Giáo trình Introduction to actuarial and financial mathematical methods 2015

Introduction to
Actuarial and
Financial
Mathematical
Methods


Introduction to
Actuarial and
Financial
Mathematical
Methods

S. J. GARRETT

AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier


Companion website: />

Academic Press is an imprint of Elsevier
125 London Wall, London, EC2Y 5AS, UK
525 B Street, Suite 1800, San Diego, CA 92101-4495, USA
225 Wyman Street, Waltham, MA 02451, USA
The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK
Copyright © 2015 Elsevier Inc. All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical,

including photocopying, recording, or any information storage and retrieval system, without permission in writing
from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies
and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency,
can be found at our website: www.elsevier.com/permissions.
This book and the individual contributions contained in it are protected under copyright by the Publisher (other than
as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new research and experience broaden our
understanding, changes in research methods, professional practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any
information, methods, compounds, or experiments described herein. In using such information or methods they
should be mindful of their own safety and the safety of others, including parties for whom they have a professional
responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for
any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any
use or operation of any methods, products, instructions, or ideas contained in the material herein.
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
For information on all Academic Press publications
visit our website at />Printed and bound in the United States
ISBN: 978-0-12-800156-1

Publisher: Nikki Levy
Acquisition Editor: Scott Bentley
Editorial Project Manager: Susan Ikeda
Production Project Manager: Jason Mitchell
Designer: Mark Rogers

To Yvette, for everything past, present, and future.






PREFACE
Mathematics is a huge subject of fundamental importance to our individual lives and
the collective progress we make in shaping the modern world. The importance of
mathematics is recognized in formal education systems around the world and indeed how
we choose to teach our own children prior to their formal schooling. For example, from
a very early age children are taught their native language in parallel to the fundamentals
of working with numbers: children learn the names of objects and also how to count
these objects.
Despite claims to the contrary, most adults do have considerable mathematical
knowledge and an intuitive understanding of numbers. Irrespective of their educational
choices and natural ability, most people can count and understand simple arithmetical
operations. For example, most know how to check their receipt at the supermarket; that
is, they understand the fundamental concepts of addition and subtraction, even if they
prefer to use a calculator to perform the actual arithmetic. As a further example, given a
distance to travel, most people would intuitively know how to calculate an approximate
time of arrival from an estimate of their average speed.
Given that mathematics is so engrained in our childhood and used in our everyday
adult lives, any book on the practical use of mathematics must begin by drawing a line
that separates the material that is assumed as prerequisite and that which the book wishes
to develop. The correct place to draw this line is difficult to determine and must, of
course, depend on the intended audience of the book. This particular book, as the title
suggests, is intended for people who ultimately wish to study and apply mathematics in

the highly technical areas of actuarial science and finance. It is therefore assumed that
the reader has a prior interest in mathematics that has manifest in some kind of formal
mathematical study to, say, the high-school level at least. It is at the level of high-school
mathematics that the line is drawn for this book.
It may be that high school was a long time ago and the mathematics learned there
has since left you. For this reason I begin by softening what could be a sharp line.
The first chapter on preliminary concepts summarizes some relevant mathematical
terminology that you are likely to have seen before. Chapters 2–6 then proceed to discuss
mathematical concepts and methods that you may also have been familiar with at some
point, possibly at high school or maybe during the early months of an undergraduate
program in a numerate subject. The material in Chapters 1–7 forms Part I which is
intended to give you the foundation for the more technical Part II.
A number of chapters close with a brief section on an example use of the ideas
developed in that chapter within actuarial science. While this book does not aim to

xi


xii

Preface

cover topics in actuarial science in any detail, these sections are included where a taste of
a topic can be given without also providing a significant amount of background material.
Not all chapters include such a section and this reflects either that the application would
be too esoteric or that sufficient “real-world” examples have already been included in
the main material of the chapter.
As you may be aware, mathematics can be an extremely formal and rigorous subject.
While such rigor is essential for the development of new mathematics and its application
to novel areas, there are many instances where formal rigor is a hindrance and a

