School of Mathematics and Statistics MATH 1021 Calculus of One Variable 2003–2021 Revised February 2021 Table of contents Acknowledgements 1 Introduction 2 1 Real and Complex Numbers 5 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Number Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 The real number line – Intervals . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Complex numbers – Cartesian form . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Arithmetic in Cartesian form . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 The set of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Polar Forms of Complex Numbers 26 2.1 Standard Polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Polar exponential form – Euler’s formula . . . . . . . . . . . . . . . . . . . . 33 2.3 Arithmetic in polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5 Roots of polynomial equations . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6 Sine and cosine in terms of exponentials . . . . . . . . . . . . . . . . . . . . 45 2.7 Complex exponential function . . . . . . . . . . . . . . . . . . . . . . . . . 46 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Functions 52 3.1 Functions – definitions and examples . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Combining functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Injective and inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 Hyperbolic functions and their inverses . . . . . . . . . . . . . . . . . . . . 63 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4 Limits and Continuity 70 4.1 Informal definition of limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 One-sided limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3 The basic limit laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Limits at infinity – Horizontal asymptotes . . . . . . . . . . . . . . . . . . . 76 4.5 Infinite limits – Vertical asymptotes . . . . . . . . . . . . . . . . . . . . . . 78 iii 4.6 The squeeze law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.7 Continuous and discontinuous functions . . . . . . . . . . . . . . . . . . . . 82 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5 Differentiation 90 5.1 The derivative at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2 The derivative as a function . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3 Basic rules of differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 Implicit differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.6 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6 Applications of Differentiation 103 6.1 Optimizing functions of one variable . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Increasing and decreasing functions . . . . . . . . . . . . . . . . . . . . . . 107 6.3 Concavity and points of inflection . . . . . . . . . . . . . . . . . . . . . . . 111 6.4 Curve sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.5 L’H pital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7 Taylor Polynomials 123 7.1 An approximation for ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2 Taylor polynomials about x= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.3 Taylor polynomials about x= a . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4 Taylor’s formula – The remainder term . . . . . . . . . . . . . . . . . . . . . 131 7.5 How good is the Taylor polynomial approximation . . . . . . . . . . . . . . 132 7.6 Proof of the remainder formula . . . . . . . . . . . . . . . . . . . . . . . . . 134 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8 Taylor Series 137 8.1 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.2 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.3 Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.4 The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.5 A series for the inverse tan function . . . . . . . . . . . . . . . . . . . . . . 148 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9 The Riemann Integral 151 9.1 Riemann sums – The area problem . . . . . . . . . . . . . . . . . . . . . . . 151 9.2 The Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.3 Calculating Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 9.4 Properties of the Riemann integral . . . . . . . . . . . . . . . . . . . . . . . 