8 Key excerpts on "Complex Numbers"

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  eBook - ePub

  Mathematical Economics

  • Arsen Melkumian(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  11   Complex Numbers

  In this chapter we introduce an extension of real numbers called Complex Numbers. At first, it is difficult to understand the real-life importance of such an apparently abstract concept as Complex Numbers. However, there are deep mathematical reasons why Complex Numbers inevitably appear in many applications. For example, matrices with real entries can have complex eigenvalues. The behavior of such well-known functions as sine and cosine is properly understood only when we introduce Complex Numbers. Most importantly, one of the key questions in time-series analysis deals with the stability of the system. This question is reduced to finding eigenvalues of a certain matrix, which describe the ongoing autoregressive process.

  Solving difference or differential equations is greatly simplified by using Complex Numbers. Further, solving quadratic equations over Complex Numbers is simpler, as we are guaranteed to have solutions. Our goal is to present all of the necessary mathematical facts about Complex Numbers to prepare the reader for textbooks in time-series analysis and other areas that frequently deal with Complex Numbers.

  11.1 The set of Complex Numbers

  The set Complex Numbers ℂ is an extension of the set of real numbers ℝ. The set of Complex Numbers contains all roots of quadratic equations. Let i be such that

  then the set of Complex Numbers can be constructed as follows:

  Complex Numbers can be represented graphically by associating z = a + bi with the point (a, b) on the complex plane. Figure 11.1 demonstrates how to locate a complex number on the complex plane.

  Now, let z = a + bi be a complex number. Then a is the real part of z and b is the imaginary part of z and we shall write

  to indicate that fact.

  Figure 11.1

  Note the following points.

  (a)  Instead of 0 + bi, we shall write bi

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  eBook - ePub

  An Introduction to Signal Processing for Non-Engineers

  • Afshin Samani(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Appendix A.2: Complex Numbers

  The numbers that we regularly work with are called real numbers. Any number from minus infinity to plus infinity that we can imagine is a real number. Numbers like −10.333, −2.5, 0, 1, 2, 3.55 and 10 powered by 1000 are all real numbers. There is a one-to-one correspondence between the points on a line and real numbers. Thus, to represent real numbers geometrically, we would just need to show them on a line. However, to describe some physical phenomena, real numbers are not adequate for quantification purposes. One example of such a physical phenomenon is the frequency representation of signals. We should be able to describe the magnitude and phase of the Fourier transform together. Here is where the concept of Complex Numbers comes handy and can resolve many of our computational problems.

  The Complex Numbers have two parts, namely, a real and an imaginary part. To geometrically represent these numbers, a single axis is not adequate—they should be presented on a plane. In other words, these numbers have two dimensions. Figure A.2.1 shows the geometrical representation of a complex number. Mathematically, a complex number (c) is written with its real part (a) and imaginary part (b) as c = a + i.b. One may then ask what i is in this notation, and that is the imaginary unit. However, there is an interesting relationship between the imaginary unit and the real numbers. The i satisfies i2 = −1 or sometimes written as

  i =

  − 1

  . This is why the i is called the “imaginary” unit. No real number satisfies this relationship, as the square of any real number is always non-negative. Very often, j can also be used to denote the imaginary unit. MATLAB recognizes both i and j as the imaginary unit, provided you have not already defined them as variables named i and j. To avoid confusion, newer versions of MATLAB introduced 1i and 1j

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  eBook - ePub

  CLEP® College Mathematics Book + Online

  • Mel Friedman(Author)
  • 2012(Publication Date)
  • Research & Education Association

    (Publisher)

  As indicated above, real numbers provide the basis for most precalculus mathematics topics. However, on occasion, real numbers by themselves are not enough to explain what is happening. As a result, Complex Numbers were developed.

  A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = The number a is the real part, and the number bi is the imaginary part of the complex number.

  Returning momentarily to real numbers, the square of a real number cannot be negative. More specifically, the square of a positive real number is positive, the square of a negative real number is positive, and the square of 0 is 0.

  i is defined to be a number with a property that i2 = –1. Obviously, i is not a real number. C is then used to represent the set of all Complex Numbers:

  C = {a + bi | a and b are real numbers}.

  ADDITION, SUBTRACTION, AND MULTIPLICATION OF Complex Numbers

  Here are the definitions of addition, subtraction, and multiplication of Complex Numbers.

  Suppose x + yi and z + wi are Complex Numbers. Then (remembering that i2 = –1):

  (x + yi) + (z + wi) = (x + z) + (y + w)i

  (x + yi) – (z + wi) = (x – z) + (y – w)i

           (x + yi) × (z + wi) = (xz – wy) + (xw + yz)i

  PROBLEM

  Simplify (3 + i)(2 + i).

  SOLUTION

  DIVISION OF Complex Numbers

  Division of two Complex Numbers is usually accomplished with a special procedure that involves the conjugate of a complex number. The conjugate of a + bi is denoted by and defined by = a – bi.

