History of Math for the Liberal Arts CHAPTER 4. The Pythagoreans. Lawrence Morales. Seattle Central Community College

1 3 4 History of Math for the Liberal Arts 5 6 CHAPTER 4 7 8 The Pythagoreans 9 10 11 Lawrence Morales 1 13 14 Seattle Central Community College MAT107 Chapter 4, Lawrence Morales, 001; Page 1

15 16 17 18 19 0 1 3 4 5 6 7 8 9 30 31 3 33 34 35 36 37 38 39 40 41 4 43 44 Table of Contents Table of Contents... Part 1: Figurative and Friendly Numbers... 4 Introduction... 4 Meet Pythagoras... 5 Pythagorean Arithmetica... 6 Figurative Numbers... 7 Friendly Numbers... 17 Part : The Pythagorean Problem... 0 Proofs in Other Civilizations... Pythagorean Triples... 4 The Pythagorean Formula for Triples... 4 The Triples of Proclus... 5 The Triples of Plato... 7 The Triples of Euclid... 9 Part 3: Irrational Numbers (Incommensurables) and the Pythagoreans... 31 Part 4: Modern Facts on Rational and Irrational Numbers... 33 Starting Simple... 33 Rational Numbers in More Detail... 34 Rational Numbers: Terminating and Repeating Decimals... 34 Non Terminating, Repeating Decimals... 37 What About the Other Direction?... 38 Irrational Numbers... 41 Proof by Contradiction... 4 Generating Hoards of Irrational Numbers... 44 Polynomial Equations... 45 The Rational Roots Theorem... 48 Using Rational Roots Theorem to Show Irrationality... 50 Part 5: Pythagorean Geometry, A Brief Discussion... 54 Conclusion... 54 MAT107 Chapter 4, Lawrence Morales, 001; Page

45 46 47 48 49 50 51 5 53 54 55 56 57 58 59 Part 6: Homework Problems... 55 Triangular Numbers... 55 Pentagonal Numbers... 57 Oblong Numbers... 57 Non-Standard Figurative Numbers... 58 Other Proofs of the Pythagorean Theorem... 58 Pythagorean Triples via the Pythagoreans... 64 Pythagorean Triples via Proclus... 65 Pythagorean Triples via Plato... 65 Pythagorean Triples via Euclid... 66 Generalizing Pythagorean Facts... 66 Rational and Irrational Numbers... 67 Writing... 68 Part 6: Chapter Endnotes... 71 MAT107 Chapter 4, Lawrence Morales, 001; Page 3

60 61 6 63 64 65 66 67 68 69 70 71 7 73 74 75 76 77 78 79 80 81 8 83 84 85 86 87 88 89 90 91 9 93 94 95 96 97 98 99 100 101 10 103 104 105 Part 1: Figurative and Friendly Numbers Introduction In the last few centuries of the second millennium B.C.E., we see many changes take place on political and economic fronts. Great civilizations such as Egypt and Babylon, powerful in the past, began to lose their power. New groups such as the Israelites (Hebrews), Assyrians, and Greeks began to emerge. As the Iron Age began, all sorts of new tools emerged that changed the way people lived their lives. Around this time, things like coins and an alphabet were being introduced, and trade became more and more a part of life. Along the coast of Asia Minor, in Greece, and in Italy, major trading towns emerged that took advantage of this new era. Along with this change came a shift in the kinds of questions people are believed to have begun to ask. For what appears to be the first time, people began to explore mathematical questions like Why is it true that the diameter of a circle precisely cuts the circle into two equal pieces? The question of Why? became much more prominent, whereas before people were content with asking How? (As in, How do I compute what is owed to me for these goods? ) At this time, it became more important for people to demonstrate that certain things were true. To this day demonstrative mathematics rules the landscape of the field modern mathematicians do no fully believe a mathematical statement until it is proven with one hundred percent certainty. (The beyond reasonable doubt threshold of proof is soundly rejected by almost all modern mathematicians.) The idea of demonstrating that something is true appears to have started with Thales of Miletus (c. 64 547 B.C.E.) 1. Some believe that he lived in Egypt for a while, gaining a knowledge of the work of the Egyptians. In Miletus, he was well known as a mathematician (among other things) and is the first known person in mathematics that is given credit for proving mathematical statements. This does not mean that others before him did prove such statements, only that they did not get credit for their efforts. He is thought to have proved the following, for example: a. A circle is bisected (cut into two equal pieces) by its diameter. b. The base angles of an isosceles triangle are equal. (Recall that an isosceles triangle is one in which two of the sides are the same length.) Figure 1 c. The vertical (opposite) angles formed by two intersecting straight lines are equal. In the figure shown, angles a and b are vertical and thus are equal, having the same number of degrees. These and other statements he proved are now considered basic facts of geometry and we accept them as true almost at face value. a b Thales is recognized for trying to find some logical reasoing that would demonstrate they were true instead of relying on intuiton to establish such facts. a = b Figure MAT107 Chapter 4, Lawrence Morales, 001; Page 4

106 107 108 109 110 111 11 113 114 115 116 117 118 119 10 11 1 13 14 15 16 17 18 19 130 131 13 133 134 135 136 137 138 139 140 141 14 143 144 145 146 147 148 149 150 151 Miletus begins a long line of mathematicians in search of proof. The history of the few hundred years of Greek mathematics is difficult to detail because so few primary sources of information are available (unlike the Egyptian and Babylonian records which we have in hand). Much of the information that we have comes from manuscripts and sources that were written well after this time period, even hundreds of years later. This means that what we have is a hypothetical account of Greek mathematics. But, from what we do have, we can try to piece together a little bit of history. One question often asked is how the Greeks were influenced by the mathematics of other civilizations like the Babylonians and Egyptans. In the past, these peoples were not given much credit for influencing Greek mathematics (widely considered an important development in the history of mathematics overall). However, recent evidence suggests that their influence may have been greater than first thought. Primarily, the Greeks themselves appear to express respect for the work of the East in their own writings 3. Other internal connections related to arithmetic and astronomy also seem to exist. 4 When studying Greek mathematics, one thing we see immediately is that early Greek mathematics is overshadowed by the achievement of Euclid (c 35 B.C.E. 65 B.C.E.). Euclid s Elements, which we will look at more closely later, is a compliation of all the important mathematics known at the time, and, although not the first of its kind, is really the only one that survives due to its superiority. Finding informatin on other mathematicians work is not an easy task. Meet Pythagoras Pythagoras, pictured here 5, is one of the best known early Greek mathematicians. He was born around 570 B.C.E. in Samos. This was about 50 years after Thales was born, so some have speculated that he studied with Thales. Others believe it was more likely that he studied under Anaximander. 6 We do not know for sure if this is true. From what we know, we think that after studying under some teacher, he went to Egypt and then eventually to Babylon for seven years. (Note the connection between Pythagoras and the mathematics we ve already studied in this course.) In Babylon, he probably learned Mesopotamian and Egyptian mystical rites, numbers, and music. 7 Afterward he went back to Greece and eventually settled in Croton where he established his philosophical/religious society. He was a mystic, Figure 3 philosopher, prophet, geometer, and sophist. 8 Pythagoras was the leader of a group of followers, now known as the Pythagoreans, which was a secret society based on following his teachings. Because of their secrecy as well as the Near Eastern practice of oral teaching, no firsthand records of their teachings, beliefs, or work exist. MAT107 Chapter 4, Lawrence Morales, 001; Page 5

