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Benjamin Fine, Anthony Gaglione, Anja Moldenhauer, Gerhard Rosenberger,
Dennis Spellman
Algebra and Number Theory
De Gruyter Textbook

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Benjamin Fine, Anthony Gaglione,
Anja Moldenhauer, Gerhard Rosenberger,
Dennis Spellman

Algebra and
Number Theory
|

A Selection of Highlights

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Mathematics Subject Classification 2010
0001, 00A06, 1101, 1201
Authors

Prof. Dr. Benjamin Fine
Fairfield University
Department of Mathematics
1073 North Benson Road
Fairfield, CT 06430
USA

Prof. Dr. Gerhard Rosenberger
University of Hamburg
Department of Mathematics
Bundesstr. 55
20146 Hamburg
Germany

Prof. Dr. Anthony Gaglione
United States Naval Academy
Department of Mathematics
212 Blake Road
Annapolis, MD 21401
USA

Prof. Dr. Dennis Spellman
Temple University
Department of Mathematics
1801 N Broad Street
Philadelphia, PA 19122
USA

Dr. Anja Moldenhauer
University of Hamburg

Department of Mathematics
Bundesstr. 55
20146 Hamburg
Germany

ISBN 978-3-11-051584-8
e-ISBN (PDF) 978-3-11-051614-2
e-ISBN (EPUB) 978-3-11-051626-5
Library of Congress Cataloging-in-Publication Data
A CIP catalog record for this book has been applied for at the Library of Congress.
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the Internet at .
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Typesetting: VTeX UAB, Lithuania
Printing and binding: CPI books GmbH, Leck
Cover image: agsandrew / iStock / Getty Images Plus
♾ Printed on acid-free paper
Printed in Germany
www.degruyter.com

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Preface
To many students, as well as to many teachers, mathematics seems like a mundane
discipline, filled with rules and algorithms and devoid of beauty and art. However to
someone who truly digs deeply into mathematics this is quite far from the truth. The
world of mathematics is populated with true gems; results that both astound and point
to a unity in both the world and a seemingly chaotic subject. It is often that these gems

and their surprising results are used to point to the existence of a force governing the
universe; that is, they point to a higher power. Euler’s magic formula, eiπ + 1 = 0, which
we go over and prove in this book is often cited as a proof of the existence of God. While
to someone seeing this statement for the first time it might seem outlandish, however if
one delves into how this result is generated naturally from such a disparate collection
of numbers it does not seem so strange to attribute to it a certain mystical significance.
Unfortunately most students of mathematics only see bits and pieces of this amazing discipline. In this book, which we call Algebra and Number Theory, we introduce and examine many of these exciting results. We planned this book to be used in
courses for teachers and for the general mathematically interested so it is somewhat
between a textbook and just a collection of results. We examine these mathematical
gems and also their proofs, developing whatever mathematical results and techniques
we need along the way. In Germany and the United States we see the book as a Masters
Level Book for prospective teachers.
With the increasing demand for education in the STEM subjects, there is the realization that to get better teaching in mathematics, the prospective teachers must both
be more knowledgeable in mathematics and excited about the subject. The courses in
teacher preparation do not touch many of these results that make the discipline so exciting. This book is intended to address this issue. The first volume is on Algebra and
Number Theory. We touch on numbers and number systems, polynomials and polynomial equations, geometry and geometric constructions. These parts are somewhat
independent so a professor can pick and choose the areas to concentrate on. Much
more material is included than can be covered in a single course. We prove all relevant results that are not too technical or complicated to scare the students. We find
that mathematics is also tied to its history so we include many historical comments.
We try to introduce all that is necessary however we do presuppose certain subjects from school and undergraduate mathematics. These include basic knowledge
in algebra, geometry and calculus as well as some knowledge of matrices and linear
equations. Beyond these the book is self-contained.
This first volume of two is called Algebra and Number Theory. There are fourteen
chapters and we think we have introduced a very wide collection of results of the type
that we have alluded to above. In Chapters 1–5 we look at highlights on the integers. We
examine unique factorization and modular arithmetic and related ideas. We show how
these become critical components of modern cryptography especially public key crypDOI 10.1515/9783110516142-201