distraction from the real application and purpose of using the mathematics. Rather than
over complicating the descriptions with excessive technical considerations, the aim of
this book is to present a concise account of the application of mathematical methods that
may be required when studying for actuarial examinations under the Institute and Faculty
of Actuaries (IFoA) or the Society of Actuaries (SoA) in the UK and the USA, respectively.
The book should also be of use for those studying under the CFA Institute, for example,
and many other professional bodies related to finance professionals. The book does not
give any formal proofs of the concepts used, although some attempt will be made to
justify many of the ideas. After studying this book the reader should expect to possess a
well-stocked tool box of mathematical concepts, a practical understanding of when and
how to use each tool, and an intuitive understanding of why the tools work.
The scope of the material discussed in this book has been heavily influenced by
the statements of prerequisite knowledge for commencing studies with the IFoA and
SoA. Certainly the book should be considered as covering all prerequisite material
required for beginning studies with the IFoA and SoA. However, I have gone further and
included some additional topics that, in my experience, students from diverse academic
backgrounds have found useful to refresh during their early studies of actuarial science
and financial mathematics at the postgraduate level.
I am grateful for the many discussions regarding the content of this book with
numerous students on the various actuarial and financial mathematics programs at the
University of Leicester, particularly Marco De Virgillis. I would also like to thank Dr
Jacqueline Butter who provided the additional perspective of someone who has gone
through an actuarial program from a background in physics and entered industry on the
other side.
Writing a book is a lonely task and I would like thank my two sons, Adam and
Matthew, and my wife, Yvette, for giving me the time and space to impose this loneliness
on myself. This book is dedicated to Yvette who is an unfailing supporter of everything
I do.
Professor Stephen Garrett
Leicestershire, UK

Spring 2015


Part One

Fundamental
Mathematics

AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier



CHAPTER 1

Mathematical Language
Contents
1.1 Common Mathematical Notation
1.1.1 Number systems
1.1.2 Mathematical symbols
1.2 More Advanced Notation
1.2.1 Set notation
1.2.2 Interval notation
1.2.3 Quantifiers and statements
1.3 Algebraic Expressions
1.3.1 Equations and identities
1.3.2 An introduction to mathematics on your computer
1.3.3 Inequalities

1.4 Questions

3
3
6
8
8
12
13
14
14
17
18
20

Prerequisite knowledge

Learning objectives

• “School” mathematics
• use of a calculator
• algebraic manipulation
• analytical solution of simple
polynomial expressions
• Familiarity with basic use of Excel

• Define, recognize, and use
• number systems
• mathematical notation including set notation
• bracket notation

• quantifiers
• equations, identities, and inequalities

In this chapter, we state and illustrate the use of common mathematical notation that will
be used without further comment throughout this book. It is assumed that much of this
section will have been familiar to you at some point of your education and is included as
an aide-mémoire. Of course, given that the book will explore many areas of the application
of mathematics, the material presented here may well prove to be incomplete. It should
therefore be considered as an illustration of the level of mathematics that will be assumed
as prerequisite, rather than a definitive list.

1.1 COMMON MATHEMATICAL NOTATION
1.1.1 Number systems
We begin by summarizing the types of numbers that exist. As this book in concerned
with the practical application of mathematics, it should be unsurprising that the set of
Introduction to Actuarial and Financial Mathematical Methods

© 2015 Elsevier Inc.
All rights reserved.

3


4

Introduction to Actuarial and Financial Mathematical Methods

−∞

5

5.2

5.6767

6

7

∞

Figure 1.1 The real number line.

real numbers forms the building blocks of most (but not quite all, see Chapter 8) of what
we will study.
A real number is a value that represents a position along a continuous number line.
For example, numbers 5 and 6 have clear positions on the number line in Figure 1.1
and so are real numbers. The number 5.2 also has a position on the number line, a fifth
between 5 and 6. Going further we see that 5.6767 is also on the line. In fact, we can
keep going and, with a sharp enough pencil, mark a number with any number of decimal
places on the number line. With this intuitive understanding, it should be clear that the
set of all real numbers includes numbers to any number of decimal places and that we
can also freely expand the number line without limit. As illustrated in Figure 1.1, real
numbers can be positive or negative. The set of all real numbers, denoted R, is therefore
seen as the fundamental collection of numbers that we might want to work with in
real-world applications.
As we can in principle define a real number with an infinite number of decimal places,
there is in some sense an “infinity of infinities” of real numbers. It should then be of
no surprise that the set R has many subsets, each with an infinite number of members.
Such subsets include