15 153 154 155 156 157 158 159 160 161 16 163 164 165 166 167 168 169 170 171 17 173 174 175 176 177 178 179 180 181 18 183 184 185 186 187 188 189 190 191 19 193 194 195 196 197 The Pythagorean Society was an elite, well known group of people, numbering in the several hundreds. They were vegetarians, shared their possessions, believed in the transmigration and reincarnation of souls, as well as having other common beliefs and practices. This group was very unique due to their belief that one achieves union with the divine through numbers or mathematics. To the Pythagoreans, you could understand all that pertained to the universe by studying numbers. God created the universe by using numbers, said the Pythagoreans, and the universe still depends on them for its continued existence. The phrase Everything is number is an accurate description of their worldview. Members of the society would generally take up the study of four subjects. They were: 1. Arithmetica the study of numbers and number theory (as opposed to basic calculations). Harmonia the study of music 3. Geometria the study of geometry 4. Astrologia the study of astronomy Later, in the Middle Ages, this became know as the Quadrivium and was adopted as the standard set of subjects to be studied while receiving a liberal arts education. (How would you like it if three out of four classes required for an A.A. degree were geometry, number theory, and astronomy?) Pythagoras divided his followers into two groups. One studied mystical religious matters while the other studied scientific matters. For the first three years of their involvement, members were listeners who were silent as they heard their teacher speak. After that time, they were allowed into the mathematikoi group, where they would receive more complete teachings as well as be allowed the opportunity to express their own opinions and even elaborate on the teachings of Pythagoras. Pythagorean Arithmetica Prior to the Pythagoreans, it is generally believed that mathematicians in other cultures were primarily interested in mathematics for its applications to such fields as surveying, commerce, architecture, etc. The Pythagoreans, however, were more focused on the mystical aspects of numbers and so seem more conscious of more abstract principles associated to geometric figures and numbers. They believed that the universe could be interpreted by the study of numbers, which existed in a self contained realm. 9 As an outgrowth of their studies and their attitudes towards mathematics and numbers, the Pythagoreans helped to create what we might call pure mathematics. Question to think about: why would it be called pure mathematics? They also helped to establish the idea of formal proof as they placed an emphasis on finding results by following a chain of logical reasoning. One belief that they held was that geometry lies beneath physical objects, and that numbers lie beneath geometry. To them, numbers are what we would call the positive integers: {1,,3,4,5,.}. They took these numbers and linked them to shape with what are called figurative numbers. MAT107 Chapter 4, Lawrence Morales, 001; Page 6

198 199 00 01 0 03 04 Figurative Numbers Figurative numbers are numbers that can be represented as geometric figures. The easiest figurative number for us to recognize is the square number. These are numbers, which can be represented with dots in the shapes of squares. Here are the first few square numbers, which you probably recognize as perfect squares. Figure 4 S n = 1 S = 4 S 3 = 9 S 4 = 16 05 06 07 08 09 10 11 1 13 14 15 You can visually see why they are called square numbers. The formula for the n th square number is given by: S n = n Another example of figurative numbers, that are not as familiar to most of us, is that of the triangular numbers. The triangular numbers are numbers that can be represented as triangles. Each number can be represented with a series of dots. The triangular numbers are 1,3,6,10,15,1,8, and are pictured below. Figure 5 T 1 =1 T = 3 T 3 = 6 T 4 = 10 16 17 The first figure (on the left) has 1 dot, the second figure has 3 dots, the third has 6 dots, etc. MAT107 Chapter 4, Lawrence Morales, 001; Page 7

18 19 0 1 3 4 5 6 7 8 9 30 31 3 33 34 To distinguish the triangular numbers from each other, we ll use the following notation to name them: T 1 = 1, the first triangular number. T = 3, the second triangular number. T 3 = 6, the third triangular number. Etc. Note that T 3 has three dots on each side of the triangle, so we say its side has length of three. In general, the n th triangular number, which we denote by T n, has n dots per side on the triangle. If you look closely, you can see how each succeeding triangular number is built by adding a diagonal row of dots to the previous triangular number. (We ll need this for a proof later.) Figure 6 T 1 =1 T = 3 T 3 = 6 T 4 = 10 35 36 37 38 39 40 41 4 43 44 45 46 47 48 49 There are some interesting facts that emerge when we study triangular numbers in more detail. For example: FACT1: The sum of two consecutive triangular numbers always equals the square whose side is the same as the side of the larger of the two triangles. Huh? Read that again and then consider the following illustration of FACT 1. Figure 7 We are looking at the triangular numbers T 4 and T 5. These are two consecutive triangular numbers, as FACT1 requires. What is the sum of these two consecutive triangular numbers? Numerically, we have the sum to be: T 4 =10 T 5 =15 MAT107 Chapter 4, Lawrence Morales, 001; Page 8

50 51 5 53 54 55 56 57 58 T 4 +T 5 = 10 + 15 = 5 = 5 Note that the sum is actually a perfect square, which we can interpret as a square with area 5 and with each side of length 5. FACT1 states this square is the same as the one whose side is the same as the larger of the two triangles. If you look at the larger of the two triangular numbers above, each side has 5 dots, which is what FACT1 is stating. Geometrically, you can see what is happening in the following diagram: Figure 8 Rotate me around + = T 4 T 5 59 60 61 6 63 64 65 66 67 68 69 70 71 7 73 74 This picture shows that when you add up these two triangular numbers, you do get a square and its side has length 5 this is the length of the side of the larger triangle that corresponds to T 5. If you take any two consecutive triangular numbers then, and then geometrically fit them together, you will get a square. FACT: The n th n( n +1) triangular number, T n, is given by the formula T n =. (This is a modern representation the Pythagoreans certainly did not have such notation.) Let s first see if this is true for a triangular number whose value we already know. We know T 5 = 15. The formula gives: T 5 ( + ) 55 1 = 5 6 = 30 = = 15 MAT107 Chapter 4, Lawrence Morales, 001; Page 9

75 76 77 78 79 80 81 8 83 84 85 86 87 88 89 90 91 9 93 94 95 96 97 98 99 300 301 30 303 304 305 306 307 308 309 310 311 31 The formula works for n = 5, as it should. You can try a few more if you want to check for yourself. Example 1 Solution: Check Point A Solution: What is the 50 th triangular number? We certainly don t want to try to draw 50 triangles, or even just one with 50 dots on each side. The value of the formula becomes apparent as we can now simply compute T 50 by using the formula carefully. T 50 ( + ) 50 50 1 = 50 51 = 550 = = 175 Hence, the 50 th triangular number has 1,75 dots that compose it. Find the 100 th triangular number, T 100. See endnotes to check your answer. 10 We can also ask the opposite question Example Solution: Is 105 a triangular number? To answer this question, we need to determine if there is a positive integer n that will make the following equation true: n( n + 1) 105 = Think About It Why is T 100 more than twice the size of T 50 if 100 is only twice the size of 50? Think About It Why is it enough to solve this equation? To solve this equation, we will eventually need the quadratic formula. Recall that the quadratic formula states that if ax + bx + c = 0, then the solutions of MAT107 Chapter 4, Lawrence Morales, 001; Page 10