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VI | Preface
tographic methods such as RSA. Three of the authors (Fine, Moldenhauer and Rosenberger) work partly as cryptographers so cryptography is mentioned and explained
in several places. In Chapters 4 and 5 we look at exceptional classes of integers such
as the Fibonacci numbers as well as the Fermat numbers, Mersenne numbers, perfect
numbers and Pythagorean triples. We explain the golden section as well as expressing
integers as sums of squares. In Chapters 6–8 we look at results involving polynomials and polynomial equations. We explain field extensions at an understandable level
and then prove the insolvability of the quintic and beyond. The insolvability of the
quintic in general is one of the important results of modern mathematics.
In Chapters 9–12 we look at highlights from the real and complex numbers leading
eventually to an explanation and proof of the Fundamental Theorem of Algebra. Along
the way we consider the amazing properties of the numbers e and π and prove in detail
that these two numbers are transcendent.
Chapter 13 is concerned with the classical problem of geometric constructions and
uses the material we developed on field extensions to prove the impossibility of certain
constructions.
Finally in Chapter 14 we look at Euclidean Vector Spaces. We give several geometric applications and look for instance at a secret sharing protocol using the closest
vector theorem.
We would like to thank the people who were involved in the preparation of the
manuscript. Their dedicated participation in translating and proofreading are gratefully acknowledged. In particular, we have to mention Anja Rosenberger, Annika
Schürenberg and the many students who have taken the respective courses in Dortmund, Fairfield and Hamburg. Those mathematical, stylistic, and orthographic errors
that undoubtedly remain shall be charged to the authors. Last but not least, we thank
de Gruyter for publishing our book.
Benjamin Fine
Anthony Gaglione
Anja Moldenhauer
Gerhard Rosenberger
Dennis Spellman

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Construction of the golden section α with compass and straightedge
from a given a ∈ ℝ, a > 0 | 70
Perfect numbers and Mersenne numbers | 71
Fermat numbers | 78

5
5.1
5.2
5.3

Pythagorean triples and sums of squares | 83
The Pythagorean Theorem | 83
Classification of the Pythagorean triples | 85
Sum of squares | 89

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VIII | Contents
6
6.1
6.2
6.3
6.3.1
6.3.2
6.4
6.5
6.5.1
6.5.2
6.5.3

Orthogonality and Applications in ℝ2 and ℝ3 | 309
Orthonormalization and closest vector | 317
Polynomial approximation | 321
Secret sharing scheme using the closest vector theorem | 323

Bibliography | 327
Index | 329

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1 The natural, integral and rational numbers
1.1 Number theory and axiomatic systems
Number theory begins as the study of the whole numbers or counting numbers. Formally the counting numbers 1, 2, … are called the natural numbers and denoted by ℕ.
If we add to this the number zero, denoted by 0, and the negative whole numbers we
get a more comprehensive system called the integers which we denote by ℤ. The focus
of this book is on important and sometimes surprising results in number theory and
then further results in algebra. Many results in number theory, as we shall see, seem
like magic. In order to rigorously prove these results we place the whole theory in an
axiomatic setting which we now explain.
In mathematics, when developing a concept or a theory it is often not possible, all
used terms, properties or claims to prove, especially existence of some mathematical
fundamentals. One can solve this problem then by an axiomatic approach. The basis
of a theory then is a system of axioms:
– Certain objects and certain properties of these objects are taken as given and accepted.
– A selection of statements (the axioms) are considered by definition as true and
evident.
A theorem in the theory then is a true statement, whose truth can be proved from the
axioms with help of true implications. A system of axioms is consistent if one can not
prove a statement of the form “A and not A”. The verification is in individual cases

often a complicated or even an unsolvable problem. We are satisfied, if we can quote
a model for the system of axioms, that is, a system of concrete objects, which meet all
the given axioms. A system of axioms is called categorical if essentially there exists
only one model. By this we mean that for any two models we always get from one
model to the other by renaming of the objects. If this is true then we have an axiomatic
characterization of the model.
In the next section we introduce the natural numbers axiomatically.