159 iv Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 10 Fundamental Theorem of Calculus 164 10.1 Integrals as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 10.2 The Fundamental Theorem of Calculus I . . . . . . . . . . . . . . . . . . . . 166 10.3 The Fundamental Theorem of Calculus II . . . . . . . . . . . . . . . . . . . 167 10.4 Leibniz Integral Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 10.5 The natural logarithm and exponential functions . . . . . . . . . . . . . . . . 171 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 11 Integration Techniques 178 11.1 Basic rules of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 11.2 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11.3 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 11.4 Partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 12 Applications of Integration 193 12.1 Further integration techniques . . . . . . . . . . . . . . . . . . . . . . . . . 193 12.2 Length of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 12.3 Area between two curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 12.4 Solids of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 A Formal Definition of Limits 208 B Geometric proof that limx→0 sinx/x= 1 212 C Linear approximations and differentials 214 D The Distance Problem 219 E Growth Rates 223 F Table of Standard Integrals 225 G Answers to Selected Exercises 226 v Acknowledgements The material in these notes has been developed over many years by the following members of the School of Mathematics and Statistics: Eduardo Altmann Mary Myerscough Sandra Britton Nigel O’Brian Chris Durrant Sharon Stephen Dave Galloway Fernando Viera Jenny Henderson Haotian Wu Andrew Mathas 1 Introduction Calculus is one of the major achievements of the 17th century. It plays a key role in almost every instance where mathematics is applied in the sciences, engineering, or in economics. Without calculus, we would not have cars, computers, televisions or mobile phones; Einstein would never have penned his theory of relativity; we would not know of the existence of DNA; we would never have landed on the moon. The list goes on. Whether you end up continuing in mathematics or majoring in another field it will be im- portant for you to learn and understand the meaning of calculus. The reason for this is quite simple. In high school you can progress simply by memorizing formulas; in university there will be times when you need to develop formulas for yourself, and this is where a proper understanding of calculus will be a definite asset. These notes are intended to supplement the lectures ofMATH1021. Your lecturers will almost certainly use different examples and they will also explain some of the material in the course slightly differently from these notes. In some places these notes go into more detail than your lectures; at other times your lecturer will go into more detail. Reading mathematics is not like reading a novel; we have to think and struggle with every sentence. We are professional mathematicians and we are not ashamed to say that in our research there have been times when we have spent more than a day trying to understand a single line of mathematics! You will be pleased to know that in this course you should not have to spend this long on a single sentence; however, there will be times when you do have to think quite hard to understand what is going on. If you do get stuck then go and ask your lecturer or tutor to explain it to you! In addition to thinking when you read mathematics you should also work through the calcu- lations yourself using pen and paper. At some places in the notes and in the Appendices we have included material which is more “advanced” than we expect you to know or understand. You are free to either read these sections or skip over them, as you wish. Tutorial problems and Exercise sheets There are plenty of problems with full solutions for you to practice. a) Worked examples with full solutions have been included in these lecture notes throughout all chapters. b) Exercises are available at the end of the chapters in these notes and answers to Selected Exercises can be found in Appendix G. 2 Introduction 3 c) Exercise sheets containing problems to be solved before the tutorial session are avail- able on the MATH1021 web page. Full solutions will be available online at the end of the corresponding week. A detailed list of mathematical