Operation with Complex Numbers

11 Key excerpts on "Operation with Complex Numbers"

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  Mathematics N4 Student's Book

  TVET FIRST

  • SA Chuturgoon(Author)
  • 2022(Publication Date)
  • Macmillan

    (Publisher)

  We can use complex numbers to solve engineering problems. For example, in electrical engineering we use them to describe reactance in AC circuits as well as active and reactive components of alternating-current quantities. In this module, we will learn about the different forms of complex numbers, how to manipulate them and how to use them to solve engineering problems. Complex numbers Operations on complex numbers in standard form: ● Addition ● Subtraction ● Multiplication ● Division (using the conjugate) Conjugate of complex numbers ● Definition ● Finding the conjugate 2.2 Performing the four mathematical operations using complex numbers Definition of imaginary numbers Definition of complex numbers Rectangular form of complex numbers Parts of a complex number (real and imaginary) Determining the square root of negative numbers Simplifying powers of i Solving quadratic equations with complex roots 2.1 Solving quadratic equations that have complex roots Drawing Argand diagrams Definition of modulus Definition of argument Modulus and argument on the Argand diagram Calculating the modulus and argument Acute angle α 2.3 Representing complex numbers on an Argand diagram Polar form Converting between forms using the analytical method: ● Rectangular to polar form ● Polar to rectangular form Real-life applications Converting between forms using a calculator 2.4 Converting between the rectangular and the polar form De Moivre’s theorem Applying De Moivre’s theorem: ● Products ● Quotients ● Powers 2.5 Applying De Moivre’s theorem to complex numbers Procedure Multiplication and division 2.6 Solving complex equations Figure 2.1: Alternating electrical quantities such as voltage and current can be expressed as complex numbers Note The full Learning Outcomes for each module are listed in the table at the back of the book.

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  Complex Analysis & Key Mathematical Concepts (Concepts & Applications)

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  Elementary operations Addition and subtraction Addition of two complex numbers can be done geometrically by constructing a parallelogram. ________________________ WORLD TECHNOLOGIES ________________________ Complex numbers are added by adding the real and imaginary parts of the summands. That is to say: Similarly, subtraction is defined by Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B , interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are 0, A and B . Equivalently, X is the point such that the triangles with vertices 0, A , B , and X , B , A , are congruent. Multiplication and division The multiplication of two complex numbers is defined by the following formula: In particular, the square of the imaginary unit is −1: A motivation for the latter equation is given below. The preceding definition of multiplication of general complex numbers is the natural way of extending this fundamental property of the imaginary unit. Indeed, treating i as a variable, the formula follows from this (distributive law) (commutative law of addition—the order of the summands can be changed) (commutative law of multiplication—the order of the factors can be changed (fundamental property of the imaginary unit). The division of two complex numbers is defined by the following formula: The real and imaginary part ( c and d , respectively) of the denominator must not both be zero for the division to be defined. Division is defined in this way in order because the ________________________ WORLD TECHNOLOGIES ________________________ product of the right hand expression with c + di (using the previous formula for multiplication) is a + bi . Thus, dividing a + bi by c + di and then multiplying it with c + di again gives back a + bi , as is familiar from real or rational numbers.

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  Quantum Computing for Computer Scientists