1.2 The natural numbers and induction
The natural numbers ℕ are presented by the system of axioms developed by G. Peano
(1858–1932). This is done as follows.
The set ℕ of the natural numbers is described by the following axioms:
(ℕ 1) 1 ∈ ℕ.
(ℕ 2) Each a ∈ ℕ has exactly one successor a+ ∈ ℕ.
(ℕ 3) Always is a+ ≠ 1, and for each b ≠ 1 there exists an a ∈ ℕ with b = a+ .
(ℕ 4) a ≠ b ⇒ a+ ≠ b+ .
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2 | 1 The natural, integral and rational numbers
(ℕ 5) If T ⊂ ℕ, 1 ∈ T, and if together with a ∈ T also a+ ∈ T, then T = ℕ.
(Axiom of mathematical induction or just induction.)
Remarks 1.1. (1) (ℕ 2) and (ℕ 4) mean that the map
σ∶ℕ→ℕ
a ↦ a+
is injective.
(2) From the Peano axioms we get per definition an addition, a multiplication and an
ordering for ℕ:

(i) a + 1 ∶= a+ ,
a + b+ ∶= (a + b)+ ,
(ii) a ⋅ 1 ∶= a,
a ⋅ b+ ∶= ab + a,
(iii) a < b ∶⇔ ∃ x ∈ ℕ with a + x = b (“a smaller than b”),
a ≤ b ∶⇔ a = b or a < b (“a equal or smaller than b”).
We need to recall some definitions.
A semigroup is a set H ≠ ∅ together with a binary operation ⋅ ∶ H × H → H that
satisfies the associative property for all a, b, c ∈ H:
(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).
The semigroup is commutative if
a ⋅ b = b ⋅ a.
In the commutative case we often write the operation as addition + instead of multiplication ⋅ .
A monoid S is a semigroup with a unity element e, that is, an element e with a ⋅ e =
a = e ⋅ a for all a ∈ S; e is uniquely determined.
Moreover, a monoid S is called a group if for each a ∈ S there exists an inverse
element a−1 ∈ S with aa−1 = a−1 a = e. The monoid or group is named commutative or
abelian if in addition
a ⋅ b = b ⋅ a for all a, b ∈ S.
We often write 1 instead of e. We also often drop ⋅ and use just juxtaposition for this
operation. If we use the addition + we often write 0 instead of e and call 0 the zero
element of S.
Theorem 1.2. (1) The addition for ℕ is associative, that is,
a + (b + c) = (a + b) + c,

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1.2 The natural numbers and induction

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and commutative, that is,
a + b = b + a.
This means, ℕ is a commutative semigroup with respect to the addition.
(2) The multiplication for ℕ is associative, that is,
a(bc) = (ab)c,
and commutative, that is,
ab = ba.
ℕ has also the unity element 1 for the multiplication. Therefore, ℕ is a commutative
monoid with respect to the multiplication.
(3) The multiplication is distributive with respect to the addition, that is,
(a + b)c = ac + bc.
(4) For a, b ∈ ℕ exactly one of the following is true:
a < b,

a=b

or

b < a.

(5) If a ≤ b and c ≤ d then a + c ≤ b + d and ac ≤ bd.
Proof. The statements follow directly from the definition and the Peano axioms. We
leave the proofs as an exercise. As an example we prove (3) using (1) and (2): Let
a, b ∈ ℕ be arbitrary and T ⊂ ℕ the set of the c ∈ ℕ with (a + b)c = ac + bc. We have
1 ∈ T because
(a + b) ⋅ 1 = a + b = a ⋅ 1 + b ⋅ 1.
Now, let c ∈ T. Then
(a + b)c+ = (a + b)c + (a + b) = ac + bc + a + b = ac + a + bc + b

= ac+ + bc+ .
Hence c+ ∈ T and so T = ℕ.
As usual we write an for ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
a ⋅ a ⋯ a and na for ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
a + a + ⋯ + a, when a, n ∈ ℕ.
n times

n times

Remarks 1.3. (1) By the development of the addition in ℕ we suggest the usual representation of natural numbers as numerals:
2 = 1+ = 1 + 1,

3 = 2+ = 2 + 1,

4 = 3+ = 3 + 1

and so on.

(2) From the Peano axioms we also get that for each natural number n there exist
exactly one natural number m with m ≤ n < m + 1. The set ℕ is therefore a set
unbounded from above.