objectives (knowledge, understanding and skills) for a given chapter is provided in the weekly Exercise Sheets. d) Board tutorial sheets will be handed out during tutorials with problems to be solved during the tutorial class. Full solutions will be available online at the end of the corre- sponding week. e) Solutions will be provided to assignments 1 and 2. f) Questions and solutions to selected past exam papers will be made available near the end of semester. Why study mathematics The study of mathematics enhances your ability to think logically and an- alytically, move from the particular to the general, work quantitatively and improve problem-solving skills. By reading and working carefully through the material in these notes you will develop the following additional generic skills: Generalise simple and familiar ideas to more complex settings. Use geometric/visual techniques to help understand new concepts. Apply simple techniques in unfamiliar situations. Estimate values by using suitable approximation techniques. Recognise that bounds on the error are an important part of any good approximation. A note about definitions A mathematical definition is a precise description of some mathematical con- cept. Historically, many concepts in mathematics have been used extensively before a precise definition of the concept has been formulated. While precision in definitions is certainly important, learning a definition off by heart, without an understanding of the concept, is unlikely to be helpful. It is important to spend some time thinking about a definition in order to gain this understanding. 4 MATH 1021 Calculus of One Variable C H A P T E R 1 Real and Complex Numbers Mathematics includes not only the study of logic, structure and geometry, but also ideas about numbers. Real numbers in particular, are fundamental to calculus and many other branches of mathematics. In this chapter we review the concepts of sets and extend previous work on numbers, particularly the real numbers, before introducing the set of complex numbers. 1.1 Sets Set notation is a convenient and precise way to write about collections of numbers. We start by talking about general sets. Definition A “set” is a collection of objects which are called “members” or “elements” of the set. Example 1.1a A set can be written as a list, for example, A= {a,b,c,d}, where A is the name of the set, a,b,c,d are the elements of the set enclosed in braces and separated by commas. If the list of elements is large, three dots may be used to mean ’and so on’. For example, the set of natural numbers may be denoted by N= {0,1,2,3, . . .}. 1.2 Number Sets Our understanding of numbers, what they are and how they work, develops from simple counting through fractions and negative numbers to an appreciation of irrational numbers and real numbers. Mathematically, different types of numbers belong to different sets. 5 6 MATH 1021 Calculus of One Variable The set of “natural numbers” {0,1,2,3,4, . . .}, is denoted by the symbol N. It is closed under the operations of addition and multiplication. That is, adding two natural numbers gives another natural number, as does multiplying them together. The set of “integers” {. . . , 4, 3, 2, 1,0,1,2,3,4, . . .}, denoted by Z, is the set of whole numbers, including both positive whole numbers, negative whole numbers and zero. The set of integers is closed under the operations of addition, subtraction and multiplication. The set of “rational numbers”, denoted by Q, is the set of all numbers of the form n/m where n and m are integers and m 6= 0. Some examples are 1 2 , 1 4 , 4 3 . Rational numbers include decimals which either terminate or repeat. Note that the integers are a subset of the rational numbers, since they are of the form n/m where m = 1. The set of rational numbers is closed under the operations of addition, subtraction, multiplication and division, provided that division by zero is excluded. The set of “real numbers”, denoted by R, includes all rational numbers and all irrational numbers. Irrational numbers cannot be expressed as n/m, where m and n are integers, al- though some may be interpreted geometrically. For example, √ 2 is the length of a diagonal of a unit square. The irrational number pi is the ratio of the circumference of a circle to the circle’s diameter. The set of “complex numbers”, denoted by C, contains all the