  • Noson S. Yanofsky, Mirco A. Mannucci(Authors)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  These fellows, being akin to i , are known as imaginary numbers. But there is more: add a real number and an imaginary number, for instance, 3 + 5 × i , and you get a number that is neither a real nor an imaginary. Such a number, being a hybrid entity, is rightfully called a complex number. Definition 1.1.1 A complex number is an expression c = a + b × i = a + bi , (1.5) where a, b are two real numbers; a is called the real part of c, whereas b is its imaginary part. The set of all complex numbers will be denoted as C. When the × is understood, we shall omit it. Complex numbers can be added and multiplied, as shown next. Example 1.1.2 Let c 1 = 3 − i and c 2 = 1 + 4i . We want to compute c 1 + c 2 and c 1 × c 2 . c 1 + c 2 = 3 − i + 1 + 4i = (3 + 1) + (−1 + 4)i = 4 + 3i . (1.6) Multiplying is not as easy. We must remember to multiply each term of the first complex number with each term of the second complex number. Also, remember that i 2 = −1. c 1 × c 2 = (3 − i ) × (1 + 4i ) = (3 × 1) + (3 × 4i ) + (−i × 1) + (−i × 4i ) = (3 + 4) + (−1 + 12)i = 7 + 11i . (1.7)  Exercise 1.1.3 Let c 1 = −3 + i and c 2 = 2 − 4i . Calculate c 1 + c 2 and c 1 × c 2 .  With addition and multiplication we can get all polynomials. We set out to find a solution for Equation (1.1); it turns out that complex numbers are enough to provide solutions for all polynomial equations. Proposition 1.1.1 (Fundamental Theorem of Algebra). Every polynomial equa- tion of one variable with complex coefficients has a complex solution. Exercise 1.1.4 Verify that the complex number −1 + i is a solution for the polyno- mial equation x 2 + 2x + 2 = 0.  This nontrivial result shows that complex numbers are well worth our attention. In the next two sections, we explore the complex kingdom a little further. Programming Drill 1.1.1 Write a program that accepts two complex numbers and outputs their sum and their product.

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  Mathematics NQF4 SB

  TVET FIRST

  • M Van Rensburg, I Mapaling M Trollope(Authors)
  • 2017(Publication Date)
  • Macmillan

    (Publisher)

  1 Module 1 Topic 1: Complex numbers Work with complex numbers Module 1 Overview By the end of this module you should be able to: • Unit 1.1: Perform addition, subtraction, multiplication and division on complex numbers in standard form. • Unit 1.2: Perform multiplication and division on complex numbers in polar form. • Unit 1.3: Use De Moivre’s Theorem to raise complex numbers to powers (excluding fractional powers). • Unit 1.4: Convert the form of complex numbers where needed so that you can do advanced operations on complex numbers (a combination of standard and polar form may be assessed in one expression). Introduction In Level 3 you were introduced to imaginary numbers and complex numbers. You also learned that a complex number is a single quantity that can be represented in standard form or rectangular form as: z = a + bi where: l z represents the complex number. l a and b are real numbers. l i represents the imaginary number. l a is the real part. l bi is the imaginary part, where b tells you how many i units there are in the complex number. Imaginary numbers The square root of a negative number does not have a solution, but we can use imaginary numbers to express these quantities as the product of i and a real number. How to calculate with i When you calculate with i , the same rules of algebra apply as to an ordinary variable such as x : l When you add or subtract imaginary numbers, you add or subtract like terms, for example: ¢ 1 i + 1 i = 2 i ¢ 9 i − 3 i = 6 i ¢ 1 i + 7 i = 8 i ¢ 3 i + 2 i − 8 i = − 3 i complex number: a single quantity that consists of a real part and an imaginary part standard form: the standard form of a complex number is z = a + bi ; also called the rectangular form New words Note: Both the real numbers in the complex number can be 0. In other words, a and b can be 0, for example: 0 + 0 i .

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  Trigonometry

  • Ron Larson(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4.1 Complex Numbers 315 Operations with Complex Numbers To add (or subtract) two complex numbers, add (or subtract) the real and imaginary parts of the numbers separately. Addition and Subtraction of Complex Numbers For two complex numbers a + bi and c + di written in standard form, the sum and difference are Sum: (a + bi) + (c + di) = (a + c) + (b + d)i Difference: (a + bi) - (c + di) = (a - c) + (b - d)i. The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the additive inverse of the complex number a + bi is -(a + bi) = -a - bi. Additive inverse So, you have (a + bi) + (-a - bi) = 0 + 0i = 0. Adding and Subtracting Complex Numbers a. (4 + 7i) + (1 - 6i) = 4 + 7i + 1 - 6i Remove parentheses. = (4 + 1) + (7 - 6)i Group like terms. = 5 + i Write in standard form. b. (1 + 2i) + (3 - 2i) = 1 + 2i + 3 - 2i Remove parentheses. = (1 + 3) + (2 - 2)i Group like terms. = 4 + 0i Simplify. = 4 Write in standard form. c. 3i - (-2 + 3i) - (2 + 5i) = 3i + 2 - 3i - 2 - 5i = (2 - 2) + (3 - 3 - 5)i = 0 - 5i = -5i d. (3 + 2i) + (4 - i) - (7 + i) = 3 + 2i + 4 - i - 7 - i = (3 + 4 - 7) + (2 - 1 - 1)i = 0 + 0i = 0 Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Perform each operation and write the result in standard form. a. (7 + 3i) + (5 - 4i) b. (3 + 4i) - (5 - 3i) c. 2i + (-3 - 4i) - (-3 - 3i) d. (5 - 3i) + (3 + 5i) - (8 + 2i) REMARK Note that the sum of two complex numbers can be a real number. The fast Fourier transform (FFT), which has important applications in digital signal processing, involves operations with complex numbers.