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4 | 1 The natural, integral and rational numbers
(3) Theorem 1.2 also allows to subtract smaller natural numbers from larger ones. If
a, b ∈ ℕ with a < b, then there is an x ∈ ℕ with a + x = b. We define the subtraction
by
x ∶= b − a

and say “x is equal b minus a”.
Example 1.4.
3 = 11 − 8 = 17 − 14,
31 = 50 − 19.
(4) The mathematical proof technique mathematical induction is based on the Peano
axiom (ℕ 5). It is a form of direct proof, and it is done in two steps.
The first step, known as the base case, is to prove the given statement A(n), which
is definable for all n ∈ ℕ, for the first natural number 1. The second step, known as
the induction step, is to prove that the given statement A(n) is true for any natural
number n implies the given statement is true for the next natural number. In other
words, if A(1) is true and if we can show that under the assumption that A(n) is
true for any n, then A(n + 1) is true, then A(n) is true for all n ∈ ℕ.
We call the mathematical induction the first induction principle or the principle of
mathematical induction (PMI).
It is clear that we may start with the mathematical induction with any natural
number n0 > 1 instead of 1, we just need a base. This can be done with the approach B(n) ∶= A(n0 − 1 + n).
Examples 1.5. (1) Claim.
n

∑k=

k=1

n(n + 1)
2

for all n ∈ ℕ.

Proof. Let A(n), n ∈ ℕ, be the asserted statement.
(a) A(1) is true because

1

∑k=1=

k=1

1(1 + 1)
.
2

(b) Assume that A(n) is true for n ∈ ℕ. We have to show that A(n + 1) is true:
n+1

n

k=1

k=1

∑ k = ∑ k + (n + 1) =
=

(n + 1)(n + 2)
,
2

n(n + 1)
+ (n + 1)
2

and this is A(n + 1).

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1.2 The natural numbers and induction

| 5

We remark that the n-th triangular number Tn , n ∈ ℕ, is defined as n(n+1)
. Geomet2
rically Tn is the number of dots composing a regular triangle with n dots on a side,
see Figure 1.1.

Figure 1.1: Geometrical representation of the triangular numbers T1 , T2 , T3 and T4 .

From this geometrical representation (see Figure 1.1) in a regular triangle we see
by completing the triangle to a square that
Tn−1 + Tn = n2 .
This easily gives directly by induction
T1 + T2 + T3 + ⋯ + Tn−1 + Tn =

n(n + 1)(n + 2)
6

for n ≥ 2.

Concerning the triangular numbers there is a nice story from the time when Gauss
got his first lessons in Calculations. He could solve an exercise, which was considered as very time-consuming, in a very short time.
The children had to calculate the sum of all numbers from 1 to 100. Gauss realized

the pattern:
1 + 100 = 101,

2 + 99 = 101,

3 + 98 = 101,

…,

50 + 51 = 101.

So he had 50 pairs of numbers, and each pair results in the sum 101. It is clear that
this adds up to
50 ⋅ 101 = 5 050.
In an analogous manner we may show that
n

∑ k2 =

k=1

n(n + 1)(2n + 1)
6

and
n

∑ k3 =

k=1

n2 (n + 1)2
= Tn2 ,
4

for n ∈ ℕ.

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6 | 1 The natural, integral and rational numbers
There is a beautiful general formula by Al-Haitam (965–1038):
n

n

n

p

i=1

i=1

p=1

i=1

(n + 1) ∑ ik = ∑ ik+1 + ∑ (∑ ik ) .
We present a geometrical proof, see Figure 1.2, which is still remarkable.

Figure 1.2: Geometrical proof of the formula
by Al-Haitam.

Recall that 1k+1 = 1k , 2 ⋅ 2k = 2k+1 , 3 ⋅ 3k = 3k+1 , ….
(2) Claim.
2n > n2

for all n ≥ 5.

Proof. Let A(n), n ≥ 5, be the asserted statement.
(a) A(5) is true because
25 = 32 > 25 = 52 .
(b) Assume that A(n) is true for n ≥ 5. Then
2n+1 = 2 ⋅ 2n > 2 ⋅ n2 = n2 + n2 > n2 + 2n + 1 = (n + 1)2 ,
because n2 > 2n + 1 for n ≥ 3 (which is left as an exercise). Hence A(n) is true for all
n ∈ ℕ with n ≥ 5.
(3) A polygon in the plane ℝ2 with n + 2 sides, n ∈ ℕ, is called convex, if the connecting
line segment between any two points of the polygon is totally within the polygon.
Examples 1.6. We give examples for convex polygons in Figure 1.3.

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1.2 The natural numbers and induction

| 7

Figure 1.3: Convex polygons.