313 314 315 316 317 318 319 30 31 3 33 34 35 36 37 38 39 330 331 33 333 334 335 Check Point B Solution: b ± b 4ac this equation are x =. We now proceed to check if 105 is a a triangular number. Step Comments n( n + 1) Starting equation 105 = 10 = n ( n + 1) Multiply both sides by. 10 = n + n Distribute the n on the right side 0 = n + n 10 Set the equation equal to 0 so that it is in standard form and we can use the quadratic formula. Note that a = 1, b = 1, and c = 10 1± 1 4(1)( 10) Apply the quadratic equation. n = (1) 1± 841 Carefully simplify the inside of the n = radical 1± 9 Take the square root n = n = 14 or n = 15 Simplify Since n must be a positive integer, we toss out the negative solution and are left to conclude that 105 must be the 14 th triangular number. Is 76 a triangular number? Think About It If we don t start with a triangular number like we did in this example, what kind of result should we expect from the quadratic formula? (See Check Point C for an example.) See endnotes to check your answer. 11 MAT107 Chapter 4, Lawrence Morales, 001; Page 11

336 337 338 339 340 341 34 343 344 345 346 347 348 349 350 351 35 353 354 355 356 357 358 359 360 361 36 363 364 365 366 367 368 369 370 371 37 373 374 375 376 377 378 379 380 381 Check Point C Is 333 a triangular number? Solution: See endnotes to check your answer. 1 So we ve used this formula for working with triangular numbers and even checked to see that it is true for n = 5. In mathematics, however, we would not call this a proof. It is simply a verification that the formula holds true for one (or maybe even several) value of n. If we want to prove that it s true for every value of n, then we ll need something more than this. (Although there are more formal ways of proof than the one we give here, this one at least provides a more general demonstration than simply testing it for one number.) Proof: We ll start by recalling each succeeding triangular number is built by adding a diagonal row of dots to the previous triangular number (see above). In particular, to create the n th triangular number, we add n dots to the previous triangular number. As in illustration of this fact, look at the picture below. To get T we add dots to T 1 To get T 3 we add 3 dots to T. To get T 4 we add 4 dots to T 3. T 1 T T 3 T 4 T 5 Figure 9 This pattern continues so that we can state that to get T n, we add n dots to T n 1. The notation for T n 1 simply means the triangular number before the n th one, T n. If we were to write out a list of the first n triangular numbers, it would look something like this T 1, T, T 3, T 4, T 4, T 5, T 6, T 7,., T n, T n 1, T n It may seem like odd notation because it is likely unfamiliar to you, but we need it for our proof. With this notation we can now use it to describe what we mean when we say that to create the n th triangular number, we add n dots to the previous triangular number. In this notation we would write T T n = n 1 + n MAT107 Chapter 4, Lawrence Morales, 001; Page 1

38 383 384 385 386 387 388 389 390 391 39 393 394 395 396 397 398 399 400 401 40 403 404 405 406 407 408 409 410 411 41 413 414 415 416 417 418 The n th triangular number is gotten by taking n dots & adding them to the previous triangular number How do we get T n 1? The same way as we do all the others: by adding (n 1) dots to the previous triangular number, which is T n (see list above). This means that T n-1 = T n-1 + n. So, we can rewrite our equation as: T n = T = T n 1 n + n + ( n 1) + n We can continue this process until we get all the way down to T 1 = 1, the first triangular number: T n = T = T = T = T n 1 n n 3 n 4 =... + n + ( n 1) + n + ( n ) + ( n 1) + n + ( n 3) + ( n ) + ( n 1) + n = 1+ +... + T n 4 + ( n 3) + ( n ) + ( n 1) + n This odd-looking string of symbols tells us how many dots there are in T n. Now we do something very clever. We take two triangular numbers of the same size (their sides have lengths of n dots) and we add them together. That is, we physically join them. Here s a picture of what that looks like: T n = n dots per side When we put these two together, you can see below that we get a rectangle, where one side has n dots and the other side has n + 1 dots. Figure 11 + Figure 10 T n = n dots per side MAT107 Chapter 4, Lawrence Morales, 001; Page 13

n dots 419 40 41 4 43 44 45 46 47 48 49 430 431 43 433 434 435 436 437 438 439 440 441 44 443 444 445 446 447 448 449 450 451 (n +1) dots From the picture, we can see we have n rows and (n + 1) columns so the total number of dots is n(n + 1). But, this is two T n s so we can express this fact as follows: T T n n = n( n + 1) Divide both sides by n( n + 1) = There it is! This is the formula we ve been using. This proof works because it does not depend on particular values of n for it to be true. Because it is done in general terms using an arbitrary number, n, it always holds. This is the first formal proof we ve seen this quarter and it comes at an appropriate place. We ve said the Pythagoreans were among the first to develop the idea of formal proof on the basis of logical reasoning and this proof uses that kind of logical reasoning to establish itself. (The Pythagoreans went about their proofs in different ways since they did not have variables and modern notation to help them out, but we can still share the spirit of proof with them.) FACT3: The n th triangular number is the sum of the first positive n integers. For example, T 10 = 1+ + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 =? 10(10 + 1) 10 11 110 We know from the previous fact that T 10 = = = = 55. If you add up 1 + + 3 + + 10, you ll see you get exactly 55. Proof: This fact basically says that T n = 1 + + 3 + 4 +... + ( n ) + ( n 1) + n. We ve already proved it! Go back into the previous proof and see where we did it. Proof #: Here s a visual proof of this fact. MAT107 Chapter 4, Lawrence Morales, 001; Page 14

45 453 454 455 456 457 458 459 460 461 46 463 464 465 466 467 468 469 470 471 47 473 474 475 476 477 478 479 480 481 48 483 484 485 486 487 488 489 490 491 This often happens. Mathematicians will be working hard at proving some statement that they think is important and in order to get to their desired destination, they either forced to or accidentally prove some other interesting fact along the way. It creates an interwoven web of mathematics that connects facts together in beautiful ways. Example 3 Solution: Figure 1 Find the sum of the first 100 positive integers. We want to determine 1++3+4+5+.+98+99+100. But, the sum of the first n positive integers is the n th triangular number, so this sum is equal to n( n +1) T n =. Hence the first 100 integers add up as follows: T 100 100(100 + 1) = 100 101 = = 5050 Where have we seen this before? There s a fun story about one of the most famous and talented mathematicians of all time that s related to this sum. Carl F. Gauss 13 (1777 1855) is reported to have been assigned the task of adding up 1 through 100 by his teacher. At the age of seven, Gauss apparently could not be bothered by the tedious task before him so instead devised a scheme close to this: Take the sum 1++3+4+.+97+98+99+100 and rearrange them in pairs, as shown below. Think About It Why does this picture prove this fact? MAT107 Chapter 4, Lawrence Morales, 001; Page 15

49 493 494 495 496 497 498 499 500 501 50 503 504 505 506 507 508 509 510 511 51 513 514 515 516 517 518 519 50 51 5 53 54 55 56 57 58 59 530 531 Now add up the pairs: You can see that here are precisely fifty pairs and they each add up to 101. Therefore the sum is 50 101=5050, just as the formula predicted! We can see the connection between his method and the formula for T n by observing that FACT4: 1 + 100 + 99 3 + 98 4 + 97 50 + 51 1 + 100 = 101 + 99 = 101 3 + 98 = 101 4 + 97 = 101 50 + 51 = 101 100(100 + 1) 100 101 100 T 100 = = = 101 = 50 101 = 5050 The sum of any consecutive odd numbers, starting with 1, is a square number. Here are some examples of what is meant by this statement: 1+3+5+7= 16 = 4 1+3+5+7+9= 5 = 5 1+3+5+7+9+11= 36 = 6 Does this always work? Consider the picture to the right. How does the picture verify this fact? The proof is left to the reader. (This is a famous line that you see throughout mathematics textbooks at the higher level. Students often joke that this is inserted whenever the author is too lazy to write out the proof him or herself.) 50 pairs of 101 Figure 13 Figure 14 MAT107 Chapter 4, Lawrence Morales, 001; Page 16