other number sets mentioned above and all the imaginary numbers to be introduced in Section 1.4. In fact we can summarise these numbers sets diagrammatically as shown in Figure 1.1. Natural Numbers, N 0, 1, 2, 3, . . . Integers, Z . . . , 2, 1, Rational Numbers, Q eg. 1 2 , 4 3 , 0.1, 7.2, 0.3˙ Real Numbers, R eg. pi, e, √5, 0.1010010001 . . . Complex Numbers, C eg. 1+2 i, 3 i . . . , with i2 = 1 Figure 1.1: Number Sets Chapter 1: Real and Complex Numbers 7 Set notation Element of a set – The symbol ∈ means “is an element of”. For example, 3 ∈ Z is read as “ 3 is an element of the set of integers”; y ∈ B is read as “y is an element of the set B” or “y is a member of the set B”. Subset of a set – The symbol should be read as “is a subset of”. For example, N Z is read as “the set of natural numbers is a subset of the set of integers” or “the set of natural numbers is contained in the set of integers”. Strictly a subset – Sometimes you may see the symbol which means that the smaller set is strictly a subset of the larger; the two cannot be equal. For example, it is most precise to write N Z as the two sets are not the same. Contains a set – The reversed symbol means “contains”. For example, R Q reads “the set of real numbers contains the set of rational numbers”. (There is also a symbol which means “contains, but is not equal to”.) If A B then B A. Not an element of a set – A forward slash thr
ough any of these symbols above means “not”. For example, 1 6∈ N is read as “ 1 is not an element of the set of natural numbers”. Not a subset – Another example, R 6 Z, is read as “the set of real numbers is not a subset of the set of integers.” There are other symbols which describe sets formed from other sets: Union of sets – The expression A∪B denotes the union of set Awith set B. The “union” of two sets is the set of elements which are members of either one or both of the sets. If an element occurs in both sets, it is only listed once in the union. For example {1,2,3,4,}∪{3,4,5,6}= {1,2,3,4,5,6} Intersection of sets – The intersection of sets A and B is written A∩B. The “inter- section” of two sets is the set of elements which are members of both of the sets. For example {1,2,3,4,}∩{3,4,5,6}= {3,4} Subtraction of sets – The symbol which is read “minus” or “without”, is used to indicate the set of elements which are in one set but not in another. That is, AB is the set of all elements which are in A but not in B. So for example, {1,2,3,4,}{3,4,5,6}= {1,2}. 8 MATH 1021 Calculus of One Variable Venn Diagrams A set can be represented in a simple, graphical way by a “Venn diagram”. Each set is drawn as a circle, a square or some other closed shape. Shapes representing sets may overlap one another if sets have elements in common. Sometimes, the elements of the sets are written on the Venn diagram but often they are not. Different parts of a Venn diagram can be shaded to illustrate different parts of the set. Venn diagrams are a useful way to represent relations between sets. Note that AB is not the same as BA. A B A∪B A B AB A B A∩B A B BA Conditions on Sets If we want to specify a set whose elements fulfil a certain condition then we do this in the way illustrated in the following examples. If we want to express that “A is the set of all rational numbers x such that x is positive”, we write A= {x ∈Q | x> 0}. The vertical slash should be read as such that. LetW be the set of words in English. Then B= {x inW |x begins with the letter “P”} reads “B is the set of all elements x of the set of English words such that x begins with P” or “B is the set of all English words that begin with P.” Chapter 1: Real and Complex Numbers 9 If we want to say that “C is the set of all integers x such that x/2 is an integer” or “C is the set of all even integers”, we write C = {x ∈ Z | x 2 ∈ Z}. If we want to say that “D is the set of all real numbers which are greater than 1 and less or equal to 1”, we write D= {x ∈ R | 1< x≤ 1}. 