Claim. The angle sum Wn of the interior angles in a convex polygon with n + 2, n ∈ ℕ,
sides is n ⋅ 180∘ .
Proof. (a) If n = 1 we have a triangle from which it is known that W1 = 180∘ .
(b) Assume that the assert statement holds for n ∈ ℕ. We consider a convex polygon with (n + 1) + 2 sides. We divide the polygon into a triangle and a polygon with
n + 2 sides as in Figure 1.4.

Figure 1.4: Polygon divided into a triangle and a polygon
with n + 2 sides.

Then
Wn+1 = Wn + 180∘ = n ⋅ 180∘ + 180∘ = (n + 1) ⋅ 180∘ .
(4) The geometric sum formula. Let q ∈ ℕ, q ≠ 1. Then
n

∑ qi =
i=1

qn+1 − q
q−1

for n ∈ ℕ.

Proof. (a) If n = 1 then
1

∑ qi = q = q ⋅
i=1

q − 1 q2 − q
=

.
q−1
q−1

(b) Assume that the asserted statement holds for n ∈ ℕ. Then
n+1

n

i=1

i=1

∑ qi = ∑ qi + qn+1 =

qn+1 − q
qn+2 − q
+ qn+1 =
.
q−1
q−1

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8 | 1 The natural, integral and rational numbers
We remark that

n

1 + ∑ qi =
i=1

qn+1 − 1
.
q−1

(5) Claim. Let M be an arbitrary set with n (distinct) elements, n ∈ ℕ. Let 𝒫(M) be the
power set of M. Then |𝒫(M)| = 2n for the number of elements in 𝒫(M).
Proof. (a) If n = 1 then M = {a} for some a, and hence, 𝒫(M) = {∅, {a}}, that is,
|𝒫(M)| = 2 = 21 .
(b) Assume that the asserted statement holds for each set with n elements. Let,
without loss of generality,
M = {a1 , a2 , … , an+1 } = {a1 , a2 , … , an } ∪ {an+1 } = M ′ ∪ {an+1 }
be a set with n + 1 elements. By the induction hypothesis
|𝒫(M ′ )| = 2n .
Now, let A ⊂ M be a subset of M. Then we have exactly one of the following cases:
Case 1. an+1 ∉ A.
Then A ⊂ M ′ , and by the induction hypothesis there are exactly 2n of such subsets.
Case 2. an+1 ∈ A.
Then A ⧵ {an+1 } ⊂ M ′ . Again, by the induction hypothesis, there are exactly 2n such
subsets.
Altogether
|𝒫(M)| = 2n + 2n = 2 ⋅ 2n = 2n+1 .
The principle of induction is equivalent to another property called the least wellordering property. This is the following. Let S be a nonempty subset of the natural numbers ℕ. Then S has a least element.
We will abbreviate the least well-ordering property by LWO.
In the next theorem below we show that the principle of mathematical induction
(PMI) is equivalent to the LWO. By equivalent, we mean here that if we assume that
the PMI is true then we can prove the LWO and if we assume the LWO is true then we
can prove the PMI.

Theorem 1.7. The principle of mathematical induction is equivalent to the least wellordering property.
Proof. To prove this we must assume first the principle of mathematical induction and
show that the well-ordering property holds and then vice versa. Suppose that the PMI
holds and let S be a nonempty subset of ℕ. We must show that S has a least element.
We let T be the set
T = {x ∈ ℕ ∣ x ≤ s, for all s ∈ S} .

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1.2 The natural numbers and induction

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Now 1 ∈ T since S is a subset of ℕ. If whenever x ∈ T it were to follow that (x + 1) ∈ T,
then by the inductive property T = ℕ but then S would be empty contradicting that S
is nonempty. Therefore there exists an a with a ∈ T and (a + 1) ∉ T. We claim that a is
the least element of S. Now a ≤ s for all s ∈ S because a ∈ T. If a ∉ S then every s ∈ S
would also satisfy (a + 1) ≤ s. This would imply that (a + 1) ∈ T which is a contradiction.
Therefore a ∈ S and a ≤ s for all s ∈ S and hence a is the least element of S.
Conversely suppose the well-ordering property holds and suppose that S is a subset of ℕ with the properties that 1 ∈ S and that whenever n ∈ S it follows that (n + 1) ∈ S.
We must show that S = ℕ. If S ≠ ℕ, then the set difference ℕ ⧵ S, that is, the set of all elements in ℕ but not in S, would be a nonempty subset of ℕ. Thus by the LWO, it has a
least element, say n. Hence (n − 1) is not in ℕ ⧵ S or (n − 1) ∈ S. But then by the assumed
property of S we get that (n − 1) + 1 = n ∈ S which gives a contradiction. Therefore ℕ ⧵ S
is empty and S = ℕ.
Based on Theorem 1.7 we have a second form of mathematical induction that we
call the second induction principle. This is also known as course of values induction or
strong induction.
Theorem 1.8. Let A(n) be a statement which is defined for all n ∈ ℕ. If A(1) is true and
if we can show that under the assumption that A(k) is true for all k ∈ ℕ with k < n for