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  Complex Numbers and Geometry

  • Liang-shin Hahn(Author)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  For example, a ib c id a± c) i b d , ( a ib · ( c id = ac ibc iad i bd = ac -bd i bc ad . omplex um rs The division of two complex numbers, a ib) / e id), involves fi nding a complex number x iy satisfying a+ ib e +id)• x iy). Hence, by the above computation, we get a+ ib (ex -dy) i(dx cy and so it is sufficient to find x and y satisfy ing ex -dy a, dx cy This system of s imultaneous equations has a u nique solutio n ac bd x c2+J2 unless e d 0. Hence be-ad y c2+J2 a ib ac bd be -ad i --­ c id c2 J2 c 2 J2 . Of course, this can be obtained by multip lying both the numerator and the denominator by c -id. But why are suc h operations justified? Isn t the addition of a real number a and an imaginary number ib to get a+ ib similar to the addition of 17m and kg to get 21 Also, x 1 0 has two solutions, and which one of them is i? Note that x 1 0 also has two solutions, the positive one is 1 and the other -1. But is it meaningful to say i is positive? Definition of Complex Numbers To answer the criticism at the end of t he last section, we now give a formal definition of co mplex numbers. B ut first let us recall the prope rties of the real number system R OMPLEX NUM ERS ND GEOMETRY I Properties concern ing add ition. Two arbitrary real numbers a and b uniquely determ ine a th ird num­ ber called th eir sum, denoted by a b, with the following propert ies: A1. Commuta tive law: a b b a for all a, b E JR. A • Associative l aw: a b c a b c for all a, b, c E JR. A Additive ident ity: There is a unique real number, de noted 0, such that a 0 = 0 a = a for all a E JR. A Ad ditive inverse: For every a E JR, there is a un ique x E JR satisfying a x x a 0. This unique solution will be deno ted by -a. II Prope rties concerning multipl icat ion. 1wo arbitrary real numbers a and b uniquely determ ine a th ird num­ ber called their product, denoted by ab, with the following proper ties: M1. Commuta tive law: ab ba for all a, b E JR. M2. As sociat ive law: ab c a bc for all a, b, c E JR.

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  Classical Complex Analysis

  A Geometric Approach(Volume 1)

