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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