any n ∈ ℕ, then A(n) is true for all n ∈ ℕ.
Proof. Let T = {n ∈ ℕ ∣ A(n) is not true} ⊂ ℕ. Assume that T ≠ ∅. Then by LWO T contains a smallest element, which means that there is an n ∈ ℕ, n > 1, with A(n) not true
but A(1) is true and A(k) is true for all k ∈ ℕ, k < n. But this contradicts our hypothesis.
Therefore A(n) is true for all n ∈ ℕ.
Corollary 1.9. The two principles for mathematical induction are equivalent.
Proof. If the second principle holds then certainly also the first one. If the first principle holds then also the second one vie Theorem 1.8.
Remarks 1.10. (1) Also for the second induction principle we may take as the base
any n0 ∈ ℕ instead of 1.
(2) Please note that mathematical induction is not an empirical method to get general
statements from calculations or observations for some numbers.
Example 1.11 (For the second principle). We define here the Fibonacci numbers fn ,
n ∈ ℕ, recursively by
f1 ∶= f2 ∶= 1

and fn+1 ∶= fn + fn−1

for n ≥ 2.

Claim. For n ≥ 2 the Fibonacci number fn is the number of all 0–1-sequences of length
n − 2 which do not contain neighboring 1’s.

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10 | 1 The natural, integral and rational numbers
Proof. Let Mn be the set of all 0–1-sequences of length n−2, n ≥ 2, which do not contain
neighboring 1’s. We have |M2 | = f2 = 1, the empty sequence, and |M3 | = f3 = 2 because
we just have the sequences 0 and 1.
Now, let n ≥ 4, and we assume that the statement is true for all k with 2 ≤ k < n. Let
(0)

(1)
(0)
Mn = Mn ∪ Mn be the disjoint union of Mn , the set of the sequences in Mn ending
(1)
with a 0, and Mn , the set of the sequences in Mn ending with a 1. Because of our
(1)
condition, in fact, each sequence in Mn has to end with 01 (recall that we do not have
neighboring 1’s). Therefore
|Mn | = |Mn | + |Mn | = |Mn−1 | + |Mn−2 | = fn−1 + fn−2 = fn .
(0)

(1)

Remark 1.12. We may establish the natural numbers constructive from the axiomatic
set theory. One starts with the axiomatically existent empty set ∅ and gets for each set
X a successor set using the axiom that there exists an infinite set, that is, there is a set
which contains with ∅ and X also the set X ∪ {x}. One then defines
0 ∶= ∅,
1 ∶= ∅ ∪ {∅} = {0},
2 ∶= 1 ∪ {1} = {0} ∪ {1} = {0, 1},
3 ∶= 2 ∪ {2} = {0, 1} ∪ {2} = {0, 1, 2},
⋮
n ∶= {0, 1, 2, … , n − 1}.
In this approach the element 0 is a natural number and each n is the set of its predecessors 0, 1, 2, … , n − 1. We remember that in our approach using the Peano axioms
the natural numbers start with 1. In the next section we use the above Remark 1.12 to
introduce briefly the integers and the rational numbers in which the zero symbol plays
a fundamental role.

1.3 The integers ℤ
Since the integers play a fundamental role in our book, we now introduce them and

deduce their existence from the natural numbers. We begin by looking at the difference of two natural numbers introduced in the last section.
x = a − b for a, b ∈ ℕ and a > b.
This means algebraically that we may solve the equation
b+x=a

for a > b in ℕ.

The motivation is to solve the equation
b+x=a
over ℕ in general. This leads to the number 0 and the negative numbers.

