1 ).(x 2 ,y 2 ) ] (in vector form) = 2 [ |(x 1 ,y 1 )||(x 2 ,y 2 )| - |(x 1 ,y 1 )||(x 2 ,y 2 )| cos ] = 2 |(x 1 ,y 1 )||(x 2 ,y 2 )| (1 - cos ) 0, giving the result. Note The triangle inequality can also be seen by drawing a picture of z 1 + z 2 . 5. We now consider the polar form of these things: If z = x+iy, we can write x = r cos ø and y = r sin ø, getting z = r cos ø + ir sin ø, so This is called the polar form of z . It is important to draw pictures in order to feel comfortable with the polar representation. Here r is the magnitude of z, r = |z| , and ø is called the argument of z , denoted arg (z). To calculate ø , we can use the fact that tan ø = y/x. Thus ø is not arctan (y/x) as claimed in the book, but by: since the arctan function takes values between - π /2 and π /2. The principal value of arg (z) is the unique choice of ø such that - π < ø π . We write this as Arg (z) - π < Arg (z) ≤ π - π ≤ A
  
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  4 Examples (a) Express z = 1+i in polar form, using the principal value (b) Same for 3 + 3 3 i (c) 6 = 6( cos 0 + i sin 0) 6.
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  • As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.
  
  Kiran Temple University Fox School of Business ‘17, Course Hero Intern
  
  • I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.
  
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  • The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.
  
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distilabel_metadata.raw_output_q_a_detector_0: The provided text is an excerpt from lecture notes on Advanced Engineering Math II, specifically covering the topic of complex numbers. The document defines complex numbers, their algebraic and geometric representations, and operations such as addition, multiplication, and conjugation. It also introduces the concept of the magnitude of a complex number and the triangle inequality. The text includes several examples and notes to illustrate these concepts.
  
  The document does not contain a specific question or a question-and-answer pair. It is primarily a set of definitions, examples, and notes.
  
  Label: no-question