Complex Numbers

12 Key excerpts on "Complex Numbers"

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  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  9 Complex Numbers 9.1 Historical introduction The quadratic equation x 2 + 1 = 0 has no solution in the real-number system because there is no real number whose square is −1. New types of numbers, called Complex Numbers, have been introduced to provide solutions to such equations. In this brief chapter we discuss Complex Numbers and show that they are important in solving algebraic equations and that they have an impact on differential and integral calculus. As early as the 16th century, a symbol √ −1 was introduced to provide solutions of the quadratic equation x 2 + 1 = 0. This symbol, later denoted by the letter i, was regarded as a fictitious or imaginary number which could be manipulated algebraically like an ordinary real number, except that its square was −1. Thus, for example, the quadratic polynomial x 2 + 1 was factored by writing x 2 + 1 = x 2 − i 2 = (x − i)(x + i), and the solutions of the equation x 2 + 1 = 0 were exhibited as x = ±i, without any concern regarding the meaning or validity of such formulas. Expressions such as 2 + 3i were called Complex Numbers, and they were used in a purely formal way for nearly 300 years before they were described in a manner that would be considered satisfactory by present-day standards. Early in the 19th century, Karl Friedrich Gauss (1777−1855) and William Rowan Hamilton (1805−1865) independently and almost simultaneously proposed the idea of defining Complex Numbers as ordered pairs (a, b) of real numbers endowed with certain special properties. This idea is widely accepted today and is described in the next section. 9.2 Definitions and field properties definition. If a and b are real numbers, the pair (a, b) is called a complex number, provided that equality, addition, and multiplication of pairs is defined as follows: (a) Equality: (a, b) = (c, d) means a = c and b = d. (b) Sum: (a, b) + (c, d) = (a + c, b + d). (c) Product: (a, b)(c, d) = (ac − bd, ad + bc).

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  Guide to Mathematical Methods

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

    (Publisher)

  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. The existence and nature of this ‘number’ i gave rise to a certain amount of mystery and scepticism, some of which lingers on through the use of the term ‘imaginary number’. We shall give a definition of a complex number which overcomes the problem of existence. In practice we think of Complex Numbers in the above form and we shall see how our definition is consistent with this. Definition 13.1 A complex number is an ordered pair ( x, y ) of real num-bers, usually written as x + i y . The use of the word ‘ordered’ in the definition is important; the ordered pairs ( x, y ) and ( y, x ) are distinct unless x = y . The set of all Complex Numbers is denoted by C and it is usual to denote a typical member of the set by the single letter z . This is often more convenient in manipulations than the fuller form x + i y . Complex Numbers can be represented as points in the Cartesian plane in an obvious way: x + i y is represented by the point with Cartesian coordinates ( x, y ). For example, the number 2 − 3i (note that we write the second number, -3, in front of the i) is plotted as the point (2 , − 3). We usually refer to the plane as the Argand diagram in this context. Figure 13.1 shows an example, with the point 2 − 3i and other points marked on it. The complex number 334 Guide to Mathematical Methods − 4i − 3i − 2i − i i 2i 3i 4i − 4 − 3 − 2 − 1 1 2 1 4 R I 4 + i − 1 + 4i − 4 − 2i 2 − 3i Figure 13.1: Argand diagram 0 + 1i is usually written as i. If x is a real number then we write it in the form of a complex number as x + 0i. Thus, in the Argand diagram, real numbers appear on the x axis, sometimes called the real axis for this reason.

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  Practical MATLAB Basics for Engineers

  • Misza Kalechman(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  A com-plex number then can be represented in the real/imaginary coordinate system called the complex plane as a point, as shown in Figure 6.1. Thus, each point on the plane represents a complex number and conversely, each complex number represents a point on the plane. The collection of all these points constitutes the complex plane. The complex plane consists of a Cartesian rectangular axes system, where the x -axis (horizontal) of the complex plane is referred as the real axis, where the real part is repre-sented, whereas the y -axis (vertical) is referred as the imaginary axis, where the imagi-nary part is represented (with i as its unit). Observe that the real numbers are just a subset of the Complex Numbers, when the imaginary part is equal to zero. Much of modern mathematics is based on Complex Numbers, and they are used extensively in science and engineering. For example, in electrical circuit theory, when dealing with impedances, the real axis is generally associated with the resistance, whereas the imaginary axis is referred as the reactance axis. Most standard MATLAB ® algebraic manipulations defined for real numbers work with Complex Numbers. There are a few exceptions between real and Complex Numbers, such as 350 Practical MATLAB ® Basics for Engineers the concept of equivalence. For the case of Complex Numbers, the only equivalent relation is the identity. Other relations such as greater than and smaller than have no meaning when dealing with Complex Numbers. For example, two Complex Numbers z 1 = x 1 + jy 1 and z 2 = x 2 + jy 2 are equal, if and only if, x 1 = x 2 and y 1 = y 2 . The concept of one complex number being greater than, or smaller than, another is meaningless. The nature of Complex Numbers can best be illustrated and visualized by analyzing the following set of quadratic equations: 1.