•
•
•
•
•
•

positive real numbers, R+
negative real numbers, R−
integers, Z
natural numbers, N
rational numbers, Q
irrational numbers, J

The meaning of the terms positive real numbers and negative real numbers should be clear,
although note that 0 is technically neither. You may however need to be reminded that
the integers are the subset of real numbers that are “whole.” For example, 0, −10, and 34
are integers, but −10.1 and 34.8 are not.
The natural numbers are easily understood as the positive integers and zero.1 For
example, 57 and −6 are both integers, but only 57 is a natural number. Natural numbers
1

Note that there is some disagreement as to whether zero is a natural number. Some authors claim that it
does not belong to the set of natural numbers, instead is a member of an additional set called the whole
numbers which are the positive integers and zero.


Mathematical Language

are useful for counting and are the first number system we work with as children. It will

prove useful to define N+ as the nonzero natural numbers.
In addition to the sets of whole and natural numbers, a rational number is any real
number that can be expressed as the fraction of two integers. It should be clear that the
set of integers are also rational numbers, for example, 32 = 32/1 and −7 = −7/1, but
so are numbers like 45/2 and −98, 736/345, 298.
In contrast, irrational numbers are those which cannot be represented as a fraction of
two integers. Irrational numbers√are numbers which have an infinite number of decimal
places, for example, π, e, and 2. Irrational numbers cannot therefore be integers or
natural numbers.
The relationship between the different sets of real numbers is summarized in
Figure 1.2. From this it is clear that the “sum” of the sets of rational and irrational
numbers form the broader set of real numbers. The set of rational numbers can be
further subdivided into integers and nonintegers; the set of integers contains the natural
numbers.
EXAMPLE 1.1
Where would 0 appear in the Venn diagram of Figure 1.2?
Solution
According to the definitions given here, zero is a real number, a rational number, an
integer, and a natural number. It will then sit inside of the circle indicated by N. However,
other authors claim that it is not a natural number and so sits inside of the circle indicated
by Z but outside of N.

R

Q

Z

N

Figure 1.2 Venn diagram of the real number systems.

J

5


6

Introduction to Actuarial and Financial Mathematical Methods

EXAMPLE 1.2
Give three examples for each of the following number systems.
a. R+
b. Z
c. N
d. Q
e. J
Solution
a. R+ is the set of positive real numbers. Examples could be 0.0001, 3.2, and 100.
b. Z is the set of integers. Examples could be −10, 0, and 35.
c. N is the set of natural numbers. Examples could be 1, 7, and 92.
d. Q is the set of rational numbers. Examples could be −2,
√ and 9,883/3. √
√ 5/6,
e. J is the set of irrational numbers. Examples could be 3, 100 − π, and 3 4.

1.1.2 Mathematical symbols
In addition to the symbols used to denote the different number systems, mathematics is
full of notation. At this stage, it is useful to list the most common items of notation that

you should be able to identify, this done in Table 1.1.
Note that the list of basic notation in Table 1.1 identifies two symbols for “is
approximately equal to,” ≈, and , and this prompts discussion of our first mathematical
subtly: The symbol ≈ is commonly used to reflect that in practical situations we are
often forced to report approximate values of exact values. For example, the mathematical
constant e is an irrational number and so has a numerical value with an infinite number
of decimal places

Table 1.1 Basic mathematical notation

=
≈
>
≥
⇒
⇔
: or |

is equal to
is approximately equal to
is greater than
is greater than or equal to
implies that
implies and is implied by
such that

≡
<
≤
⇐

→
...

is identical to
is approximately equal to
is less than
is less than or equal to
is implied by
goes to
and continues

!

factorial

|x|

the modulus of x, i.e., |x| =

Note that a strike through indicates its negation, e.g., = denotes “is not equal to.”