1.3 The real number line – Intervals The real number line – Every real number can be located on the "real number line". For example: 0 1 3 2 pi 4 It is straightforward to sketch sets that are written using interval notation on the real number line. Note that an open dot is used if the end point of the interval is not included in the set. If the endpoint is part of the set, then a closed dot is used. Interval notation – Sets of real numbers which lie between two end points can be represented using "interval notation". For example D= {x ∈ R | 1< x≤ 1}= ( 1,1] A curved bracket is used to show that an endpoint (such as 1 in this example) is not included in the set and a square bracket is used when the endpoint is part of the set. Open interval – An interval where neither endpoint is part of the set is called an "open interval". a b (a, b) = {x ∈ R | a< x< b} The interval (a,b) = {x ∈R | a< x< b} and the point (a,b) in the Cartesian plane are written in exactly the same way. They are not, however, the same thing. It is usually clear from the context whether (a,b) represents a point or an interval. Closed interval – If both endpoints are part of the interval it is called a "closed inter- val". 10 MATH 1021 Calculus of One Variable a b [a, b] = {x ∈ R | a≤ x≤ b} It is also possible that one endpoint will be in the set and the other will not be. For example, a b (a, b] = {x ∈ R | a< x≤ b} a b [a, b) = {x ∈ R | a≤ x< b} Semi-infinite intervals – There is special notation for sets of the number line that extend infinitely in one direction or the other. (a,∞) = {x ∈ R | x> a}; ( ∞,a) = {x ∈ R | x< a} [a,∞) = {x ∈ R | x≥ a}; ( ∞,a] = {x ∈ R | x≤ a} Note that ∞ is not a number, rather, it represents infinity. Both ∞ and ∞ always take a round bracket. Examples 1.3a i) A= [7,29] = {x ∈ R | 7≤ x≤ 29} 0 7 29 [7, 29] ii) S= (2,∞) = {x ∈ R | x> 2} 0 2 (2, ∞) iii) V = ( 3, 1)∪ [2,5] = {x ∈ R | 3< x< 1 or 2≤ x≤ 5} 0-3 -1 2 5 ( 3, 1)∪ [2, 5] iv) T = ( ∞,0)∪ (0,∞) = {x ∈ R | x 6= 0} = R{0}. As you can see there may be a number of ways of writing down a set. 0 R{0} Chapter 1: Real and Complex Numbers 11 Modulus or absolute value The "modulus" or "absolute value" |x| of a real number x gives the distance on the real number line from x to zero. The modulus of x is defined in this way: |x|= { x if x≥ 0, x if x< 0. For example |5|= 5 and | 10|= 10. The distance between two numbers on the number line can also be expressed using modulus. The distance between x and y is given by |x y|= |y x|. For example, the distance between 3 and 4 is |3 ( 4)| = |3+ 4| = 7 which is what we intuitively expect to be the distance from 4 to 3. Alternatively we could have written | 4 3|= | 7|= 7. 1.4 Complex numbers - Cartesian form Suppose that you are asked to solve the equation x2+1= 0. Your first response might be to say that there will be two solutions as it is a quadratic equation. Very quickly you might write down the line x2 = 1. At that point you might conclude, correctly, that there are no real solutions to the equation, because in the real number system, we cannot take square roots of negative numbers. But what if we agree that there exists a number x such that x= √ 1 Such a number does indeed exist, although it is not a real number. It is known as an "imagi- nary number". We denote it by i (although some branches of engineering use j instead) and we’ll assume that the usual rules for algebraic manipulation apply. Imaginary unit The number denoted by i that satisfies the condition i2 = 1 is called the imaginary unit. It follows that i= √ 1. The equation x2+ 1 = 0 now has two imaginary solutions, namely i and i. To check that x=± i are solutions, substitute into the equation x2+1= (±i)2+1= 1+1= 0. 12 MATH 1021 Calculus of One Variable What about the equation x2+9= 0 In this case x2+9= 0 = x=±√ 9=±√ 1×9=±√ 1 √ 9=±3 i. It is easy to show by substitution into x2+9= 0 that x=±3 i are both solutions. Properties of i – The imaginary unit satisfies the following useful relations: i2 = ( i)2 = 1 i3 = (i2. i) = ( 1. i) = i i4 = (i2. i2) = ( 1).( 1) = 1 i8 = (i4. i4) = (1.1) = 1, and so on. We are now in a position to introduce a new number set: Imaginary numbers Any non–zero real multiple of i is called a purely imaginary number or just imaginary number. The square of an imaginary number is a negative real number. For example 3i, 20i, i/5 and pii are all imaginary numbers, and their squares (3i)2 = 9, ( 20i)2 = 400, ( i/5)2 = 1/25, (pi i)2 = pi2, are all negative real numbers. Complex numbers Suppose now that you are given this equation to solve: x2 4x+5= 0. Completing the square and rearranging gives (x 2)2 = 1; that is, x 2=±i or x = 2± i. These solutions can also be obtained by applying the familiar quadratic formula: x= 4±√16 20 2 = 4±√ 4 2 = 4±√ 1√4 2 = 4±2 i 2 = 2± i. Chapter 