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  Introductory Modern Algebra

  A Historical Approach

  • Saul Stahl(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  real. These Complex Numbers can be added and subtracted as polynomials. Thus,

  The multiplication of Complex Numbers also resembles that of polynomials, except that each occurrence of i2 is replaced by −1. Thus,

  The division of Complex Numbers mimics the well-known process of rationalizing denominators. Thus,

  Surprisingly, all of these arithmetical operations can be given very interesting visual, or geometric, interpretations. To accomplish this, we represent each complex number a + bi by the point (a, b) of the Cartesian plane. The point (a, b) is called the Cartesian representation of the complex number a + bi. Given two Complex Numbers a + bi and c + di, let their Cartesian representations be P = (a, b) and Q = (c, d) (Figure 2.1 ). Their sum

  Figure 2.1

  Complex addition

  is represented by the point R = (a + c, b + d). However,

  and

  Consequently, PR || OQ and QR || OP and so OPRQ is a parallelogram. Thus we see that the addition of Complex Numbers resembles that of vectors. These considerations are summarized as follows.

  Proposition 2.1 Let O denote the origin of the Cartesian plane and let P and Q be the Cartesian representations of the Complex Numbers a + bi and c + di, respectively. If the sum of the two Complex Numbers is represented by the point R, then the quadrilateral OPRQ is a parallelogram.

  To give the multiplication of Complex Numbers a visual interpretation, it is convenient to begin by establishing some conventions. In the sequel, the general complex number a + bi will frequently be abbreviated as z. If either a or b is 0, it is omitted from a + b

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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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  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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  Mathematics NQF3 SB

  TVET FIRST

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

    (Publisher)

  1 Module 1 Topic 1: Complex Numbers Representing Complex Numbers in a form appropriate to the context Module 1 Learning Outcomes This module will show you how to do the following: • Unit 1.1: Write imaginary numbers in i -notation. • Unit 1.2: Simplify and perform addition, subtraction, multiplication and division on single-term imaginary numbers. • Unit 1.3: Construct Argand diagrams to find and represent Complex Numbers in standard/ rectangular form. • Unit 1.4: Represent Complex Numbers in polar form with modulus and positive argument. Figure 1.1: Sunglasses are created using the principles of imaginary and Complex Numbers. Figure 1.2: This small colour CCD spy camera is about 2 cm high and the electronic state in the materials is described by Complex Numbers. Figure 1.3: Mandelbrot fractals are created on a computer using Complex Numbers. Unit 1.1: Writing imaginary numbers in i -notation Pre-knowledge Figure 1.4: A graphical (visual) representation of the number system including Complex Numbers Real numbers ( ℝ ) Rational numbers ( ℚ ) Integers ( ℤ ) Whole numbers ( ℕ 0 ) Natural numbers ( ℕ ) Irrational numbers ( ℚ′ ) 4 __ 5 8 __ 9 0,2 1 2 3 4 5 − 6 − 8 − 1 − 4 ___ 7 √ __ 2 √ __ 3 √ __ 5 π Complex Numbers z = 2 + 3 i 2 Module 1 Table 1.1: Revision of the different types of numbers Type Explanation Example Natural numbers ( ℕ ) All positive numbers used for counting and can be represented by the number line. ℕ = { 1; 2; 3; 4; 5; 6; 7; … } 1 2 3 4 5 6 7 Figure 1.5: Number line for natural numbers Whole numbers ( ℕ 0 ) All natural numbers including zero ℕ 0 = { 0; 1; 2; 3; 4; 5; 6; 7; … } Integers ( ℤ ) Positive and negative numbers (excluding decimals) ℤ = { …; − 3; − 2; − 1; 0; 1; 2; 3; … } Rational numbers ( ℚ ) Any number that can be written as a ratio hence the name rational. It can be written as a common fraction in the form: a __ b ; b ≠ 0. These include whole numbers, terminating decimals and non-terminating recurring decimals.

