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Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1 Lecture Notes
Introduction to the Natural Numbers
Winter 2017
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
What Is Mathematics?
Here are some definitions, found after a quick search on the
internet:
1. Mathematics is the science that deals with the logic of shape,
quantity and arrangement.
2. Mathematics is the study of topics such as quantity, structure,
space, and change.
3. Aristotle defined mathematics as “the science of quantity,”
and this definition prevailed until the 18th century.
4. An early definition of mathematics in terms of logic was
Benjamin Peirce’s “the science that draws necessary
conclusions” (1870)
5. Another logicist definition of mathematics is Russell’s “All
Mathematics is Symbolic Logic” (1903)
Chapter 1 Lecture Notes Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
More Definitions
I
Intuitionist definitions, developing from the philosophy of
mathematician L.E.J. Brouwer, identify mathematics with
certain mental phenomena. An example of an intuitionist
definition is “Mathematics is the mental activity which
consists in carrying out constructs one after the other.”
I
Formalist definitions identify mathematics with its symbols
and the rules for operating on them. Haskell Curry defined
mathematics simply as “the science of formal systems”.
I
Generalist definition: “the science of numbers and their
operations, interrelations, combinations, generalizations, and
abstractions and of space configurations and their structure,
measurement, transformations, and generalizations.”
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
The Abstract Nature of Mathematics
Pure mathematics consists entirely of such assertions as that, if
such and such a proposition is true of anything, then such and
such another proposition is true of that thing... It’s essential not to
discuss whether the proposition is really true, and not to mention
what the anything is of which it is supposed to be true... If our
hypothesis is about anything and not about some one or more
particular things, then our deductions constitute mathematics.
Thus mathematics may be defined as the subject in which we
never know what we are talking about, nor whether what we are
Bertrand Russell
saying is true.
I would say that mathematics is the science of skillful operations
with concepts and rules invented just for this purpose.
Eugene Wigner, The Unreasonable Effectiveness of
Mathematics in the Natural Sciences, 1960
Chapter 1 Lecture Notes Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
The Usefulness of Mathematics
Philosophy is written in that great book which ever lies
before our eyes (I mean the universe) but we cannot
understand it if we do not first learn the language and
grasp the characters in which it is written. It is written in
the language of mathematics, and the characters are
triangles, circles and other geometrical figures, without
which it is humanly impossible to comprehend a single
word of it, and without which one wanders in vain
through a dark labyrinth.
Galileo, The Assayer, 1623
The ‘real’ mathematics, of the ‘real’ mathematicians, the
mathematics of Fermat and Euler and Gauss and Abel
and Riemann, is almost wholly useless.
G. H. Hardy, A Mathematician’s Apology, 1940
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Mathematics: Science or Art?
Mathematics, rightly viewed, possesses not only truth,
but supreme beauty: a beauty cold and austere, like that
of sculpture, without appeal to any part of our weaker
nature, without the gorgeous trappings of painting or
music, yet sublimely pure, and capable of a stern
perfection such as only the greatest art can show. The
true spirit of delight, the exaltation, the sense of being
more than Man, which is the touchstone of the highest
excellence, is to be found in mathematics as surely as in
poetry.
Bertrand Russell, A History of Western Philosophy, 1945
Chapter 1 Lecture Notes Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Some Interesting Books About Mathematics
1. Symmetry by Hermann Weyl, 1952
2. The Road to Reality, A Complete Guide to the Laws of the
Universe by Roger Penrose, 2004
3. Symmetry and the Monster by Mark Ronan, 2006
4. Dr. Euler’s Fabulous Formula by Paul J. Nahin, 2006
5. The Black Swan: The Impact of the Highly Improbable by
Nassim Nicholas Taleb, 2007
6. Is God a Mathematician? by Mario Livio, 2009
7. Mathematics, An Illustrated History of Numbers, edited by
Tom Jackson, 2012
8. Our Mathematical Universe by Max Tegmark, 2014
9. A Beautiful Question, Finding Nature’s Deep Design by Frank
Wilczek, 2015
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
What Are the Natural Numbers?
We assume you are familiar with the natural numbers,
1, 2, 3, . . . , etc.
The set of natural numbers is denoted by N. In set notation we
write
N = {1, 2, 3, 4, . . . , n, . . . }
There are two operations with which we assume you are also
familiar:
1. Addition: for m, n ∈ N the sum m + n is also in N.
2. Multiplication: for m, n ∈ N the product m · n, or simply m n,
is also in N.
The key property of N is that if n ∈ N then so is n + 1. As a
consequence there is no largest natural number.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
What Are the Integers?
A larger set of numbers is obtained by including the number 0 and
the negatives of natural numbers. This larger set of numbers is
called the set of integers and is denoted by Z. That is,
Z = {0, ±1, ±2, ±3, . . . , ±n, . . . }
This larger set of numbers is now also closed under subtraction: if
m, n ∈ Z, then m − n is also in Z. We assume you are familiar with
the following key properties of Z :
1. for m ∈ Z, m + 0 = m,
2. for m ∈ Z, m + (−m) = 0,
3. if m > 0 and n > 0, then mn > 0,
4. if m > 0 and n < 0, then mn < 0,
5. if m < 0 and n < 0, then mn > 0.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Divisibility of Natural Numbers
We say the natural number m is divisible by the natural number n
if there is a natural number q such that m = n q. For example, 15
is divisible by 3 since 15 = 3 · 5. For m, n ∈ N, we can say m is
divisible by n if and only if
m
∈ N.
n
Note that every natural number m is divisible by 1 and itself, since
m = 1 · m. If m = a · b, for natural numbers a, b then m is divisible
by a (and b), and we call a a factor, or divisor, of m. Factoring a
number means to write it out as a product of factors. For example,
144 = 12 · 12 = 3 · 48 = 9 · 16 = 3 · 3 · 2 · 2 · 2 · 2
are all possible factorizations of 144.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
What Is a Prime Number?