= x for x ≥ 0
= −x for x < 0


Mathematical Language

e = 2.71828182845904523536028747135266249775724709369995 . . .
The practical use of e therefore requires one to truncate this to a manageable number
of decimal places, say three or four. This truncation is an approximation of the actual

value and we write
e ≈ 2.7183
Similarly, the mathematical constant π is an irrational number with value
π = 3.14159265358979323846264338327950288419716939937510 . . .
In practice one might use
π ≈ 3.1416
In contrast, the symbol is used when a mathematical method is used explicitly to
generate an approximation. For example, as we shall see in Chapter 5, it is often possible
to develop a method to approximate the value of an equation. In this case, one would
acknowledge that the method is not intended to deliver the exact numerical value by
using the symbol . The practical use of this symbol can be seen in Chapters 5 and 13,
and in particular Eq. (5.9), for example.
The other items of notation in Table 1.1 are assumed to be self explanatory.

EXAMPLE 1.3
Interpret the following mathematical statements in words and give two examples in each
case. You should work in the set of real numbers, R.
a. y > 5.4
b. z ≤ 10
c. x + 2 > 4
d. y = x and y = z ⇔ x = z
e. x ≈ 12
f. |q| = 7
Solution
a. y is greater than 5.4. For example, y = 5.41 or y = 6.
b. z is less than or equal to 10. For example, z = 10 or z = 9.6.
c. x + 2 is greater than 4. For example, x = 2.1 or x = 3.
d. y = x and y = z implies and is implied by x = z. Any identical values of x, y, and z
are examples of this.
e. x has approximate value 12 . For example, x = 0.503866281774 or x = 0.4986827.

f. The modulus of q is 7. For example, q = 7 or q = −7, that is q = ±7.

7


8

Introduction to Actuarial and Financial Mathematical Methods

1.2 MORE ADVANCED NOTATION
1.2.1 Set notation
Table 1.2 lists the basic items of set notation. We have loosely used the term set when
discussing the number systems, for example, we have discussed “subsets of the set of real
numbers,” without a proper definition of what a set actually is.
For all intents and purposes in this book, a set is simply understood as a collection of
distinct objects. The set of real numbers, R, is interpreted as the collection of all possible
real numbers. Any particular real number is a member of the set of real numbers, denoted
by ∈. For example, 4.56 ∈ R is read as “4.56 is a member of the set of real numbers.”
Any set formed from a collection of particular real numbers is considered to be a subset,
denoted ⊂, of real numbers. For example, {π, 4.56, 456/23} ⊂ R. From Figure 1.2 it
should be clear that
R+ ⊂ R

R− ⊂ R

Q⊂R

J⊂R

Z⊂R

N⊂R

Z⊂Q

N⊂Q

N⊂Z

Consider the sets
A = {−1, 1, 2}

B = {0, 2, 3}

and

C = {−3, −2, 4}.

(1.1)

It is clear that A and C are subsets of the sets of real numbers, rational numbers,
and integers, and B is a subset of real numbers, rational numbers, integers, and natural
numbers. None is a subset of the irrational numbers. The mathematical shorthand for
these statements would be
A⊂R

A⊂Q

A⊂Z

A⊂N

A⊂J

B⊂R

B⊂Q

B⊂Z

B⊂N

B⊂J

C⊂Z C⊂N

C⊂J

C⊂R C⊂Q

Of course, set theory is not limited to discussing number systems and we can work
with sets of any objects. Where appropriate in this book we will use capital letters
to refer to sets and lower case letters to refer to a particular member, that is element,
of a set.

Table 1.2 Basic set notation

{}
∅
∩

\

a set
empty set
set intersection
relative complement

∈
⊂
∪

is a member of
is a subset of
set union


Mathematical Language

EXAMPLE 1.4
Give all possible values of x that would satisfy the following statements concerning the
sets in Eq. (1.1).
a. x ∈ A
b. x ∈ A and x ∈ B
c. x ∈ B and −x ∈ C
Solution
a. x ∈ A ⇒ x = −1, 1, or 2.
b. x ∈ A and x ∈ B ⇒ x − 2
c. x ∈ B and −x ∈ C ⇒ x = 2 and 3.

It is possible to form larger sets by “adding” two sets using the union operation, ∪.