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1.3 The integers ℤ

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Historically it took a long time before the negative numbers were accepted. In ancient times and in parts of the middle ages mathematicians like Al Khwarizmi (ca.
780–850), Cardano (1501–1576) and Vieta (1540–1603) considered the negative numbers as forbidden or worked with them solely symbolically believing that these negative numbers do not exist. This can be seen from the many quadratic equations over
ℕ which in their feeling are forbidden or not-existent.
We consider the set
ℕ × ℕ = {(a, b) ∣ a, b ∈ ℕ},
the set of pairs of natural numbers.
If a > b and c > d, then we may have the equation a − b = c − d in ℕ, or equivalently,
a + d = c + b. This is the inspiring background for introducing the integers. We define
an addition and multiplication on ℕ × ℕ as follows:
(a, b) + (c, d) = (a + c, b + d),
(a, b) ⋅ (c, d) = (ac + bd, ad + bc).
With respect to the above equation a + d = b + c whenever a > b, c > d and a − b = c − d,

we introduce a relation on ℕ × ℕ:
(a, b) ∼ (c, d) ⇔ a + d = c + b.
This is an equivalence relation. Certainly, (a, b) ∼ (a, b) and (c, d) ∼ (a, b) if (a, b) ∼
(c, d), that is, the relation is reflexive and symmetric. It is also transitive, that is, if
(a, b) ∼ (c, d) and (c, d) ∼ (e, f ) then (a, b) ∼ (e, f ) because from a + b = c + d and c + d =
e + f we get a + b = e + f .
Let ℤ ∶= ℕ × ℕ/∼ be the set of equivalence classes. The addition and the multiplication of the pairs in ℕ × ℕ induce a well defined addition and multiplication on ℤ, that
is, if (a, b) ∼ (a′ , b′ ) and (c, d) ∼ (c′ , d′ ), then
(a, b) + (c, d) ∼ (a′ , b′ ) + (c′ , d′ )
and
(a, b) ⋅ (c, d) ∼ (a′ , b′ ) ⋅ (c′ , d′ ).
We leave the proof for this as an exercise. Together with these operations ℤ becomes
a commutative ring with unity.
Remark 1.13. We remind that a ring is a set R ≠ ∅ equipped with two binary operations + ∶ R × R → R and ⋅ ∶ R × R → R satisfying the following three sets of properties
for all a, b, c ∈ R:
– R is a commutative group under addition, that is,
(1) (a + b) + c = a + (b + c).

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12 | 1 The natural, integral and rational numbers

–

(2) a + b = b + a.
(3) There is a zero element 0 ∈ R such that a + 0 = a for all a ∈ R.
(4) For each a ∈ R exists −a ∈ R such that a + (−a) = 0.
We call −a the negative element of a.
R is a semigroup under multiplication, that is,

(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

–

The multiplication is distributive with respect to the addition, that is,
a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)

and

(b + c) ⋅ a = (b ⋅ a) + (c ⋅ a).
A commutative ring with unity 1 is a ring R, in which the semigroup under multiplication is also commutative and has the unity element 1.
In the case ℤ the zero element is the class represented by (1, 1), and the unity
element is the class represented by (2, 1).
We briefly write (a, b) for the equivalence class represented by (a, b). The equivalence class (a, b) has for a > b a unique representative of the form (n + 1, 1), where
n = a − b, and for b > a a unique representative of the form (1, m + 1), where m = b − a.
This gives a possibility to embed ℕ into ℤ. We define the map
φ∶ℕ→ℤ
n ↦ (n + 1, 1).
The map φ is injective and satisfies
φ(a + b) = φ(a) + φ(b)

and

φ(ab) = φ(a)φ(b)

for a, b ∈ ℕ. This means, that φ is an embedding.
Therefore we may identify ℕ with φ(ℕ) ⊂ ℤ, and we write after the identification
n = (n + 1, 1) for n ∈ ℕ.
We write −n for the equivalence class (1, n + 1). This is reasonable because
(n + 1, 1) + (1, n + 1) = (1, 1),

the zero element of ℤ. Therefore we now write 0 for the class (1, 1).
In this sense, ℤ is the disjoint union
ℤ = ℕ− ∪ {0} ∪ ℕ,
where ℕ− = {−n ∣ n ∈ ℕ}.
With this, the addition and multiplication in ℤ is just according to customs.