1: Real and Complex Numbers 13 These solutions are not purely imaginary, although they do involve an imaginary number. The solutions 2+ i and 2 i are called complex numbers. Cartesian form of a complex number A complex number expressed in the form a+ ib is said to be in Cartesian form. Real numbers are a special case of complex numbers when b= 0. Imaginary numbers are a special case of complex numbers when a= 0. The complex number 2+ i in the above example is written in Cartesian form with a= 2 and b= 1. Real and Imaginary parts Given a complex number in Cartesian form z= a+ ib: The real number a is called the real part of z and we write Re(z) = a. The real number b is called the imaginary part of z and we write Im(z) = b. Examples 1.4a i) If z= 2+ i then Re(z) = 2 and Im(z) = 1. ii) If w= 3+8i then Re(w) = 3 and Im(w) = 8. iii) If z= 1 2 5i then Re(z) = 1 2 and Im(z) = 5. iv) The purely imaginary number z= 7i can be written as z= 0 7i. Therefore Re(z) = 0 and Im(z) = 7. v) For the real number z= 4= 4+0i we have Re(4) = 4 and Im(4) = 0. Quadratic equations If we allow complex numbers as solutions to quadratic equations with real coefficients then every such quadratic equation will always have two solutions, and they will be either both real or both complex. 14 MATH 1021 Calculus of One Variable We can see this in general if we look at the quadratic formula. The solution to the quadratic equation ax2+bx+ c= 0, where a, b and c are reals, is given by x= b± √ b2 4ac 2a . Whether ax2+bx+ c= 0 has (purely) real or complex roots depends on the sign expression b2 4ac which is known as the discriminant of the quadratic. x is { real if b2 4ac≥ 0 complex if b2 4ac< 0. Example 1.4b The solutions of x2+6x+25= 0 must be complex since b2 4ac= 64< 0. Using the quadratic formula, the solutions are found to be 3+4i and 3 4i. These com- plex numbers are related; they are complex conjugates of each other. This will be examined further in the next section. 1.5 Arithmetic in Cartesian form Complex numbers can be added or multiplied together, subtracted from one another or di- vided by one another. Consider two complex numbers z= a+bi and w= c+di. Here the real part of z is a and the imaginary part of z is b; the real part of w is c and the imaginary part of w is d. Addition z+w = (a+bi)+(c+di) = (a+ c)+(b+d)i Rule: Add real parts to real parts and imaginary parts to imaginary parts. Example 1.5a (3 4i)+(1+2i) = 3+1+( 4+2)i = 4 2i Chapter 1: Real and Complex Numbers 15 Subtraction z w = (a+bi) (c+di) = (a c)+(b d)i Rule: Subtract real parts from real parts and imaginary parts from imaginary parts. Example 1.5b (3 4i) (1+2i) = 3 1+( 4 2)i= 2 6i Multiplication zw = (a+bi)(c+di) = ac+adi+bci+(bd)i2 = (ac bd)+(ad+ cb)i Rule: Expand the brackets in the normal way, remembering that i2 can be simplified to 1, and collect terms into real and imaginary parts. Example 1.5c (3 4i)(1+2i) = 3 4i+6i 8i2 = 3+2i+8= 11+2i Complex conjugate and division To divide one complex number by another we have to introduce the complex conjugate of a complex number. Complex conjugate The complex conjugate of the number z = a+ ib is the complex number defined by z= a ib. 16 MATH 1021 Calculus of One Variable The geometric interpretation of the complex conjugate z is the reflection of z about the real axis. The following properties of the complex conjugate can be easily proved from the definition, Properties of conjugates z+w= z+w zw= zw zn = zn If z= a+ ib then zzˉ= (a+ ib)(a ib) = a2+b2. Examples 1.5d i) 3+5i= 3 5i 2 7i= 2+7i ii) Verify the first property of conjugates in the box above when z= 1+2i and w= 3+ i z+w= (1+2i)+(3+ i) = 4+3i= 4 3i z+w= (1+2i)+(3+ i) = (1 2i)+(3 i) = 4 3i iii) If z is a real number then z= z. For example, if z= √ 2 then z= √ 2. iv) If z is a purely imaginary number then z= z. For example 3i= 3i. Division If w 6= 0 then to find z w we multiply both top and bottom by the complex conjugate of w. z w = z w w w = (a+bi) (c+di) (c di) (c di) = ac adi+ cbi (bd)i2 c2 cdi+ cdi d2i2 = (ac+bd)+(cb ad)i c2+d2 = ac+bd c2+d2 + cb ad c2+d2 i This process is similar to rationalising the denominator of a quotient of surds. Multiplying by the complex conjugate of the divisor produces a real number in the denominator and allows the number to be written in the Cartesian form a+bi. Chapter 1: Real and Complex Numbers 17 Example 1.5e 5 10i 1+2i = (5 10i)(1 