53 533 534 535 536 537 538 539 540 541 54 543 544 545 546 547 548 549 550 551 55 553 554 555 556 557 558 559 560 561 56 563 564 565 566 567 568 569 570 571 57 There are even more patterns imbedded in the triangular numbers. For example, what pattern do you see below? 3 1 = 1 = T 1 1 1 3 3 3 + + + 3 3 3 1 = 9 = T + 3 + 3 3 3 = 36 = T + 4 4 MAT107 Chapter 4, Lawrence Morales, 001; Page 17 3 3 = 100 = T Can express your pattern in words? 14 (The proof is a bit tough, so we ll omit it here.) All of these facts deal with specific properties of the triangular numbers and are nice examples of what we would call (simple) number theory, the study of numbers. This is basically what the Pythagoreans would call arithmetica in the quadrivium. This, you can see, is much different than simply adding numbers together to keep track of animals, people, or possessions. In this way, the Pythagoreans are different from the Egyptians and Babylonians. The Pythagoreans also studied other figurative numbers such as squares, rectangles, pentagons, and higher numbers, using these shapes to classify numbers. Friendly Numbers Another area of study for the Pythagoreans related to numbers was friendly (or amicable) numbers. Two numbers are amicable if each is the sum of the proper divisors of the other. For example, 84 and 0 are friendly numbers. The proper divisors of 0 are 1,,4,5,10,11,0,,44,55,110. The sum of these is 84. On the other hand, the proper divisors of 84 are 1,,4,71,14. The sum of these is 0. Because of their relationship to each other, this pair of numbers achieved a mystical aura 15 about them and later superstition said that two talismans with these two numbers on them would symbolize a perfect friendship between the two people who wore them. These numbers also played a role in sorcery, astrology, magic, horoscopes, etc. It was not until Pierre de Fermat (in 1646) that another pair of friendly numbers was found. They are 17,96 and 18,416. The divisors of 17,96 are: {1,, 4, 8, 16, 3, 46, 47, 9, 94, 1081, 184, 368, 188, 376, 75, 16, 434, 8648} The divisors of 18,416 are: {1,, 4, 8, 16, 1151, 30, 4604, 908} Oh, but wait. It was found that Arab al Banna 16 (156 131) discovered this exact pair in 1300, well before Fermat was born. You will not always hear his name associated with this pair of friendly numbers, which is a common occurrence in mathematics history someone who Challenge Find the proper divisors of 1184 and 110 and verify that these are friendly numbers warning, do not undertake this unless you have lots of time to blow.

573 574 575 576 577 578 579 580 581 discovered something does not get credit for it while someone else who lived later does. Other great names in the history of mathematics have searched for friendly pairs including Descartes and Euler. But the search for friendly numbers is not confined to the great minds of mathematics. In 1866, 16 year old Nicolo Paganini found that 1184 and 110 are friendly numbers. 58 583 584 585 586 587 588 589 590 591 59 593 594 595 596 597 Fermat 17 Descartes 18 Euler 19 Figure 15 Figure 16 Euler himself gave 30 pairs of them in 1747 and later extended his list to 59 pairs. He was a calculating monster! Just for fun, here s a pair of friendly numbers that was discovered recently (000) by Pedersen 0 : Friendly Number1: 1019106988809033651544435080308971597933410769937913048147409037917819405 16859658747514109161744634973401968904501746180788089576747149 Friendly Number: 101934506983145064851550193810773447990753096538718179759957437998660440 91895019397550855886898518316607038140496708540148794730069613851 Yes, those are both individual numbers, so long that they overflow onto two lines of the page. MAT107 Chapter 4, Lawrence Morales, 001; Page 18

598 599 600 601 60 603 604 605 606 607 608 609 610 611 61 613 614 615 616 617 618 Any suggestions as to how they were found? Discoverer # of pairs Te Riele/Pedersen 399076 As of May, 000, there have been at least 657,000 pairs of Pedersen 168608 friendly numbers found. Here is a list of the top 10 discovery leaders. Borho&Battiato 37785 Einstein 16935 As you can see, there has been some busy mathematics at work. Wiethaus 10401 Of course, there is no way they can check these large numbers Te Riele 785 by hand. Instead, number theorists look for patterns and rules Einstein&Moews 447 ( theorems ) that allow them to search for such numbers with some level of efficiency. In this day and age, the computer Borho&Hoffmann 3471 doesn t hurt either! Moews&Moews 614 Ball 1938 Keep in mind that Pythagoras (and/or his followers) found ONE Garcia 147 pair of friendly numbers that we know of. But that contribution helped to get the question out there for people to think about as well as helped to establish number theory as its own branch of mathematics. The Pythagoreans also appeared to have studied perfect, deficient, and abundant numbers. These are topics that we ll leave for the reader to explore alone if interest is generated. MAT107 Chapter 4, Lawrence Morales, 001; Page 19

619 60 61 6 63 64 65 641 64 643 644 645 646 647 648 Part : The Pythagorean Problem The most famous result that Pythagoras is famous for is, of course, the Pythagorean Theorem. We ve all seen it before. In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse (longest side). In modern notation, we write: a + b = 66 67 68 69 630 631 63 633 634 635 636 637 638 c Almost everyone gives credit to Pythagoras for first discovering this fact. By discovering we mean he (or his followers?) was the first to prove that this relationship holds in all right triangles. We already know that the Babylonians at least knew of the relationship (Plimpton 3) even though they may not have proved it. The Egyptians were also aware of the relationship and used ropes to measure out proper lengths of right triangles. 639 The actual Think About It 640 proof that they gave is unknown to us. It is likely that it was very geometrical What does the figure and involved taking a geometric object, dissecting it into shown have to do with the pieces, and then rearranging the pieces into a new figure that Pythagorean Theorem? demonstrated the Pythagorean relationship. Here s one such proof for you to consider. For many of you, this may very well be the very first time you ve ever seen this famous theorem proven. Proof: Figure 17 a a b c c We consider the two following pictures b b a a b a b c a b c c a 649 650 651 b b b c a b a Figure 18 c b c a b MAT107 Chapter 4, Lawrence Morales, 001; Page 0

65 653 654 655 656 657 658 659 660 661 66 663 664 665 666 667 668 669 670 671 67 673 674 675 676 677 678 679 680 681 68 683 684 685 Both squares are to be considered to be the same size. The length of a side is (a+b). The values for a, b and c in each picture are to be considered consistent in their lengths across both pictures. The square on the left has an area of ( a + b) since the area of any square is its side squared. But the figure on the left is also made up of six pieces two small squares and four triangles. If we add up their areas we should get the area of the whole square. The area of the squares are a and b. There are also four triangles to consider. Thus the area of the triangle on the life, A L, is: 1 A L = a + b + 4 a b 1 The 4 a b comes from the fact that there are 4 equally sized right triangles in the left picture and the area of any right triangle is one half its base times its height. Now if we take another square of the same size and dissect it differently as shown in the picture to the right, we still get the same area, namely ( a + b). However, there are only 5 pieces in this dissection (one square with area c and four triangles), so we ll add them up to get the area on the right, A R : 1 A R = 4 a b + c However the two areas are the same, since AL = AR = ( a + b). They were, after all, both squares of the same size so we equate the two and then simplify from there. A 1 1 a + b + 4 a b = 4 a b + c L = A a + b = c This finishes the proof as it established the relationship that we all know and love. There are literally thousands of proofs of the Pythagorean Theorem, many of them published in books devoted just to this subject. Even a former president of the United States is credited with R MAT107 Chapter 4, Lawrence Morales, 001; Page 1