For example,
A ∪ B = {−1, 1, 2} ∪ {0, 2, 3} = {−1, 0, 1, 2, 3}
The union operation forms a new set that consists of all members of the original two
sets. Note that 2 ∈ A and 2 ∈ B but it is only listed once in the resulting union of A and
B. This is because a set is a list of distinct elements. The idea of a union can be extended
to three or more sets in the obvious way.
A ∪ B ∪ C = {−1, 1, 2} ∪ {0, 2, 3} ∪ {−3, −2, 4}
= {−1, 0, 1, 2, 3} ∪ {−3, −2, 4} = {−3, −2, −1, 0, 1, 2, 3, 4}
Furthermore, we can form the set that consists of the common elements of two sets
using the intersection notation, ∩. For example,
A ∩ B = {−1, 1, 2} ∩ {0, 2, 3} = {2}
A set with no elements is called an empty set for obvious reasons, and is denoted by
∅. For example, since A and C have no common elements
A ∩ C = {−1, 1, 2} ∩ {−3, −2, 4} = ∅
The intersection of three or more sets, for example, A ∩ B ∩ C, has an obvious
meaning.
In terms of the number system, we can write the following statements with the union
and intersection notation
Q∪J=R

Q∩J =∅

The complement of a set can be understood in broad terms as the set of items
outside of the set. However, in order to define the items outside of a set, we need

9


10

Introduction to Actuarial and Financial Mathematical Methods

to define the space of items that the set exists in. For example, the complement of
the set of irrational numbers is the set of all items that are not irrational numbers;
without somehow specifying that we actually meant “the complement of the irrational
numbers within the set of real numbers,” there is nothing stopping us listing cats, dogs,
and apples alongside the set of rational numbers as members of the complement! For
this reason it is useful to define the absolute complement of a set within some broad
space of all possible elements , and the relative complement of two sets that are both
within .
If it is clear that we are concerned only with real numbers, then the space of all
possible elements is limited to the set of real numbers and = R. Now that is defined,
we can consider the absolute complement of subsets of . The absolute complement
¯ or Ac . For example, in the space of real numbers,
of set A is denoted by either A
c
J = Q.
In contrast, the relative complement of two sets provides a means of subtracting one
set from another, assuming that both sets exist in . In general, for A, B ⊂ , we define
the relative complement of A in B as
B \ A = {x ∈ B : x ∈
/ A}
Using Tables 1.1 and 1.2, we can translate this to words as “the relative complement
of A in B are those things, x, in B such that are not in A.” Even more simply, it is
what remains of set B after having removed those items also in A. The analogue to the
subtraction B − A should be clear.
EXAMPLE 1.5
Using the sets in Eq. (1.1), determine the relative complement of A in B.
Solution
The relative complement of A = {−1, 1, 2} in B = {0, 2, 3} is all the elements in B that

are not in A. Therefore,
B \ A = {0, 2, 3} \ {−1, 1, 2} = {0, 3}

Back to our motivating example of number systems, we can broaden our space
to include both the real and imaginary number systems (see Chapter 8), and define the
relative complement of the irrational numbers in the real numbers,
R\J=Q


Mathematical Language

EXAMPLE 1.6
Interpret the following mathematical statements in words and give an example in each
case. You assume that = R.
a. x ∈ R
b. y ∈ Z
c. z ∈ {0, 1, 2, 3} ∪ {5, 6}
d. y ∈ Z ∩ R+
e. R+ ∪ {0} ∩ Z = N
f. B \ B = ∅
Solution
a. x is a real number. For example, x = 1.53.
b. y is an integer. For example, y = 9.
c. z is a member of set formed from the union of the two sets {0, 1, 2, 3} and {5, 6}, i.e.,
z is from {0, 1, 2, 3, 5, 6}. For example, z = 2.
d. y is a member of the set formed from the intersection (i.e., overlap) of the integers and
positive real numbers. For example, y = 892.
e. The intersection of the set of the union positive real numbers and zero with the set
of integers is the set of natural numbers. For example, 3 is a positive real number (one
can label it on the positive half of the number line), it is an integer, and is also a natural

number.
f. The complement of set B within itself is the empty set. That is, there are no elements
outside of B than are simultaneously also in B.