  • I-Hsiung Lin(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  CHAPTER 1 Complex Numbers This opening chapter is to introduce the basic knowledge about the complex numbers in three aspects and the interactions among them. Sketch of the Content Algebraic : The imaginary solution i = √ − 1 of the equation t 2 + 1 = 0 creates the complex numbers z = x + iy which obey the same basic laws of arithmetics as the reals do (Sec. 1.3). The main distinction between them lies on the fact that the complexes cannot be ordered, but compensated by the conjugate operation z → ¯ z (Sec. 1.4.1). This enables us to interrelate both by the operation | z | 2 = z ¯ z and to introduce the inequalities among various | z | (Secs. 1.4.1 and 1.4.2). An instant consequence is that every polynomial with complex coefficients is always solvable; in particular, z n − 1 = 0 has exactly n -distinct roots (Sec. 1.5). Geometric : z = x + iy can be understood not only as a point or a vector in the Euclidean plane R 2 , but also as a planar motion composed of one-way stretch and rotation (Secs. 1.1, 1.2, and 1.4). It comes naturally that z = x + iy can be represented as the point ( x, y ) in R 2 which is thus renamed as the complex plane C , or in the polar form re iθ ( r = | z | , θ = arg z ) with the origin 0 as pole and the positive x -axis as the polar axis. And then, they can be used effectively to describe planar geometric objects or to solve geometric problems (Secs. 1.4.3 and 1.4.4). Therefore, almost every algebraic operation about complex numbers has its illuminative geometric meaning. Topological (point-set properties): Owing to | x | , | y | ≤ | z | = x 2 + y 2 ≤ | x | + | y | ( z = x + iy ), the limit concepts of real sequences and series and their properties can be carried verbatim over to the complex ones except the one appeared in (3) of (1.7.3), where arg z n → arg z 0 needs to be treated carefully. So do the concepts of open, closed, compact, and connected sets for both R 2 and C (Sec. 1.8). The addition of the point at infinity ∞ to C 1

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  Complex Analysis

  • Theral O Moore, Edwin H Hadlock;;;(Authors)
  • 1991(Publication Date)
  • WSPC

    (Publisher)

  Chapter 1 Complex Numbers 1.1 Complex Numbers In the calculus, it became clear that there is a very important structure in the real number system other than the algebraic operations of addition and multiplication. The concepts of limit and continuity depend upon this structure so that, indeed, it is basic to all of calculus. (Derivatives, definite integrals and sums of infinite series may be defined as limits.) These limits may be defined in terms of the less than relation (<), called an order relation. We use R to denote the set of all real numbers and to emphasize both the algebraic and limit structures in R, we use to denote the system of real numbers . The student has met complex numbers as roots of quadratic equations with neg-ative discriminants. He probably wrote a complex number z in the form z = a + ib where a and 6 are in R. The complex number z was determined by the ordered pair (a, b) of real numbers, and conversely. This suggests that we may use the notation (a, 6) for the complex number (a + z'6), if we like. In practice, the notation a + ib is desirable, and we shall return to it. But for smoothness in our definitions, we shall use the ordered pair notation. (Also, the student should be introduced to this scheme, since it is often used.) We shall use R 2 to denote the set of all ordered pairs of real numbers where it is understood that if (x, y) and (w, v) are members of # 2 , then (#, y) = (w, v) iff x = u and y = v. (1.1) We use iff as an abbreviation for if and only if throughout this book. 1 2 CHAPTER 1. COMPLEX NUMBERS Definition 1.1.1 The complex number system is the system (R 2 , +, •, | |) where for each pair z = (x, y) and w = (u, v) of members of R 2 y z + w = (x + u, y + v), (1.2) zw = 2 • w = (xu — yv, xv + yu), (1.3) and |z| = y x 2 + y 2 (the non-negative square root). (1.4) In this system, each member z of R 2 is called a complex number, and z is called the absolute value of z or the modulus of z.

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  Guide to Mathematical Methods

  • John Gilbert, Camilla Jordan, David A Towers(Authors)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  CHAPTER 13 Complex numbers By the end of this chapter you will have • been introduced to complex numbers; • learnt how do complex arithmetic; • solved equations using complex numbers; • represented complex numbers as points in the plane; • studied de Moivre’s Theorem. Aims and Objectives 13.1 Introduction Consider the four number sets, N , Z , Q and R . Each one has algebraic short-comings, illustrated by the following table: Equation Solution in: N Z Q R x + 1 = 0 no yes yes yes 2 x − 1 = 0 no no yes yes x 2 − 2 = 0 no no no yes x 2 + 1 = 0 no no no no Each of the sets N , Z , Q , and R includes the previous set and extends the class of equations for which solutions exist. The set C of complex numbers completes the extension to include solutions of the fourth equation, and indeed a much wider class of equations. This is demonstrated in a remarkable theorem, due to Gauss and known as the Fundamental Theorem of Algebra, which proves that every polynomial equation with coefficients in C has a solution in C . 332 Complex numbers 333 The significance of the complex numbers lies not only in the algebraic properties described above. If we think of the set of real numbers as one-dimensional, with the numbers corresponding to points on a line, then the set of complex numbers is two-dimensional with the numbers corresponding to points in a plane. This provides a mathematical tool for modelling physical problems involving two variables, as problems of a single complex variable. This tool is both powerful in its applications and elegant in its purely mathematical nature. We start by investigating the algebraic properties of complex numbers. 13.2 The algebra of complex numbers Complex numbers first appeared in the sixteenth century in the work of the Italian mathematician Bombelli. They arose in connection with the solutions of equations, cubic as well as quadratic, and appeared in the form x + i y (13.1) where x and y are real numbers and i is a ‘number’ whose square is − 1.