686 687 688 689 690 691 69 693 694 695 696 697 698 699 700 701 70 703 704 presenting a unique proof of this theorem. See the endnotes for some websites with some interesting proofs to examine. 1 Proofs in Other Civilizations Many civilizations had proofs of the Pythagorean Theorem, including the Chinese. The picture is the Chinese hsuanthu, which is believed to show one of the oldest demonstrations of the Pythagorean Theorem in history. It is dated by some people to as far back as 1100 B.C.E. Another related proof comes from Bhaskara (1114-1185), a Hindu mathematician. His proof is demonstrated interactively on the internet. 3 His proof, like many of his time and culture, consisted solely of a picture with the word Behold written nearby 4. Figure 19 below is the basic picture that goes with Bhaskara s proof. Figure 19 Figure 0 705 706 707 708 709 710 711 71 713 714 715 716 717 Other proofs of the Pythagorean Theorem revolve around a common theme: dissect a square into pieces that somehow demonstrate the Pythagorean Theorem. (This is why they are often called dissection proofs. ) In most cases, the figure is dissected into pieces to show that it is equivalent to having three new squares, one with sides of length a, one with sides of length b, and one with sides of lengths c. Furthermore, these three squares must be dissected and/or arranged to show that the area of the squares with sides of lengths a and b, when added together, are exactly equal to the area of the square with sides of length c. In the figure below, the square with side of length a and the square with side of length b would each be cut up into several pieces. Those pieces would then be arranged so that they fit exactly into the square with sides of length c. The hard part, of course, is determining how to cut up the first two squares so that all the pieces exactly fit into the third! MAT107 Chapter 4, Lawrence Morales, 001; Page

718 719 70 71 Area of a square with sides Of length a Area of a square with sides Of length b a + b = c Figure 1 c c 7 Area of a square with sides Of length c a a b b MAT107 Chapter 4, Lawrence Morales, 001; Page 3

73 74 75 76 77 78 79 730 731 73 733 734 735 736 737 738 739 740 741 74 743 744 745 746 747 748 749 750 751 Pythagorean Triples Closely related to this theorem is the problem of finding integers, a, b, and c that satisfy the Pythagorean Theorem. We ve already seen that the Babylonians had a tablet (Plimpton 3) that listed several of these. We called them Pythagorean Triples. The set of three numbers (3,4,5) is commonly known as such a Pythagorean triple. The Pythagorean Formula for Triples The Pythagoreans are given credit for finding a formula for generating these trios of numbers. In our modern notation it would look like the following: m + 1 m 1 = + m This formula is good for any odd value of m 3 and generates a series of triples. This formula essentially assigns the following values to a, b, and c: a = m m 1 b = m + 1 c = Example 4 Find the Pythagorean triple that corresponds to m = 3. Solution: 3 + 1 3 1 = + 3 9+ 1 9 1 = + 3 5 = 4 + 3 Well, look at that! m = 3 produces the triple (3,4,5) MAT107 Chapter 4, Lawrence Morales, 001; Page 4

75 753 754 755 756 757 758 759 760 761 76 763 764 765 766 767 768 769 770 771 77 773 774 775 776 777 778 779 780 781 78 Check Point D Find the Pythagorean triple that corresponds to m = 5. Solution: See the endnotes to check you answer. 5 One interesting thing to note is that if (3,4,5) is a triple, so is (6,8,10) and (9,1,15), etc. In fact, any multiple of the triple (3,4,5) is also a triple. We can represent this generic triple as (3k,4k,5k), where k is a positive integer. Proof: (3k,4k,5k) is a Pythagorean Triple. We need to show that these satisfy a + b = c + = (3 ) + 4 ( ) a b k k = 9k + 16k = 5k = 5k 5k = (5 k) = c (3 k) + (4 k) = (5 k) b = n c = n + n This shows that (3k,4k,5k), which is any multiple of (3,4,5), is also a Pythagorean Triple. The Triples of Proclus The writer Proclus (411-485) claims that Pythagoras devised an algorithm for obtaining triples. 6 In our own notation, it would amount to finding a, b, and c with the following equations: a = n + 1 + n + 1 where n is an integer and n 1. (n does not have to be odd for these equations to work, as we show below.) MAT107 Chapter 4, Lawrence Morales, 001; Page 5

783 784 785 786 787 788 789 790 791 79 793 794 795 796 797 798 799 800 801 80 803 804 805 806 807 808 809 810 811 81 813 814 815 816 Example 5 Solution: Check Point E Find the Pythagorean Triple for n =. We simply use the formulas as given: a = () + 1 = 5 b = () c = () + () = (4) + 4 = 1 + () + 1 = 8 + 4 + 1 = 13 This gives the triple (5,1,13) which can easily be checked. Find the Pythagorean Triple for n = 10 Solution: See the endnotes to check your answer. 7 To convince ourselves that these three expressions for a, b, and c do indeed give us Pythagorean Triples, we propose the following proof: Proof: We want to show that the Proclus equations for a, b, and c satisfy the Pythagorean Theorem. To verify this, we need to show that they satisfy the equation a + b = c. We will substitute the corresponding formulas for a, b, and c into this equation to see what we get. Read carefully and follow along as we go back and review lots of polynomial algebra. We ll do both sides of the equation separately so can compare what s happening. Step a + b 1 c ( n + 1) + ( n + n) ( n + n + 1) (n + 1)(n + 1) + (n + n)(n + n) 3 (n + n + 1)(n + n + 1) 4n + n 4n + 8n + 8n + 4n + 1 5 4n + 8n + 8n + 4n + 1 4 3 4 3 4n + 8n + 8n + 4n+ 1 = 4n + 8n + 8n + 4n+ 1 4 3 3 + n + n + 1+ 4n + 4n + 4n 4 4 You fill in what goes here when multiplying 4 3 4 3 Step 1: Starting point. Step : Substitute Proclus equations into a, b, and c. Step 3: What does it mean to square something? Answer: To multiply it by itself. Step 4: Multiply the polynomials by each other very, very carefully. Step 5: Combine like terms. Step 6: Note that we have the same thing on Think About It each side, so we can conclude that the quantities in Step 1 were also If these equations did not satisfy the equal, concluding this proof. Pythagorean Theorem, what would you get in Step 5? MAT107 Chapter 4, Lawrence Morales, 001; Page 6

817 818 819 80 81 8 83 84 85 86 87 88 89 830 831 83 833 834 835 836 837 838 839 840 841 84 843 These two methods of generating Pythagorean triples (from the Pythagoreans and as described by Proclus) appear to be equivalent since they seem to generate the same list of triples. Here s a table that illustrates this fact: Table 1 Triples From Original Equation Triples from Proclus Equations m a b c n a b c 1 1 3 4 5 5 1 13 3 3 4 5 3 7 4 5 4 4 9 40 41 5 5 1 13 5 11 60 61 6 6 13 84 85 7 7 4 5 7 15 11 113 8 8 17 144 145 9 9 40 41 9 19 180 181 10 10 1 0 1 11 11 60 61 11 3 64 65 1 1 5 31 313 13 13 84 85 13 7 364 365 Unfortunately, neither of these methods generates all possible Pythagorean triples. The method of Plato 8 will demonstrate that clearly by finding a triple that neither of these can generate. The Triples of Plato Plato s method says that you can generate a triple (a,b,c) with the following equations: a = p Example 6 b = p c = p Think About It What makes these two methods essentially equivalent? 1 + 1 What Pythagorean Triple is generated by p = 4? Figure MAT107 Chapter 4, Lawrence Morales, 001; Page 7

