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MATH 1021讲解、辅导Java/Python程序

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

. 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 through 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, A\B 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 A\B is not the
same as B\A.
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:
01 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

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:
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
25i 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
(34i)(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.

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