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1.4 The rational numbers ℚ

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We now transfer the ordering in ℕ to an ordering in ℤ. We define a < b in ℤ if there
exists an n ∈ ℕ with a + n = b, and a ≤ b if a = b or a < b. This ordering is compatible
with the addition and the multiplication in ℤ, and we get directly from the definition
and the respective statements in ℕ the following.
Theorem 1.14. Let a, b, c, d ∈ ℤ.
(1) If a ≤ b, c ≤ d, then a + c ≤ b + d.
(2) If a ≤ b, 0 ≤ c, then ac ≤ bc.
(3) If a ≤ b, c < 0, then bc ≤ ac.
Remarks 1.15. (1) Instead of a < b we also write b > a (“b is bigger than a”) and instead of a ≤ b also b ≥ a (“b is equal or bigger than a”).
(2) We now want to consider the first algebraic property of ℤ. We know that ℤ is a
commutative ring with unity 1. We call a commutative ring with unity 1 ≠ 0 an
integral domain or just a domain, if there are no nontrivial zero divisors in R, that
is, if a, b ∈ ℝ ⧵ {0} then ab ≠ 0. We have the following.
Theorem 1.16. ℤ is an integral domain.
Proof. Let a, b ∈ ℤ, a ≠ 0 ≠ b. Then ab ≠ 0.

1.4 The rational numbers ℚ

We also may construct the rational numbers from the integers with the help of an
equivalence relation. We are guided here by the known representation as fractions
a
with a, b ∈ ℤ, b ≠ 0. We use in a naive manner the following as a motivation:
b
a c
=
b d

if and only if ad = cb.

With help of this idea we introduce the rational numbers from the integers in a compact, but exact manner.
We consider the Cartesian product
A = ℤ × ℤ∗

with ℤ∗ = ℤ ⧵ {0}.

We define on A a relation by
(a, b) ∼ (c, d) ⇔ ad = cb
for (a, b), (c, d) ∈ A.
This relation is an equivalence relation. Certainly, (a, b) ∼ (a, b) and (c, d) ∼ (a, b)
if (a, b) ∼ (c, d), that is, the relation is reflexive and symmetric. It is also transitive, that
is, if (a, b) ∼ (c, d) and (c, d) ∼ (e, f ) then (a, b) ∼ (e, f ) because from ad = bc and cf = ed
we get af = eb. Using that ℤ is an integral domain.

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14 | 1 The natural, integral and rational numbers
Let ℚ ∶= A/∼ be the set of equivalence classes for this relation. We write

the equivalence class {(a′ , b′ ) ∈ A ∣ (a′ , b′ ) ∼ (a, b)} represented by (a, b).
The map

a
b

to present

φ∶ℤ→ℚ
n
n↦
1

is injective. This we see as follows.
If φ(n) = φ(m) then n1 = m1 , and this means (n, 1) ∼ (m, 1), that is, n ⋅ 1 = m ⋅ 1 and
hence, n = m.
We now may introduce an addition and multiplication on ℚ as follows:
a c ad + cb
+ =
b d
bd

a c ac
⋅ =
.
b d bd

and

These addition and multiplication are well defined, that is, if

(a, b) ∼ (a′ , b′ ) and

(c, d) ∼ (c′ , d′ ),

then
a c a′ c ′
+ =
+
b d b′ d′

and

a c a′ c ′
⋅ =
⋅ .
b d b′ d′

This is easy for the multiplication, and we leave this as an exercise. We present
the proof for the more complicated addition:
a c a′ c ′
ad + cb a′ d′ + c′ b′
+ = ′+ ′ ⇔
=
⇔ (ad + cb)b′ d′ = (a′ d′ + c′ b′ )bd
b d b
d
bd
b′ d′
⇔ adb′ d′ + cbb′ d′ = a′ d′ bd + c′ b′ bd.
This last equation holds in ℤ because ab′ = a′ b and cd′ = c′ d. Therefore the addition

and the multiplication are well defined. With this ℚ becomes a field.
Remark 1.17. We recall that a field is a set K ≠ ∅ equipped with two binary operations
+ ∶ K × K → K and ⋅ ∶ K × K → K satisfying the following three sets of properties:
– K is a commutative group under addition.
– K ∗ = K ⧵ {0} is a commutative group under multiplication.
– The multiplication is distributive with respect to the addition, that is,
a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)

for all a, b, c ∈ K.

The multiplicative inverse for ba , a ≠ 0 ≠ b, is ba .
The map
φ∶ℤ→ℚ
n
n↦
1

is not only injective, it also respects the addition and the multiplication, that is,

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