844 845 846 847 848 849 850 851 85 853 854 855 856 857 858 859 860 861 86 863 864 865 866 867 868 869 870 871 87 873 874 875 876 877 878 879 880 881 88 883 Solution: Check Point F Solution: Using the equation we get: a = (4) = 8 b = 4 c = 4 1 = 15 + 1 = 17 The triple generated is (8,15,17). This is not on either of the two lists before which shows there are more triples out there than the Pythagoreans (apparently) knew about. What Pythagorean Triple is generated by p = 10? See the table to check your answer. Here is a table that lists the first few triples generated with Plato s equations. You can see that it has very little in common with the other lists. Most striking is that all of the values for a in Plato s table are even numbers, while all the values for a in the Pythagorean tables are odd. Proof: Think About It Why does one table given even values for a while the other gives odd values of a? The equations of Plato satisfy the Pythagorean Theorem. Table Triples From Plato's Equations p a b c 1 4 3 5 3 6 8 10 4 8 15 17 5 10 4 6 6 1 35 37 7 14 48 50 8 16 63 65 9 18 80 8 10 0 99 101 11 10 1 1 4 143 145 13 6 168 170 To prove this, we proceed as before. We need to substitute Plato s equations for a,b, and c and then simplify what we get to make sure that we get equal expressions on both sides of a equal sign. MAT107 Chapter 4, Lawrence Morales, 001; Page 8

884 885 886 887 888 889 890 891 89 893 894 895 896 897 898 899 900 901 90 903 904 905 906 907 908 909 910 911 91 Step a + b 1 c ( p ) + ( p 1) ( p +1) ( p )( p) + ( p 1)( p 1) 3 ( p + 1)( p + 1) 4 ( p p + 1) 4 p + p 4 What do you get here? p 4 + p + 1 5 p 4 + p + 1 p + p + 1 = 4 p + p + 1 4 Step 1: Starting point. Step : Substitute Plato s equations into a, b, and c. Step 3: What does it mean to square something? Answer: To multiply it by itself. Step 4: Multiply the polynomials by each other very, very carefully Step 5: Combine like terms. Step 6: Note that we have the same thing on each side, so we can conclude that the quantities in Step 1 were also equal, concluding this proof. You would think that between these two methods (we ll count the two Pythagoras methods as one since they produce equivalent lists) would generate all of them. After all, we get both evens and odds for value of a. Unfortunately, they don t do the job. One of the first (perhaps the very first) mathematician to give a method to generate all Pythagorean triples was Euclid. The Triples of Euclid Euclid is one of the great names of Greek mathematics. His work, The Elements, is one of the classics of mathematics and is the basis of modern high school geometry (yes, you have him, in part, to thank for your high school geometry class). Euclid s method is slightly different from the others we ve seen so far. For this method, we let g and h be positive integers with g > h. Euclid shows that a Pythagorean triple (a,b,c) is generated with the following equations: a = gh b = g c = g h + h Figure 3 MAT107 Chapter 4, Lawrence Morales, 001; Page 9

913 914 915 916 917 918 919 90 91 9 93 94 95 96 97 98 99 930 931 93 933 934 935 936 Example 7 Generate a Pythagorean Triple with g = 4 and h = 3. Solution: Check Point G Solution: Using Euclid s equations we get the following: a = (4)(3) = 4 b = 4 c = 4 3 + 3 = 7 = 5 This generates the triple (7,4,5), which does not appear to be in any of the tables we have seen so far. Generate a Pythagorean Triple with g = 10 and h = 7. See the endnotes for the answer. 9 Unfortunately, trying to prove that this method generates all Pythagorean Triples is a bit tough in a course like this, so we won t be able to explore it here. But we can show that the equations Euclid gives satisfy the Pythagorean Theorem, a + b = c. That proof is left for the reader in the exercises. MAT107 Chapter 4, Lawrence Morales, 001; Page 30

937 938 Part 3: Irrational Numbers (Incommensurables) and the Pythagoreans As you may know, a rational number is a number that can be written in the form b a, where a and 939 b are integers and b is not 0. They are essentially the well behaved fractions: no decimals, 940 941 94 943 944 945 946 947 948 949 950 951 95 953 954 955 956 957 958 959 960 961 roots, or other weirdness just nice integers. (While studying the Pythagoreans, we will restrict ourselves to the positive integers since they did not recognize negative numbers.) An irrational number is a number that is not rational. That is, it cannot be written as the ratio of two integers. Irrational numbers are also called incommensurables. Some believe that the discovery that incommensurable numbers exist is the most important discovery of the Pythagoreans. 30 It was important not only to mathematics but also to the Pythagoreans and their beliefs. They assumed that a ratio was limited to containing only integers, which is simply an extension of their belief that everything is [whole] number. If there exist numbers that are not whole numbers, or ratios of whole numbers, then their fundamental philosophy, on some level, was being challenged. It is not unreasonable to speculate that such a discovery would have been disturbing to them. To understand why the Pythagoreans would have made such an assumption, it is helpful to explore what they meant when they talked about rational numbers. We need to keep in mind that the Pythagoreans did not have modern symbols for numbers. To them, a number represented the length of a line segment. It was therefore associated with a geometric object. Given any two line segments (numbers), they believed that it is always possible to find a third line segment that would evenly divide into both of the original line segments. For example, let s say the following figures represent enlarged line segments of different lengths: 96 963 964 965 The Pythagoreans believed that if you searched long and hard enough, you could find a smaller line segment that divided each of these two line segments up. Here s what that might look like: 966 967 968 969 970 971 97 You can see here that this smaller (darker) line segment will evenly divide the large segment into 1 pieces while it will divide the shorter segment into 8 pieces. The Pythagoreans believed that it was only a matter of time before they could find such a dividing segment, even if it had to be very, very small. For example, the following two line segments have a very small common measure: MAT107 Chapter 4, Lawrence Morales, 001; Page 31

973 974 975 976 When we look for a common measure, we find it to be much smaller, but still available 977 978 979 980 981 98 983 984 985 986 987 988 989 990 991 99 993 994 995 996 997 998 999 1000 1001 100 1003 1004 1005 1006 1007 1008 1009 With this common measure, two numbers are said to be commensurable (rational). Ratios that cannot be expressed by whole numbers with a common measure like this (which they did not originally believe existed) are now said to be incommensurable. The word incommensurable is alogos in Greek (which means without ratio ) or arrhetos (inexpressible). The Greek word logos (speech or word) represented the ratio of two numbers. The discovery of alogos, usually translated as irrational, was a challenge to their theory of proportions and may have led them to reconsider it completely. 31 (By they way, key terms in our mathematics vocabulary come from very geometric interpretations. Why do we read x as x squared? It is because a square with a length of x has an area of x. Thus, the measure for area gets associated with a square and the term sticks. The same goes for x 3. The volume of a cube with length x is x 3 so we say x cubed. This is how the Greeks viewed numbers as line segments. Their squares of numbers were thought of as geometrical square objects. Finally, a number multiplied by itself three times was viewed as a geometric object a cube. Because of their views and interpretations of numbers, the Greeks had a tremendous impact not only on how we do mathematics but also on how 1 we talk about mathematics. Their influence lasts until today.) It is uncertain how the Pythagoreans discovered the incommensurables. 1 Most believe it was due to their investigation of a square with unit length. c (Unit length means the side has a length of 1.) They might have asked the question, What is the length of the diagonal of this square? A quick application of the Pythagorean Theorem shows that c =. (You should do this on scratch paper to convince yourself Figure 4 that it is true.) However, the Pythagoreans would have been interested in finding two commensurable lengths (whole numbers) that could be placed into a ratio that would represent the exact value of c. (Remember that the Babylonians had a method of finding a fractional estimate of a square root. The third estimate of using this method is 17 /1 1.416666...., while the actual value of is closer to 1.414 ) As they tried to find this pair of commensurable numbers, they could not do so and eventually proved that the pair did not exist at all. A proof of the incommensurability of is given later in the chapter. MAT107 Chapter 4, Lawrence Morales, 001; Page 3

