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Harcourt Mathematics 12 INT!tOLDL!CTIQM TO IVln' PROOF ~~Fhw)r s )Iv n P Whether economic natnematlcs. vou WIII use D establish r of logic and mathematics, a proof is, uence of well-formed formulas. Our erstanding of geometric proof has its roots Mlt history. In 300 B.c., the Greek tician Euclid put together all the ooks, Euclid ma I in t. In the investigate CHAPTER 1: FERMAT’S LAST THEOREM Around 1637, Pierre de Fermat, a lawyer and amateur mathematician, conjectured that if n is a natural number greater than 2, the equation xn ϩ yn ϭ zn has no solutions where x, y, and z are non-zero integers. (Solutions where some of the integers are zero are possible but not interesting, and are known as trivial solutions.) Fermat wrote the statement in the margins of his Latin edition of a book called Arithmetica, written by the Greek mathematician Diophantus in the third century A.D. Fermat claimed, “I have discovered a truly marvellous proof of this, which, however, this margin is too small to contain.” The result has come to be known as Fermat’s Last Theorem because it was the last of his conjectures to remain unre- solved after his papers were published. It became famous among mathematicians because for hundreds of years many great mathematicians attempted to prove it and achieved only partial results. Investigate Fermat’s Last Theorem is closely related to the Pythagorean Theorem, and it is known that x2 ϩ y2 ϭ z2 has integer solutions. For example, 32 ϩ 42 ϭ 52. The numbers (3, 4, 5) are called a Pythagorean triple. There are an infinite number of Pythagorean triples. Just pick two numbers a and b with a Ͼ b and set x ϭ 2ab, y ϭ a2 Ϫ b2, and z ϭ a2 ϩ b2. Then x2 ϩ y2 ϭ z2. Try it. Then prove that it works for any a and b. Similar to Fermat’s Last Theorem is Leonhard Euler’s conjecture that there are no non-trivial solutions to x4 ϩ y4 ϩ z4 ϭ w4. This question remained unresolved for over 200 years until, in 1988, Naom Elkies found that 2 682 4404 ϩ 15 365 6394 ϩ 18 796 7604 ϭ 20 615 6734 Since x2 ϩ y2 ϭ z2 has infinitely many solutions and x4 ϩ y4 ϩ z4 ϭ w4 has at least one solution, it is hard to believe that xn ϩ yn ϭ zn has no solution. DISCUSSION QUESTIONS 1. Does x3 ϩ y3 ϭ z3 have a non-trivial solution? 2. Why is there no need to consider negative integer solutions to xn ϩ yn ϭ zn? That is, if we knew there were no solutions among the positive integers, how could we be sure there were no solutions among the negative integers? 3. When Fermat’s Last Theorem was finally proven, its proof made headlines in newspapers around the world. Do you think the attention was justified? ● 2 CHAPTER 1 t echnology Section 1.1 — What Is Proof? APPENDIX P.485 The concept of proof lies at the very heart of mathematics. When we construct a proof, we use careful and convincing reasoning to demonstrate the truth of a mathematical statement. In this chapter, we will learn how proofs are constructed and how convincing mathematical arguments can be presented. Early mathematicians in Egypt “proved” their theories by considering a number of specific cases. For example, if we want to show that an isosceles triangle has two equal angles, we can construct a triangle such as A A the one shown and fold vertex B over onto vertex C. In this example, ∠B ϭ ∠C; but that is only for this triangle. What if BC is lengthened or shortened? Even if we construct hundreds of triangles, can we conclude that ∠B ϭ ∠C for every isosceles triangle imaginable? B fold C C, B Consider the following example. One day in class Sunil was multiplying some numbers and made the following observation: 12 ϭ 1 112 ϭ 121 1112 ϭ 12 321 11112 ϭ 1 234 321 11 1112 ϭ 123 454 321 He concluded that he had found a very simple number pattern for squaring a num- ber consisting only of 1s. The class immediately jumped in to verify these calcu- lations and was astonished when Jennifer said, “This pattern breaks down.” The class checked and found that she was right. How many 1s did Jennifer use? From this example, we can see that some patterns that appear to be true for a few terms are not necessarily true when extended. The Greek mathematicians who first endeavoured to establish proofs applying to all situations took a giant step forward in the development of mathematics. We follow their lead in establishing the concept of proof. In the example just considered, the pattern breaks down quickly. Other examples, however, are much less obvious. Consider the statement, The expression 1 ϩ 1141n2,where n is a positive integer, never generates a perfect square. Is this statement true for all values of n? Does this expression ever generate a perfect square? We start by trying small values of n. 1.1 WHAT IS PROOF? 