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  NCV4 Mathematics

  • J Daniels, N Solomon, J Daniels, N Solomon(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  Chapter 1 Complex Numbers After completing this chapter, you will be able to: 1.1 Work with Complex Numbers 1.1.1 Perform addition, subtraction, multiplication and division on Complex Numbers in standard form. 1.1.2 Perform multiplication and division on Complex Numbers in polar form. 1.1.3 Use De Moivre’s theorem to raise Complex Numbers to powers (excluding fractional powers). 1.1.4 Convert the form of Complex Numbers where needed to enable performance of advanced operations on Complex Numbers (a combination of standard and polar form may be assessed in one expression). 1.2 Solve problems using Complex Numbers 1.2.1 Solve identical Complex Numbers in rectangular/standard form using the concept of simultaneous equations. 1.2.2 Use Complex Numbers to solve equations that cannot be solved using the real number system by applying: • Factorisation • Quadratic formula. 2 Mathematics: Hands-On Training Introduction In level 3 you learnt about two new types of numbers: • Imaginary numbers • Complex Numbers. In level 4 you will elaborate on solving problems based on Complex Numbers. Pre-knowledge Below is a summary of the relationships between several types of numbers, illustrated with a flat, two-dimensional Venn diagram. Complex Numbers ( a ± bi or a ± bj ) 2 + 3 i ; 3 – 2 ; –2 + 4 j Real numbers (  ) Rational numbers (  ) Integers (  ) ... –3; –2; –1; 0; 1; 2; 3; ... Whole numbers (  0 ) 0; 1; 2; 3; 4 ... Natural numbers (  ) 1; 2; 3; 4 ... –4 Irrational numbers (  ') 1,414 213... π 2 5 ... ; – ; 0; ; 1 ; – ; 4; 317; ... –8 3 2 3 1 2 –8 3 14 255 Imaginary numbers 3 i − 2 –513 5 4 – Below is a short description of the different types of numbers. Definition of Complex Numbers The complex number system defines sums of real and imaginary numbers, a + bi , where a and b are real numbers: • the imaginary unit is i • a is called the real part • bi is called the imaginary part.

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  Numbers and Proofs

  • Reg Allenby(Author)
  • 1997(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  NUMBERS AND PROOFS sin {§ The above is only the tip of the iceberg. We can go on to consider general functions of a complex variable, their derivatives and their integrals over contours in the complex plane. Such activity has applications in many areas of applied mathematics - even in civil engineering (where it is important in studying fluid seepage under dams). Back in pure mathematics, an estimate for the number of primes less than the integer n can be obtained via the Prime Number Theorem which may be proved by considering complex integrals and logarithms. But these are other stories. Summary Following their introduction by means of a 'problem', Complex Numbers were used informally by many mathematicians before it was deemed necessary to try to put them on a firmer footing. Simply defining a complex number to be 'something of the form a + ib (or a + M) where a and b are real numbers and i represents y/— Γ can lead to (temporary) difficulties - which can be eliminated by using Hamilton's definition of Complex Numbers using ordered pairs. Sticking to the a + ib notation we illustrated the 'trick' for simplifying a fraction of two Complex Numbers, and observed that the Complex Numbers satisfy the basic rules of arithmetic. We then stated the very important Fundamental Theorem of Algebra and deduced that, in each polynomial equation with real coefficients, the roots appear in complex conjugate pairs. The important concepts of modulus and argument were introduced via polar representation of Complex Numbers, and their more prominent properties - in particular the triangle inequality - were established. The neat polar form of complex multiplication leads, naturally, to De Moivre's theorem with its applications to finding roots and obtaining trigonometrical identities.

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  Analysis

  An Introduction

  • Richard Beals(Author)
  • 2004(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 The Real and Complex Numbers The previous chapter was somewhat informal. Starting in this chapter we develop the subject systematically and (usually) in logical order. This does not mean that every step in every chain of reasoning will be written out and referred back to the axioms or to results that have already been established. Such a procedure, though possible, is extremely tedious. The goal, rather, is to include enough results – and enough examples of reasoning – so that it may be clear how the gaps might be filled. 2A. The Real Numbers Our starting point is the real number system IR. This is a set that has two algebraic operations, addition and multiplication, and an order relation < . Let a , b denote arbitrary elements of IR. Addition associates to any pair a , b a real number denoted a + b , while multiplication associates a real number denoted a · b or simply ab . That < is a relation simply means that certain ordered pairs ( a , b ) of elements of IR are selected, and for these pairs (only) we write a < b . These operations and the order relation satisfy the following axioms, or conditions, in which a , b , c denote arbitrary elements of IR. A1 ( a + b ) + c = a + ( b + c ). A2 a + b = b + a . A3 There is an element 0 such that, for all a, a + 0 = a. A4 For each a ∈ IR there is an element − a ∈ IR such that a + ( − a ) = 0 . M1 ( ab ) c = a ( bc ). M2 ab = ba . M3 There is an element 1 = 0 in IR such that, for all a, a · 1 = a. M4 For each a such that a = 0 , there is an element a − 1 ∈ IR such that a · a − 1 = 1 . D ( a + b ) c = ac + bc ; a ( b + c ) = ab + ac . O1 For any a and b, exactly one of the following is true: a < b, b < a, or a = b. O2 If a < b and b < c, then a < c. 15 16 The Real and Complex Numbers O3 If a < b, then a + c < b + c. O4 If a < b and 0 < c, then ac < bc. O5 If 0 < a and 0 < b, then there is a positive integer n such that b < a + a + a + ··· + a (n summands).