The natural number n 6= 1 is called a prime number if n is only
divisible by 1 and itself. The first few prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, . . .
If a number n 6= 1 is not prime then it is called composite. For
example, 144 is a composite number since it is divisible by 2 and 3,
among others; 49 is a composite number since it is divisible by 7.
But 101 is prime. To confirm this we need only check that 101 is
not divisible by any of the natural numbers 2, 3, 4, 5, 6, 7, 8, 9 or
10. Why can we stop at 10?
Note: the number 1 is neither prime nor composite.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Theorems and Proofs
A theorem is a general statement proved by a chain of reasoning.
The logical reasoning that starts with the hypotheses of the
theorem and ends with its conclusion is called a proof. A canonical
historical example of this type of logical development is furnished
by Euclid’s Elements, which develops conclusions about lines,
triangles and other geometric figures by logical deduction from
axioms or propositions. One goal of this course is to make you
comfortable with this aspect of mathematics. An example of a
theorem that you may be familiar with is the Mean Value Theorem:
If the function f is continuous on the closed interval
[a, b] and differentiable on the open interval (a, b), then
there is a number c ∈ (a, b) such that
f 0 (c) =
f (b) − f (a)
.
b−a
Chapter 1 Lecture Notes Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
What is a Lemma?
A lemma is the establishment of a preliminary fact that will be
used to prove a theorem. Here is a lemma that we shall use to
prove that there is no largest prime number:
Lemma 1.1.1: every composite number m has a divisor that is a
prime number.
Proof: since m is composite it has divisors a and b such that
m = a · b and neither a nor b is 1. Thus 1 < a < m. If a is prime
we are finished. If not, then a is composite and it has divisors c
and d, which must also be divisors of a, such that a = c · d and
neither c nor d is 1. Thus 1 < c < a < m. If c is prime, we are
finished. If not . . . we can find a smaller divisor e of c (and m)
such that 1 < e < c < a < m. But this cannot continue
indefinitely, since each new divisor of m must be at least one less
than the previous divisor. So at some stage a divisor of m is not
composite, it is a prime number.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
There Are Infinitely Many Prime Numbers
Theorem 1.1.2: there is no largest prime number.
Proof 1: let p be a prime number. Let
m = (2 · 3 · 5 · 7 · 11 · 13 · 17 · 19 · 23 · · · · · p) + 1.
That is, m is 1 plus the product of all the prime numbers less than
or equal to p. If m is a prime number, then it is larger than p, and
we are finished. If m is a composite number, then by Lemma 1.1.1,
it must have a prime factor, say q. But q 6= 2 because 2 is not a
divisor of m; q 6= 3, since 3 is not a divisor of m. In fact q is none
of the prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, . . . , p,
since none of these is a divisor of m. So q must be a prime larger
than p.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Proof by Contradiction
Here is another way to prove Theorem 1.1.2. It involves what’s
known as proof by contradiction, or reductio ad absurdum.
Proof 2: suppose there is a largest prime p. Let
m = (2 · 3 · 5 · 7 · 11 · 13 · 17 · 19 · 23 · · · · · p) + 1.
That is, m is the product of all the prime numbers, plus 1. Then m
is not prime, because it is bigger than the largest prime, and none
of the primes
2, 3, 5, 7, 11, 13, 17, 19, 23, . . . , p,
divides m. This contradicts Lemma 1.1.1. Thus there is no largest
prime number.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
Chapter 1: Introduction to the Natural Numbers
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Twin Primes
The study of prime numbers and their properties started in
antiquity and continues to this day. In fact there are still many
simple questions, and some not so simple, about prime numbers
that have still not been answered.
For example, consider twin primes, that is, two consecutive odd
numbers which are both primes. The first few paris of twin primes
are 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31.
Are there infinitely many paris of twin primes? The answer to this
question is still not known.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
Chapter 1: Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
The Goldbach Conjecture
Another outstanding question about prime numbers is Goldbach’s
Conjecture:
Every even natural number bigger than 2 is the sum of
two primes.
It is s called a conjecture because the statement has not yet been
proved. Nor has it been disproved. For example,
6 = 3 + 3, 20 = 3 + 17, 40 = 17 + 23,
and less obviously,
22, 901, 764, 048 = 22, 801, 763, 489 + 100, 000, 559.
But whether there is an even number for which this is impossible
has not yet been determined.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Chapter 1: Introduction to the Natural Numbers
Distribution of Prime Numbers
We have proved that there are infinitely many prime numbers. But
they do become rarer as they get larger. That is, if we let π(x) be
the number of primes less than x, then it has been proved that
π(x)
= 1.
x→∞ x/ ln(x)
lim
This result is known as the Prime Number Theorem; it was proved
near the end of the 19th century by J. Hadamard. It can be
restated as
x
π(x) ≈
.
ln x
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla
1.0: Introduciton
1.1: Prime Numbers
1.2: Unanswered Questions
Chapter 1: Introduction to the Natural Numbers
The Riemann Zeta Hypothesis
Now that Fermat’s Last Theorem has been proved, the most
famous unsolved problem in mathematics is the Riemann Zeta
Hypothesis, which is connected, albeit in a far from obvious way,
with the distribution of primes. Let
∞
X
1
,
ζ(s) =
ns
n=1
where s can be a complex number. The Riemann Zeta Hypothesis
states that if the real part of s is between 0 and 1, and
ζ(s) = 0,
then the real part of s equals 1/2. No counter example has yet
been found; but neither has the hypothesis been proved.
Chapter 1 Lecture Notes Introduction to the Natural Numbers
MAT246H1s Lec0101 Burbulla