  Also,

  (a + bi)(a – bi) = a2 + b2

  The usual procedure for division is to multiply and divide by the conjugate as shown below. Remember that multiplication and division by the same quantity leaves the original expression unchanged.

  If a is a real number, then a can be expressed in the form a = a + 0i. Hence, every real number is a complex number and R C

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  eBook - ePub

  Noise and Vibration Analysis

  Signal Analysis and Experimental Procedures

  • Anders Brandt(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Appendix A Complex Numbers

  Complex Numbers are frequently used in signal analysis. A complex number, c is defined as

  (A.1)

  where the real numbers a and b are called the real part and imaginary part, respectively, of c . The number j , the imaginary number, also sometimes denoted i , is equal to the square root of −1. Of course, this does not (at least immediately) provide any insight into the use of Complex Numbers, so we shall here show some fundamental use of Complex Numbers.

  First we define the complex conjugate , c * , of c , by

  (A.2)

  A useful picture of Complex Numbers is obtained if we plot the real and imaginary parts of c as x and y coordinates, respectively, in a coordinate system as in Figure A.1 . Complex Numbers represented by Equation (A.1) is often called rectangular form , or the Euclidian form.

  Figure A.1 The complex plane

  From Figure A.1 it directly follows that the complex number, c , may be written using trigonometric functions, as

  (A.3)

  from which it follows that

  (A.4)

  and

  (A.5)

  The expression of the complex number, c , in Equation (A.3) is often called the trigonometric form . The factor A , is also the square root of the amplitude squared of the complex number, c , which is obtained by

  (A.6)

  There is a third common notation for expressing c ; the Euler form or polar form . Here, c is written as in Equation (A.7) , which can readily be seen as a special notation.

  (A.7)

  where A and ϕ are equal to those in Equation (A.3) . The polar form also has a simplified notation commonly used in, for example, electrical engineering. Here, c is written as

  (A.8)

  where the symbol ∠ is read ‘angle.’

  When we use Complex Numbers in signal analysis, there are mainly two operations of interest. The first is a summation of two Complex Numbers, say c 1 =a 1 +j b 1 and c 2 =a 2 +j b 2 . An example of this case is when we have two sound waves with a certain common frequency, and the two sounds are added together at a certain point. Since the sound information contains both amplitude and phase, it becomes a complex addition, see also below where we describe how Complex Numbers are used to describe sinusoids. With the addition of two Complex Numbers, the rectangular form is most suitable and the sum, c

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  eBook - ePub

  Mechanical Vibration

  Analysis, Uncertainties, and Control, Fourth Edition

  • Haym Benaroya, Mark Nagurka, Seon Han(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Appendix A

  Mathematical Concepts for Vibration

  “We are reminded of the essential rules .”

  This appendix presents mathematical concepts that may be helpful to those studying vibration. It is not meant to be a substitute for courses covering linear algebra, calculus, and differential equations. We assume that the reader has had some experience with the topics.

  A.1 Complex Numbers

  A complex number z = x + iy is the sum of a real part x and an imaginary part iy , with x and y both real and

  i =

  - 1

  . The complex conjugate of z is x - iy . Complex Numbers, which arise in many scientific disciplines, are needed in the general solution polynomial equations, including of quadratic equations. Before the 18th century, Complex Numbers and the square root of minus one were in regular use.

  1

  The equation

  2

  e

  i θ

  =

  cos

  θ + i

  sin

  θ

  is credited to Euler and is known as Euler ’s formula .

  A.1.1 Complex Number Operations

  Two Complex Numbers are added or subtracted by simply adding or subtracting their corresponding real and imaginary parts. For example, the sum of 8 + 3i and 6 + 3i is 14 + 6i .

  Complex number multiplication is similar to real number multiplication. For example,

  ( x + i y ) ( u + i v ) = ( x u - y v ) + i ( x v + y u ) .

  In division of Complex Numbers, the convention is to express the fraction such that there is no imaginary part in the denominator. This can be accomplished by multiplying the numerator and denominator of the complex fraction by the complex conjugate of the denominator u - iv to find

  (A.1)

  x + i y

  u + i v

  =

  ( x u + y v ) + i ( - x v + y u )

  u 2

  +

  v 2

  Example A.l Rewrite the complex fraction

  1 - 2 i

  7 + 3 i

  into its real and imaginary parts.

  Solution:

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  eBook - ePub

  Digital Signal Processing 101

  Everything You Need to Know to Get Started

  • Michael Parker(Author)
  • 2017(Publication Date)
  • Newnes

    (Publisher)

  Chapter 2

  Complex Numbers and Exponentials

  Abstract

  This chapter introduces the importance of Complex Numbers and exponentials which are an integral part of digital communications and digital signal processing (DSP). The reason of delving deep into this area leads the readers to two-dimensional number plane, which help to understand DSP. The operators such as addition, subtraction, and multiplication take people through the complex side of numbers which are explained in two dimensions. The use of complex conjugate and complex exponential—in the number plane—allows readers understand the relationship of basic characteristics of numbers and its treatment. The chapter further looks into measuring angles in radians, which leads to the use of pi and the equivalence of the degrees and radians in measuring angles. DSP therefore uses this concept in later stages to understand the clockwise and anticlockwise movement around the circle.