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  Maths: A Student's Survival Guide

  A Self-Help Workbook for Science and Engineering Students

  • Jenny Olive(Author)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  exercise 10. d .3 Now have a go at solving these equations yourself. (1) Given that a and b are real numbers, and that z = 1 – 2 j is a root of the equation z 4 – 3 z 3 + az 2 + bz – 30 = 0, find a and b and the other three roots. (2) Given that a and b are real numbers, and that z = j is a root of the equation z 4 + az 3 + bz 2 – 4 z + 13 = 0, find the values of a and b and the other three roots. (3) Given that a and b are real numbers, and that z = 1 – j is a root of the equation z 4 + az 3 + bz 2 + 1 = 0, find a and b and the other three roots. 10.E Finding where z can be if it must fit particular rules In Chapter 3, when we were working with functions, we often found that we had to restrict the choice of possible values for x in order to make the functions work as we wanted. One example of this is given by f ( x ) = 4 – x . To make f ( x ) real, we have to have x ≤ 4. This means that we have to restrict the possible values of x to just one part of the number line which makes up the horizontal axis. The same kind of thing can happen with applications of complex numbers. To make things work in the way that we want, we may often find that we have to restrict the possible values of z . Since z is made up of both an x and y component from its real and imaginary parts, this restriction may lead to the exclusion of any area or region of the complex plane because it may affect the possible values of both x and y . (The complex plane is the flat surface shown on an Argand diagram.) Physical quantities which have both magnitude and direction, and which are acting in a flat surface, can often be represented very conveniently by complex numbers. Examples of such applications are two-dimensional problems involving lines of electric or magnetic force, or streamlines in fluid flow. If the physical quantities you are considering need three dimensions to describe them, complex numbers will no longer be any use.

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  The Shape of Algebra in the Mirrors of Mathematics

  A Visual, Computer-Aided Exploration of Elementary Algebra and Beyond (With CD-ROM)

  • Gabriel Katz, Vladimir Nodelman;;;(Authors)
  • 2011(Publication Date)
  • WSPC

    (Publisher)

  77 Chapter 3 The Complex Numbers and Other Fields The Shape of Algebra 78 Mathematicians, like children, have a few favorite “toys”. They get very attached to them and enjoy the distinctive ways in which their favorites act and interact. Occasionally, mathematicians dream of a better, smarter toy that is capable of performing new, previously impossible tricks. So they design a miracle “toy-theory”. But mathematicians also want the new creation to have the same lovable characteristics shared by the old favorites. This tension between the old and new qualities of the design moves mathematics ahead. Kids often enjoy breaking toys: not only they want to see “what is inside”, but also they enjoy witnessing the failures of their favorites, pushing them to their structural limits. And so do mathematicians. As a mathematician embarks on constructing a new theory, he often thinks: “My old favorite construction is capable of doing A, B and C, but I also want it to do D, E, F, and even more ... Which new requirements can I impose on my construction before it breaks, in other words, before the good old properties A, B, C become destroyed by the new requirements D, E, F, …?”. Across the centuries, the history of constructing new number systems from the old ones is a vivid testimony of this kind of mathematical engineering. First, let us discuss the thinking behind the extension of the numerical universe of integers to the familiar universe of rational numbers . We can perform addition, subtraction and multiplication with integers. These operations obey the usual (lovable) rules of commutativity , associativity and distributivity .

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