It is important to convince yourself of the truth of each proof as you work through the book, and make sure you have a complete understanding. Apr 03, 2018 former mcdonalds worker does a number theory proof duration. With an indirect proof, instead of proving that something must be true, you prove it indirectly by showing that it cannot be false. A second form of indirect argument, argument by contraposition, is based on the logical equivalence between a statement and its contrapositive. Reading the book, i think i have to this by contradiction like assuming that the number of primes is. Therefore, the organization of the notes mostly follows this book. Proofs are all about logic, but there are different types of logic. In this book, all numbers are integers, unless speci. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. The conjunction of the formulas p and q is the formula \p and q, which is written symbolically as p q.
Science fiction book about everyone vanishing except two guys and a young boy. This book is designed for a one semester course in discrete mathematics. Division given two integers, say a and b, the quotient ba may or may not be an integer e. Theorems, corollaries, lemmas, and methods of proof wiley. Greek number theory was exclusively interested in natural numbers. A number that is not rational is called irrational and cannot be written as a fraction, pq, where p and q are both integers. Each illustrates an aspect of set theory that youve learned so far. Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs reinforces the foundations of calculus and algebra explores how to use both a direct and indirect proof to prove a theorem presents the basic properties of real numbers. Properties of integers proving existential statements universal statements and basic techniques of direct proof common mistakes getting proofs started disproof by counterexample proof by contradiction. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. My understanding is that intuitionists reject such proofs because they rely on the law of the excluded middle, which they dont accept. Direct proofs of lehmussteiners theorem are proposed. That does not reduce its importance, and if anything it enhances its fascination.
I have seen girads proof and types recommended a few times but other than am wondering. Proof of the prime number theorem joel spencer and ronald graham p rime numbers are the atoms of our mathematical universe. To prove algebraic concepts example 2 page 356 in everyday situations example 3 page 356 to prove concepts in number theory example 4 page 357. Book of proof by richard hammack virginia commonwealth university this textbook is an introduction to the standard methods of proving mathematical theorems. When reading mathematical books you will notice that the word implies appears. An introduction to higher mathematics ebooks directory. Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs reinforces the foundations of calculus and algebra explores how to use both a direct and indirect proof to prove a theorem. For p q to be true both p and q must be true, otherwise it.
Number theory euclids number theory the overall structure elements viiix as in earlier books, euclid probably based much of his work on the discoveries of others, but the organization and presentation was his own. Former mcdonalds worker does a number theory proof duration. One kind of indirect proof, argument by contradiction, is based on the fact that. The thirteen books of euclids elements translation and commentaries by t. Strategies that are covered include direct and indirect proof, biconditional. Euler is best remembered for his contributions to analysis and number theory, especially for his use of infinite processes of various kinds infinite sums and products, continued fractions, and for establishing much of the modern notation of mathematics. What unites them is that they both start by assuming the denial of the conclusion. An alternative strategy for proving a b called indirect proof or. Indirect proof in algebra and geometry ck12 foundation.
An introduction to higher mathematics whitman people. It is written for an audience of mathematics majors at virginia commonwealth university, and is intended. For example, propositions 5 and 6 in euclid book i are about the equality of the base angles of an isosceles triangle see note 2 below. Direct proof is possible if we have agreed axioms and definitions to start from and an agreed method a logical argument that enables us to proceed logically step by step from what we know to what we do not know, but think is true. When your task in a proof is to prove that things are not congruent, not perpendicular, and so.
This relationship between d and n is symbolized d n. Does there exist a direct and constructive proof of the infinitude of primes. In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to. Eduardo is proving the theorem above by contradiction. Long story short, deductive proofs are all about using a general theory to prove something specific. Most of number theory has very few practical applications. Another common type of reasoning is indirect reasoning, which you have likely done outside of math class. Proof by contradiction a proof by contradiction is a proof that works as follows. A simple, constructive proof, based mainly on euclid, book iii, is given in. Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs reinforces the foundations of calculus and algebra explores how to use both a direct and indirect proof to prove a theorem presents the basic properties of real numbers discusses how to use mathematical. Number theory book universitas pendidikan indonesia. The greater number is amultipleof the lesser when it is measured by3 the lesser number. Based on the assumption that p is not true, conclude something impossible.
Many talented mathematicians before euler had failed to discover the value of the sum of the reciprocals of the squares. Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic. This is being written as a textbook for math 502, logic and set theory, and. By showing this assumption to be logically impossible, you prove the original conclusion true.
The interested reader can refer to the book proofs from the bookthere is an interesting reason behind this intriguing title, which has the six famous proofs of the infinitude of primes. Proof by contradiction also known as indirect proof or the method of reductio ad absurdum is a common proof technique that is based on a very simple principle. Proofs and mathematical reasoning university of birmingham. It was first proved by euclid in his work elements.
