Hints and Solutions for
A Gentle Introduction to the Art of
Mathematics
Version 3.1
Joseph Fields
Southern Connecticut State University
,ii
Copyright Ⓧ
c 2013 Joseph E. Fields. Permission is granted to
copy, distribute and/or modify this document under the terms of
the GNU Free Documentation License, Version 1.3 or any later
version published by the Free Software Foundation; with no In-
variant Sections, no Front-Cover Texts, and no Back-Cover Texts.
There are currently 4 optional versions of this text. The different
options are distinguished with flags following the version number.
Thus you may see a plain “version 3.1” (no flags) or any of the fol-
lowing: “version 3.1S”, “version 3.1N”, or “version 3.1SN”. The S
flag is for those who prefer the symbol ∼ to represent logical nega-
tion. (The default is to use the symbol ¬ for logical negation.) The
N flag distinguishes a version in which the convention that1 is
the smallest natural number is followed throughout. The absence
of the N flag indicates the original version of the textin which
the naturals do not contain 0 in chapters 1 through 4 and (after
the Peano axioms are introduced) that convention is changed to
the more modern rule.
,Chapter 1
Introduction and notation
1.1 Basic sets
Exercises — 1.1
1. Each of the quantities indexing the rows of the following table is in one
or more of the sets which index the columns. Place a check mark in a
table entry if the quantity is in the set.
N Z Q R C
17
π
22/7
−6
e0
1+i
√
3
i2
Note that these sets contain one another, so if you determine that a
number is a natural number it is automatically an integer and a rational
number and a real number and a complex number. . .
1
, 2 CHAPTER 1. INTRODUCTION AND NOTATION
2. Write the set Z of integers using a singly infinite listing.
What the heck is meant by a “singly infinite listing”? To help you
figure this out, note that
. . . − 3, −2, −1, 0, 1, 2, 3, . . .
is a doubly infinite listing.
3. Identify each as rational or irrational.
(a) 5021.2121212121 . . .
(b) 0.2340000000 . . .
(c) 12.31331133311133331111 . . .
(d) π
(e) 2.987654321987654321987654321 . . .
rat,rat,irr,irr,rat
A Gentle Introduction to the Art of
Mathematics
Version 3.1
Joseph Fields
Southern Connecticut State University
,ii
Copyright Ⓧ
c 2013 Joseph E. Fields. Permission is granted to
copy, distribute and/or modify this document under the terms of
the GNU Free Documentation License, Version 1.3 or any later
version published by the Free Software Foundation; with no In-
variant Sections, no Front-Cover Texts, and no Back-Cover Texts.
There are currently 4 optional versions of this text. The different
options are distinguished with flags following the version number.
Thus you may see a plain “version 3.1” (no flags) or any of the fol-
lowing: “version 3.1S”, “version 3.1N”, or “version 3.1SN”. The S
flag is for those who prefer the symbol ∼ to represent logical nega-
tion. (The default is to use the symbol ¬ for logical negation.) The
N flag distinguishes a version in which the convention that1 is
the smallest natural number is followed throughout. The absence
of the N flag indicates the original version of the textin which
the naturals do not contain 0 in chapters 1 through 4 and (after
the Peano axioms are introduced) that convention is changed to
the more modern rule.
,Chapter 1
Introduction and notation
1.1 Basic sets
Exercises — 1.1
1. Each of the quantities indexing the rows of the following table is in one
or more of the sets which index the columns. Place a check mark in a
table entry if the quantity is in the set.
N Z Q R C
17
π
22/7
−6
e0
1+i
√
3
i2
Note that these sets contain one another, so if you determine that a
number is a natural number it is automatically an integer and a rational
number and a real number and a complex number. . .
1
, 2 CHAPTER 1. INTRODUCTION AND NOTATION
2. Write the set Z of integers using a singly infinite listing.
What the heck is meant by a “singly infinite listing”? To help you
figure this out, note that
. . . − 3, −2, −1, 0, 1, 2, 3, . . .
is a doubly infinite listing.
3. Identify each as rational or irrational.
(a) 5021.2121212121 . . .
(b) 0.2340000000 . . .
(c) 12.31331133311133331111 . . .
(d) π
(e) 2.987654321987654321987654321 . . .
rat,rat,irr,irr,rat