1010 Part 4: Modern Facts on Rational and Irrational Numbers 1011 101 The discovery of irrational numbers by the Pythagoreans is certainly one of their most important 1013 contributions in the history of mathematics. These numbers have since been studied and 1014 dissected by thousands (if not millions) of mathematicians through the ages as they have strived 1015 to understand the basic structure of number systems. In this section, we will look at some of the 1016 interesting facts about rational and irrational numbers, concentrating mainly on more modern 1017 approaches to these numbers. So while we are not necessarily doing historical mathematics, 1018 we can appreciate the fact that groups such as the Pythagoreans and their work have survived and 1019 been extended for thousands of years. 100 101 Starting Simple 10 103 The positive whole numbers are numbers that belong to the set: {1,,3,4,.}. These are used for 104 basic counting and are sometimes called the natural numbers. 105 106 If we take any two natural numbers and add them together, we get another natural number. 107 3+ 4= 7, and 7 is a natural number. Likewise, multiply two whole numbers together and we get 108 another whole number. These two properties are actually pretty important in number theory and 109 we say that the natural numbers are closed under addition and the natural numbers are closed 1030 under multiplication. This simply means that when you take any two natural numbers and either 1031 add them or multiply them, the result will be a member of the natural numbers. 103 1033 To illustrate this property of closure from another point of view, consider the set {1,,3}, having 1034 only three numbers in it. If you take any two of these numbers and add them together, do your 1035 get another number in that set? Well, 1+ = 3, and 3 is certainly in the set. However, 1+ 3= 4, 1036 but 4 is not in the set. So, this set is not closed under addition. 1037 1038 The natural numbers are certainly not closed under all basic operations in arithmetic. For 1039 example, they are not closed under subtraction. An example of this is 1 8= 7. While 1 and 8 1040 are certainly in the natural number set, when you subtract them you get a negative number, 1041 which is not a member of the natural number set, so we don t have closure under subtraction. In 104 fact, in order to get closure, we have to add all of the negative numbers as well as 0 to cover our 1043 backs. So in one sense, we could say that a quest to obtain closure under subtraction leads to the 1044 creation of the set of integers. 1045 1046 The last basic mathematical operation, division, also provides problems in the context of the 1047 natural numbers. Take any two natural numbers and divide them and rarely will you get another 1048 natural number. For example, while 6 3= does the job, 3 4 certainly does not. Therefore, 1049 the natural numbers are not closed under division. In fact, if you want closure under division, 1050 one way to do it is to expand the natural numbers to include fractions, or what we shall call the 1051 rational numbers. These are numbers that can be written in the form a, where a and b are b integers and b is not zero. We see here another case where a desire to extend the property of 105 1053 1054 closure leads to the creation of a new set of numbers, the rational numbers. MAT107 Chapter 4, Lawrence Morales, 001; Page 33

1055 1056 1057 1058 1059 1060 1061 Rational Numbers in More Detail The definition that we gave above for rational numbers is one that is deserving of some extra commentary. First of all, notice that the definition states that a rational number is one that can be put in the form a b. It isn t quite enough to simply say that rational numbers are those of the form a b 106 because there are infinitely many ways to express a given fraction. (For example, 3 4 can be 1063 1064 1065 1066 1067 1068 written as 6 8 or 3 or in infinitely many other ways.) The reason that this is not enough is 4 that we don t want our definition of rational numbers to depend on the particular way that we decide to express that number. 3 Second, observe that we require that b is non zero. This requirement is in place to avoid division by 0, which has no mathematical meaning. (While 0 is generally 0, except when n is 0, the n 1069 1070 1071 107 1073 1074 1075 1076 1077 1078 1079 1080 value of 0 n is not defined in our basic mathematical system.) Rational Numbers: Terminating and Repeating Decimals One of the more interesting characteristics of rational numbers is that when you divide two integers and obtain a decimal representation for the number you get one of two results: 1) The decimal part either terminates. ) The decimal part does not terminate but repeats at some point. Think About It Why is 0 n undefined in our basic mathematical system? 1081 An example of the first kind is 3 which is exactly equal to 0.75. The decimal part of this 4 terminates at the 5, once and for all. To see an example of the second kind, one only look at the 108 1083 1084 1085 1086 1087 1088 1089 1090 familiar fraction 1. The decimal value of this numbers is 0.33333333., where the 3 s never 3 stop or terminate but repeat indefinitely. We sometimes write this number as 0.3, where the bar over the 3 indicates what part of the number repeats. If you re at all curious like I am, you wonder which rational numbers have decimal representations that terminate and which ones have decimal representations that repeat instead of terminate. MAT107 Chapter 4, Lawrence Morales, 001; Page 34

1091 To consider those that terminate, let s take the example of 675 0.675 =. Any terminating 10000 109 1093 1094 1095 1096 1097 1098 1099 1100 1101 110 1103 1104 1105 1106 1107 1108 1109 1110 1111 111 1113 1114 1115 1116 1117 1118 1119 decimal can be written as a fraction with a denominator that is a power of 10, simply because we are operating in a base 10 system. For example, 0.37 = 37 1000 and 1151 0.1151 =. 1000000 In the case of 0.675, we can see that the denominator, 10000 can be rewritten as follows: 675 675 = 10000 10 10 10 10 675 = 5 5 5 5 What is noteworthy here is that the denominator factors into several s and 5 s, both prime numbers. Any terminating decimal, since it can be written as a fraction with a power of 10 as a denominator, has the same property. Namely, the prime factors of the denominator are and 5 only. Even if we write the fraction in lowest terms, we still see the same property emerge. For example: 675 69 = 10000 400 69 = 4 100 69 = 100 Once again, we notice that the denominator of the reduced fraction has only and 5 as its prime factors. We are now prepared to state which rational numbers have terminating decimals: FACT 1 A rational fraction a/b (in lowest terms) has a terminating decimal representation if and only if the integer b has no prime factors except for and 5. (Please note that in order for this fact to hold true, the fraction must be in lowest terms. Otherwise, all bets are off.) 110 So, while 1 has a terminating decimal representation, 1 does not as its denominator has neither 3 a nor 5 as a factor of its denominator. 111 11 MAT107 Chapter 4, Lawrence Morales, 001; Page 35