2i) (1+2i)(1 2i) = 5 20i+20i2 1 2i+2i 4i2 = 15 20i 5 = 3 4i Equality of complex numbers Two complex numbers are equal to each other if and only if both their real and imaginary parts are equal. In other words, if z = a+ bi and w = c+ di, then z = w if and only if a = c and b= d. 1.6 The set of complex numbers We have seen that a real number is a particular type of complex number, one with zero imaginary part. The complex numbers include real numbers and form a set which contains the set of real numbers and hence all of the other number sets we have mentioned. Set of complex numbers The "set of complex numbers" C is the set of all numbers of the form a+ bi where a and b are real numbers and i2 = 1. Using set notation we can write: C= {a+bi | a,b ∈ R, i2 = 1}. As we have shown in Figure 1.1, we also have N Z Q R C . The set of complex numbers, like the set of real numbers, is closed under addition, subtrac- tion, multiplication and division. This means that the sum of two complex numbers is another complex number, and so on. 18 MATH 1021 Calculus of One Variable The set of complex numbers is not ordered Complex numbers lack an important property of the real numbers. The set of real numbers is "ordered"; that is, if we have any two real numbers x and y we can say that either x> y or x< y or x= y. Because of this property we are able to represent real numbers on the real number line. The set of complex numbers is not ordered. Consider the two complex numbers 2 3i and 1+5i. It does not make sense to write 2 3i > 1+5i or 3i < 1+5i. However, we may write Re(2 3i) > Re( 1+5i) and Im(2 3i) < Im( 1+5i) because the real part and the imaginary part of a complex numbers are both real numbers. Because the set of complex numbers is not ordered, complex numbers cannot be represented as points on a line. Instead, complex numbers are represented as points on a plane. The complex plane The "complex plane" or "Argand diagram" allows complex numbers to be represented graph- ically. The horizontal axis in the complex plane is called the "real axis". All real numbers lie on the horizontal axis in the complex plane; positive numbers to the right of the origin, negative numbers to its left. The vertical axis is known as the "imaginary axis". All purely imaginary numbers lie on the vertical axis. Each point in the complex plane corresponds to a single complex number. For example: -1 2 3i 2i 2+3i 2 2i 1+3i Imaginary axis Real axis Chapter 1: Real and Complex Numbers 19 The modulus of a complex number In Section 1.3 we defined the "modulus" or "absolute value" |x| of a real number x as the distance on the real number line from x to zero, given by |x|= { x if x≥ 0, x if x< 0. In the case of complex numbers the modulus is also defined to give the distance from the origin: Modulus For a complex number z= a+bi the modulus is defined by |z|= √ a2+b2. By Pythagoras’ Theorem, this formula gives the distance from z to the origin of the complex plane. b a+ ib |z|= √ a2 + b 2 a b We can express |z| in terms of z and its complex conjugate as follows, z z= (a+bi)(a bi) = a2+b2 = |z|2 = |z|=√z z. Properties of the modulus For all complex numbers z= a+ ib and w= c+ id, we have a) |zw|= |z| |w|, b) |z/w|= |z|/|w|, c) |z+w| ≤ |z|+ |w|. This is called the triangle inequality, d) |z w| ≥ |z| |w|. 20 MATH 1021 Calculus of One Variable To show that (a) is true, we calculate |zw| and |z| |w| separately and show they are equal. |zw|= |(a+ ib)(c+ id)| = |(ac bd)+ i(ad+bc)| = √ (ac bd)2+(ad+bc)2 = √ a2c2+b2d2 2abcd+a2d2+b2c2+2abcd = √ a2c2+b2d2+a2d2+b2c2, while |z| |w|= √ a2+b2 √ c2+d2 = √ a2c2+b2d2+b2c2+a2d2 = |zw|. We will show (c) algebraically (a geometric argument can also be used, and this is left as an exercise). Note that since all quantities are non-negative, proving |z+w| ≤ |z|+ |w| is equivalent to proving |z+w|2 ≤ (|z|+ |w|)2. Now |z+w|2 = |(a+ c)+ i(b+d)|2 = (a+ c)2+(b+d)2 = a2+b2+ c2+d2+2(ac+bd). Similarly, (|z|+ |w|)2 = |z|2+ |w|2+2|z| |w|= a2+b2+ c2+d2+2|z| |w|. So we must prove that ac+bd ≤ |z| |w|. Now |z| |w|= √ a2+b2 √ c2+d2 = √ (a2+b2)(c2+d2) = √ a2c2+b2d2+a2d2+b2c2 = √ (ac+bd)2 2abcd+a2d2+b2c2 = √ (ac+bd)2+(ad bc)2 ≥ √ (ac+bd)2 = |ac+bd| Now every real number k is less than or equal to its own absolute value |k|. Hence ac+bd ≤ |ac+bd| ≤ |z| |w|, and the proof is complete. The proofs of (b) and (d) are left as an exercise. Chapter 1: Real and Complex