The basic set operations discussed here are summarized visually in Figure 1.3.

A

B

A

A∩B

A

B

A∪B

B

A

Figure 1.3 Basic set operations illustrated with Venn diagrams.

B

11

12

Introduction to Actuarial and Financial Mathematical Methods

We now leave aside explicit mention of set theory for the while and return to this
in Chapter 9 on probability theory. Unless otherwise stated, you should assume that all
mathematical quantities represent real numbers in all that follows.

1.2.2 Interval notation
Throughout this book we will make extensive use of interval bracket notation. In particular,
we will use the following bracket notation
• [a, b] denotes the interval {x : a ≤ x ≤ b}
• [a, b) denotes the interval {x : a ≤ x < b}
• (a, b] denotes the interval {x : a < x ≤ b}
• (a, b) denotes the interval {x : a < x < b}
where the term interval can be interpreted as subset of the real number line. Using
Tables 1.1 and 1.2 to translate these statements into words, it should be clear that the
interval [a, b] is read as “the set of numbers x such that x is between and including a and
b.” In contrast, the interval (a, b) is read as “the set of numbers x such that x is between
but not including a and b.” The interpretation of the intervals [a, b) and (a, b] follows in
a similar manner. The key point, of course, is that a square bracket denotes an inclusive
endpoint of the interval, and a rounded bracket does not.
We refer to an interval that does not include its endpoints as an open interval. For
example, (1, 5) consists of all numbers x such that 1 < x < 5 and is open. A closed interval,
however, does include its endpoints. For example, [10, 102] consists of all numbers x such
that 10 ≤ x ≤ 102 and is closed.
When working with an endpoint at infinity, a closed interval is meaningless and
±∞ should appear only next to a rounded bracket: (−∞, ∞), (−∞, b], [a, ∞). The
interpretation of this is that ∞ is not a number that one can draw on a number line,

rather it represents that we can keep on using more and more of the number line without
imposing any bound.
EXAMPLE 1.7
Interpret the following mathematical statements in words and give two examples in each
case.
a. x ∈ [100, ∞)
b. y ∈ (0, 10]
c. p ∈ [0, 1]
d. z ∈ (−9.9, −9.8)
Solution
a. x is such that 100 ≤ x < ∞. For example, x = 100 or x = 564.3.
b. y is such that 0 < y ≤ 10. For example, y = 0.1 or y = 10.
c. p is such that 0 ≤ p ≤ 1. For example, p = 0 or p = 1.
d. z is such that −9.9 < z < −9.8. For example, z = −9.87 or z = −9.82.


Mathematical Language

1.2.3 Quantifiers and statements
There are two mathematical quantifiers which, when combined with the notation
described previously, form a powerful means of writing a wide variety of mathematical
statements in a concise way. These are
• ∀, read as “for all”
• ∃, read as “there exists”
The quantifier ∀ is often referred to as the universal quantifier, and ∃ as the existential
quantifier. The meaning of both should be immediately apparent, although their power
might not be. To hint at the power of the two quantifiers in simplifying statements, we
begin with an example:
EXAMPLE 1.8
Demonstrate the intuitive fact that it is possible to find a rational number that approximates

the value of π to any finite level of accuracy. Use concise mathematical notation to express
that this is true for all real numbers.
Solution
We list the approximations to the value of the irrational number π to an increasing number
of decimal places, expressed as a rational number:
π ≈ 3.14

= 157/50

π ≈ 3.142

= 1571/500

π ≈ 3.1416
...

= 3927/1250

π ≈ 3.1415926536 = 3, 926, 990, 817/1, 250, 000, 000
...
That this is true for all real numbers (not just the irrational π) is expressed by
∀x ∈ R,

∀ ∈ R+

∃r ∈ Q : |x − r| <

The mathematical statement given in the solution to Example 1.8 is translated to
words as
for all x in the set of real numbers and for all in the set of positive real numbers, there exists r in

the set of rational numbers such that the absolute value of the difference between x and r is smaller
than the value of

Some thought should convince you that this statement is a reflection of our process
for approximating π. However, aside from that this is an interesting mathematical fact,
the benefits of using the concise mathematical statement formed from the two quantifiers
should be immediately apparent.

13