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  Complex Numbers in Geometry

  • I. M. Yaglom, Henry Booker, D. Allan Bromley, Nicholas DeClaris(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R I I Geometrical Interpretation of Complex Numbers §7. Ordinary Complex Numbers as Points of a Plane 18a The development of the theory of Complex Numbers is very closely connected with the geometrical interpretation of ordinary Complex Numbers as points of a plane, which apparently was first mentioned by the Danish surveyor K. Wessel (1745-1818) but became widely known chiefly through the works of the famous mathematicians K. F. Gauss (1777-1855) and A. Cauchy (1789-1857). This interpretation arises from the fact that the point of a plane with rectangular cartesian coordinates x and y or polar coordinates r and φ corresponds to the complex number (see Figure 1): z = x + iy = r(cos φ -- i sin ψ) Here, obviously, real numbers z = x + 0 · i = r(cos 0 + i sin 0) correspond to points of the x axis, the real axis o numbers of modulus r = 1 correspond to points of the circle S with center at O and radius 1, the unit circle. Opposite Complex Numbers z = x + iy and — z = —x — iy correspond to points symmet-rical about the point O (the number 0 corresponds to the origin isa Th e contents of Sections 7, 8, 13 and 14 have many points of contact with the books of R. Deaux, Introduction to the Geometry of Complex Numbers (F. Ungar Publishing Co.), 1956 and H. Schwerdtfeger, Geometry of Complex Numbers (University of Toronto, Oliver and Boyd), 1962; the latter book also contains much material which could supplement the contents of Section 11 of the present book. 26 §7. Ordinary Complex Numbers as Points of a Plane 27 »' r , - x l ...iïz FIG. 1 O); conjugate Complex Numbers z = x + iy = r(cos φ + i sin φ) and z = x — iy = r[cos(— φ) + ¿ sin(— φ)] correspond to points symmetrical about the line o (throughout this book we shall consider a line to mean a straight line).

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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)

  Two Complex Numbers are equal if and only if each of their real and imaginary parts is separately equal. We can show that this is true in the following way. Let z 1 = a + bj and z 2 = c + dj . Then, if a = c and b = d , it is certainly true that z 1 = z 2 . Now we have to show that if z 1 = z 2 then a = c and b = d . We can see that this must be true geometrically because Complex Numbers have direction as well as length, and therefore the only way that two of them can be equal, and so lie exactly on top of each other, is if their real and imaginary parts are separately equal. We can also see this rather nicely in the following way, using algebra. If z 1 = z 2 then a + bj = c + dj so a – c = dj – bj = j ( d – b ). Squaring both sides of this equation gives ( a – c ) 2 = j 2 ( d – b ) 2 so ( a – c ) 2 = –( d – b ) 2 . Therefore ( a – c ) 2 + ( d – b ) 2 = 0. But remember that a , b , c , and d are all real numbers. Because of this, we can say that ( a – c ) 2 ≥ 0 and ( d – b ) 2 ≥ 0. Therefore, the only way for it to be possible that ( a – c ) 2 + ( d – b ) 2 = 0 is that each of ( a – c ) and ( d – b ) are equal to zero. Therefore a = c and d = b . This property of Complex Numbers is of huge importance in their application to physical situations, because it means that any equation involving Complex Numbers is actually made up of two separate equations. We are, in a sense, getting two for the price of one. example (1) To see an example of this in action, we will solve the equation z 2 = 5 + 12 j . Let z = a + bj . (We know the solution must be complex because its square is a complex number.) We have ( a + bj ) 2 = 5 + 12 j so a 2 – b 2 + 2 abj = 5 + 12 j . This can only be true if both the real parts and the imaginary parts are separately equal. Equating the real parts gives a 2 – b 2 = 5 (1) Equating the imaginary parts gives 2 ab = 12 (2) 438 Complex Numbers

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