  Keywords

  Complex conjugate; Complex exponential; Digital signal processing; Euler equation; Polar representation

  Complex Numbers are one of those things many of us were taught a long time ago and have long since forgotten. Unfortunately, they are important in digital communications and digital signal processing (DSP), so we need to resurrect them.

  What we were taught and some of us vaguely remember is that a complex number has a “real” and “imaginary” part, and the imaginary part is the square root of a negative number, which is really a nonexistent number. This right away sounds fishy, and while it's technically true, there is a much more intuitive way of looking at it.

  The whole reason for “Complex Numbers” is that we are going to need a two dimensional number plane to understand DSP. The traditional number line extends from plus infinity to minus infinity, along a single line. To represent many of the concepts in DSP, we need two dimensions. This requires two orthogonal axes, like a North–South line and an East–West line. For the arithmetic to work out, one line, usually depicted as the horizontal line, is the real number line. The other vertical line is the imaginary line. All imaginary numbers are prefaced by “j”, which is defined as the square root of −

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  eBook - ePub

  Foundations of Vibroacoustics

  • Colin Hansen(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  A Review of Complex Numbers and Relevant Linear Matrix Algebra The purpose of this appendix is to provide a brief review of Complex Numbers and many of the particular results of linear algebra used in this book. For more extensive treatments on linear algebra, the reader should consult any of the standard textbooks, such as Cullen (1991); Schneider and Barker (1989). There are also a number of specialised software packages that deal explicitly with linear algebra manipulations, such as GNU Octave. A.1    Complex Numbers A complex number is a mathematically convenient way of representing a physical quantity that is defined by an amplitude and a phase. It is also a convenient way of including the variation with time of a sinusoidally varying quantity. A complex number consists of a real and imaginary part, with the imaginary part identified by the symbol, j. For example, a complex number, F ^ may be written as F ^ = a + j b = Re{ F ^ } + j Im{ F ^ } = | F ^ |e j β, where a is the real part and b is the imaginary part. The amplitude or modulus of the complex number is | F ^ | = a 2 + b 2 and its phase is β = tan –1 (b/a). A time varying quantity, F, varying sinusoidally at radian frequency, ω = 2 πf, where f is the frequency in Hz, can be represented as: F = (a + j b) e j ω t = F ^ e j ω t = | F ^ | e j (ω t + β) = | F | e j (ω t + β) (A.1) For two complex. numbers, u = u ^ e j ωt =| u |e j(ωt + α) and F = F ^ e j ωt =. | F |e j(ωt + β) : F u = | F | e j (ω t + β) × | u | e j (ω t + α) = | F u | e j (2 ω t + β + α) = | F ^ u ^ | e j (2 ω t + β + α) (A.2) and: F u = | F | e j (ω t +[--. =PLGO-SEPARATOR=--]β) | u | e j (ω t + α) = | F u | e j (β − α) = | F ^ u ^ | e j (β − α) (A.3) A.2    Matrices and Vectors An (m × n) matrix is a collection of mn numbers (complex or real), a ij, (i = 1, 2,…, m, j = 1, 2,…, n), written in an array of m rows

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  eBook - ePub

  Digital Signal Processing

  Fundamentals and Applications

  • Li Tan(Author)
  • 2007(Publication Date)
  • Academic Press

    (Publisher)

  F Some Useful Mathematical Formulas Form of a complex number:

  (F.1)

  (F.2)

  (F.3)

  Conversion from the polar form to the rectangular form:

  (F.4)

  Conversion from the rectangular form to the polar form:

  (F.5)

  We usually specify the principal value of the angle such that −180° < θ ≤ 180°. The angle value can be determined as: (that is, the complex number is in the first or fourth quadrant in the rectangular coordinate system); (that is, the complex number is in the second quadrant in the rectangular coordinate system); and (that is, the complex number is in the third quadrant in the rectangular coordinate system). Note that Complex Numbers:

  (F.6)

  (F.7)

  (F.8)

  Complex conjugate of a + jb:

  (F.9)

  Complex conjugate of Ae

  j θ

  :

  (F.10)

  Complex number addition and subtraction:

  (F.11)

  Complex number multiplication:

  Rectangular form:

  (F.12)

  (F.13)

  Polar form:

  (F.14)

  Complex number division:

  Rectangular form:

  (F.15)

  Polar form:

  (F.16)

  Trigonometric identities:

  (F.17)

  (F.18)

  (F.19)

  (F.20)

  (F.21)

  (F.22)

  (E.23)

  (F.24)

  (F.25)

  (F.26)

  (F.27)

  (F.28)

  (F.29)

  Series of exponentials:

  (F.30)

  (F.31)

  (F.32)

  (F.33)

  L’Hospital’s rule :

  If results in the undetermined form or , then

  (F.34)

  where and .

  Solution of the quadratic equation: For a quadratic equation expressed as

  (F.35)

  the solution is given by

  (F.36)

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