3 1141(1)2 ϩ 1 ϭ 1142, which is not a square (use your calculator to verify this) 1141(2)2 ϩ 1 ϭ 4565, which is not a square 1141(3)2 ϩ 1 ϭ 10 270, which is not a square 1141(4)2 ϩ 1 ϭ 18 257, which is not a square Can we conclude that this expression never generates a perfect square? It turns out that the expression is not a perfect square for integers from 1 through to 30 693 385 322 765 657 197 397 207. It is a perfect square for the next integer, which illustrates that we must be careful about drawing conclusions based on calculations alone. It takes only one case where the conclusion is incorrect (a counter example) to prove that a statement is wrong. We can use calculations or collected data to draw general conclusions. In 1854, John Snow, a medical doctor in London, England, was trying to establish the source of a cholera epidemic that killed large numbers of people. By examining the location of infection and analyzing the data collected, he concluded that the source of the epidemic was contaminated water. The water was obtained from the Thames River, downstream from sewage outlets. By shutting off the contaminated water, the epidemic was controlled. This type of reasoning, in which we draw general conclusions from collected evidence or data, is called inductive reason- ing. Inductive reasoning rarely leads to statements of absolute certainty. (We will consider a very powerful form of proof called inductive proof later in this book.) After we collect and analyze data, the best we can normally say is that there is evidence either to support or deny the hypothesis posed. Our conclusion depends on the quality of the data we collect and the tests we use to test our hypothesis. In mathematics, there is no dependence upon collected data, although collected evidence can lead us to statements we can prove. Mathematics depends on being able to draw conclusions based on rules of logic and a minimal number of assumptions that we agree are true at the outset. Frequently, we also rely on definitions and other ideas that have already been proven to be true. In other words, we develop a chain of unshakeable facts in which the proof of any state- ment can be used in proving subsequent statements. In writing a proof, it is important to explain our reasoning and to make sure that assumptions and definitions are clearly indicated to the person who is reading the proof. When a proof is completed and there is agreement that a particular statement can be useful, the statement is called a theorem. A theorem is a proven statement that can be added to our problem-solving arsenal for use in proving subsequent statements. Theorems can be used to help prove other ideas and to draw conclusions about specific situations. Theorems are derived using deductive reasoning. Deductive reasoning allows us to prove a 4 CHAPTER 1 statement to be true. Inductive reasoning can give us a hypothesis, which might then be proved using deductive reasoning. As an example of inductive reasoning, note that if we write triples of consecutive integers, say (11, 12, 13), exactly one of the three is divisible by 3. If a number of such triples are written (say two or three by everyone in the class), we can observe that every triple has exactly one number that is a multiple of 3. This provides strong evidence for us to conclude inductively that every such triple contains exactly one multiple of 3, but it is not proof. We will consider deductive proof in the other sections of this chapter. Deductive reasoning is a method of reasoning that allows for a progression from the general to the particular. Inductive reasoning is a method of reasoning in which specific examples lead to a general conclusion. Exercise 1.1 Part A In each of the following exercises, you are given a mathematical statement. Using inductive reasoning (that is, testing specific cases), determine whether or not the claim made is likely to be true. For those that appear to be true, try to develop a deductive proof to support the claim. 1. All integers ending in 5 create a number that when squared ends in 25. Test for the first ten positive integers ending in 5.
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