Contradiction and contraposition in a direct proof you start with the hypothesis of a statement and make one deduction after another until you reach the conclusion. That just might be an obsolete aspect of this book. But compared with infinity, any number, no matter how large, is less than a drop in the bucket. An introduction to higher mathematics download link. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role. Aunitis that by virtue of which each of the things that exists is called one. Therefore, that essay is where my reading of takeutis proof theory ends. No attempts will be made to derive number theory from set theory and no knowledge of calculus will be assumed. In set theory books, the authors simply prove theorems in a normal mathematical way, so perhaps in 1987, a specifically proof theoretical attack on set theory was too difficult. Proof about the quotient remainder theorem by indirect proof. Letting pn denote the number of primes p b n, gauss conjectured in the early. To prove that p is true, assume that p is not true. Euclids theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. Number theory concerns the former case, and discovers criteria upon which one can decide about divisibility of two integers.
Indirect proofs are sort of a weird uncle of regular proofs. Browse the amazon editors picks for the best books of 2019, featuring our favorite. Each theorem is followed by the notes, which are the thoughts on the topic. In 250, and, particularly, in number theory, the mathematical statements that we. Number theory wikibooks, open books for an open world. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. Euler used infinite series to establish and exploit some remarkable connections between analysis and number theory. Discrete mathematics proof by contradiction youtube. Its useful to note that the direction of the proof cannot be done directly, and requires some indirect technique eg. Exactly the same sequence of steps can be used as the heart of a proof by contradiction for the given statement. What are some good introductory proof theory books and what are some good nonintroductory proof theory texts as well. A number of examples will be given, which should be a good resource for further. I was recently bemoaning the lack of approachable proof theory textbooks to a colleague whos from that world, but unfortunately he couldnt offer any better suggestions for introductory books.
In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent. Its a principle that is reminiscent of the philosophy of a certain fictional detective. Proof theory was created early in the 20th century by david hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics in arithmetic number theory, analysis and set theory. The textbook is right, the indirect proof is applicable, that is, prove the contrapositive.
It is written for an audience of mathematics majors at virginia commonwealth university, and is intended to prepare the students for more advanced courses. Erdos answered that he reckoned we should do as hardy and littlewood. Browse other questions tagged numbertheory elementarynumbertheory or ask your own question. Indirect proof, also called proof by contradiction, assumes the hypothesis if. Indirect proof definition of indirect proof by the free. Oct 19, 2016 the interested reader can refer to the book proofs from the bookthere is an interesting reason behind this intriguing title, which has the six famous proofs of the infinitude of primes. Today, pure and applied number theory is an exciting mix of simultaneously broad and deep theory, which is constantly informed and motivated. May 18, 2015 we take a look at an indirect proof technique, proof by contradiction. Strayer, elementary number theory published by pws publishing company, 1994. Euclids proof of the infinitude of primes is an indirect proof a. Chapter 1 contains several important topics needed to understand next chapters, namely proof techniques direct and indirect proofs. Knock the first domino over and they all will fall. A treatise on formal proofs umd department of computer science. Long story short, deductive proofs are all about using a.
Preface this book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Any advice on how to preemptively identify that i should use an indirect instead of misleading myself into a direct proof trap. Theorems, corollaries, lemmas, and methods of proof. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second. Were on problem number four, and they give us a theorem. The fact that a proof in number theory exists that uses topology. A proof by construction that uses cases and an indirect. Platoos dominoes one for each natural number an infinite row, 0, 1, 2, e of dominoes, one domino for each natural number. So the way you prove by a contradiction, youre like, well what if.
Learn exactly what happened in this chapter, scene, or section of geometric proofs and what it means. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Euclidean and noneuclidean geometry have a common basic set. Dear jamie, supreme integral with feynmans trick duration. In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. We take a look at an indirect proof technique, proof by contradiction.
When we use the indirect proof method, we assume the opposite of our theory to be true. For example, here are some problems in number theory that remain unsolved. For any natural number n, the sum of all natural numbers less than n is not equal to n. I am really interested in mathematical proof theory but do not know what a good place to start would be. Andrews, evan pugh professor of mathematics at pennsylvania state university, author of the wellestablished text number theory first published by saunders in 1971 and reprinted by dover in 1994, has led an active career discovering fascinating phenomena in his chosen field number theory. They are closely related, even interchangeable in some circumstances, though proof by contradiction is more powerful. For those who wish to use this as a reference book, an index of theorems will be given. Let k 0 be the lowest numbered domino that remains standing.
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