113 Example 8 114 115 Solution: 116 117 118 119 1130 1131 113 1133 1134 1135 Example 9 1136 1137 Solution: 1138 1139 1140 1141 114 1143 1144 1145 1146 1147 1148 1149 1150 1151 115 1153 1154 1155 1156 Check Point H Solution: Use Fact 1 above to determine if 37 160 has a terminating decimal. To settle this issue, we really only need to find the prime factorization of 160. Any method you ve learned in the past will do. Here s one possible way to find such a factorization. 160 = 8 0 3 = 4 5 3 = 5 5 = 5 Since the only prime factors in the denominator are and 5, then we know that this fraction does have a decimal representation that terminates. Use Fact 1 above to determine if 455 600 has a terminating decimal. To settle this issue, we really only need to find the prime factorization of 600. 600 = 6 100 = 3 100 At this point we can stop since we see that 3 is part of the prime factorization. Since and 5 are not the only primes present, we can conclude that this fraction does not have a decimal representation that terminates. Use FACT 1 above to determine if 30 315 Check the endnote for the answer. 33 has a terminating decimal. As humans faced questions like how to divide a loaf of bread among 5 people, the natural idea of fractions arose. MAT107 Chapter 4, Lawrence Morales, 001; Page 36

1157 1158 1159 1160 1161 116 1163 1164 1165 1166 1167 1168 1169 1170 1171 117 1173 1174 1175 1176 Non Terminating, Repeating Decimals We now know what kinds of rational numbers have decimal representations that terminate. That takes care of one of the stated possibilities for rational numbers. Recall that one of the following is true: 1) The decimal part either terminates. ) The decimal part does not terminate but repeats at some point. Certainly, if the decimal part of a fraction does not terminate then the first half of the second possibility holds. But what about the second, more noteworthy part? Namely, that it repeats? Why should we expect that it would repeat, rather than simply keep going in some chaotic, unpredictable manner? In this section, we ll try to look at why this is true. The following fractions all have decimal representations that not only do not terminate, but also repeat at some point: 1 = 0.16 6 1 = 0.3 3 3 = 0.7 11 733 = 0.814 900 To see why the decimal portions repeat, we will look at a particular example. Consider 3 7. To find the decimal representation of this number, we can divide as normal: 0.485714 7 3.000000 The 3 in the first step reappears and so the pattern will repeat after this point. 8 0 14 60 56 40 35 50 49 10 07 30 8 MAT107 Chapter 4, Lawrence Morales, 001; Page 37

1177 1178 1179 1180 1181 118 1183 1184 1185 1186 1187 1188 1189 1190 1191 119 1193 1194 While things seem to go along pleasantly for a while, we notice that several steps into the division, the 3 reappears and resets the clock, so to speak. After this point, you will see the pattern repeat itself all over again. It need not be that the first step is the one that reappears and hence causes things to repeat. It could take place anywhere. Try to divide 09/700 and you ll see it s a 6 that repeats at some point. In our extended example above, since we are dividing by 7, the possible remainders that could arise in any particular step are 1,, 3, 4, 5, and 6. Note that the remainders in this case are,6,4,5,1, and 3 and we ve cycled through all the possible remainders before we get back to 3 and notice that we have not managed to get the decimal to terminate. We are then plunged back into the process anew. We can approach the general case of a/b in a similar way. That is, when b is divided into a, the only possible remainders that we could get are 1,, 3, 4,,b-, and b-1. So, a recurrence of the division process is certain. When the division process recurs a cycle is started and the result is a periodic decimal. 34 1195 We have, at this point, demonstrated that any rational number a can be expressed as a b 1196 terminating decimal or as a non-terminating, repeating decimal. That is, if we have a rational 1197 number, then we know that it has to behave in one of these two ways. 1198 1199 What About the Other Direction? 100 101 We still have one more question to consider. That is, if we have a decimal number that 10 terminates or repeats, do we necessarily know that we have a rational number? This is very 103 different that saying what we already have; namely, that if you have a rational number then it 104 must terminate or repeat. We are considering the opposite direction of this whole mess and 105 mathematics treats them as very separate questions. (The force behind this distinction is basic 106 logic, the tool which mathematics uses to establish and prove facts and theorems in this field. We 107 will ignore the subtleties of this matter here, but it should be pointed out that there are delicate 108 issues that need to be paid attention to in cases such as these.) 109 110 The first case, when it terminates, is easy to dispose of. For example, 0.31 terminates and we 111 know that we can read this as three hundred twenty one thousandths. This translation into 11 words, and our existence in the base-ten system, allows us to write this number as 31 1000. 113 114 The second case is a little more interesting. For example, if you have the number 0.345, do you 115 know for sure that you could write this number in the form of a, where a and b are integers and b b is non-zero? 116 117 118 To answer this question, we consider a formal example. MAT107 Chapter 4, Lawrence Morales, 001; Page 38

119 10 11 1 13 14 15 16 17 18 19 130 131 13 133 134 135 136 137 138 139 140 141 14 143 144 145 146 147 148 149 150 151 15 153 154 155 156 157 158 159 160 Example 10 Solution: If possible, write 0.345 as a rational number. To do this, we start by letting x = 0.345. That is, x = 0.34545454545... Now consider that 10x = 3.4545454545... (Check it if you are not convinced.) Also, consider that 1000x = 345.45454545... (Again, you should check this.) Now we subtract these two results from each other: 1000x 10x = 345.45454545... 3.4545454545... 990x = 34 This is the key step because it completely eliminates the repeating part of the number, leaving us with only whole numbers with which to work. This equation can be solved for x by dividing both sides by 990 and we get: 34 x = 990 However, we said at the beginning that x = 0.345. So, we have shown that: 0.345 = 34 990 This means that 0.345 is indeed rational. The central idea in this last example is that if we are given a repeating decimal, we should be able to multiply the number by two different powers of 10, subtract the results to eliminate the repeating decimal, and thens solve for the actual fraction that is being sought. The hardest part is to figure out what powers of 10 to multiply by, how to describe the general case, and how it should be handled. Let s try another example first. Example 11 Solution: Write 0.1378378 as a rational fraction. We start by letting x = 0.1378378... Our first power of 10 that we will multiply by is designed to isolate the repeating part of the decimal to the right of the decimal point. In this case, the 1 is not part of the repeating part and so those two digits need to move to MAT107 Chapter 4, Lawrence Morales, 001; Page 39

161 16 163 164 165 166 167 168 169 170 171 17 173 174 175 176 177 178 179 180 181 18 183 184 185 186 187 188 189 190 191 19 193 194 195 196 197 198 199 1300 1301 130 the left of the decimal point. Or, another way to state it is that the decimal point must move two places to the right. We do this by multiplying by 100, so we consider: 100x = 1.378378378... The other power of 10 that we will multiply by is designed to allow us to subtract the two powers so that the result will eliminate the repeated part. To do this, consider: 100000x = 1378.378378... Notice that this has three more zeros than our previous multiple (100) precisely because we need to move the decimal point three more places to the right since three digits repeat. It might be more obvious why this works to see them subtracted: If we divide, we find that 100000x 100x = 1378.378378... 1.378378... = 1357 99900x = 1357 1357 x = 99900 A quick check will show that this is, indeed, equal to 0.1378378. This last example gives us a strategy for how to pick our powers of ten that we will ultimately use to get our desired result. The first power of ten should isolate the repeating part of the decimal on the right side of the decimal point. The second power of ten should move the decimal point an additional number of spaces that correspond to how many digits are repeating in the given number. We ll do one more example before moving on: Example 1 Solution: Write 0.387198198198 as a rational fraction. We start by letting x = 0.387198198198... Note that four digits repeat here. We isolate 198198198 on the right by letting: 1000x = 387.198198198... To move the decimal an additional four places to the right we consider: MAT107 Chapter 4, Lawrence Morales, 001; Page 40

1303 1304 1305 1306 1307 1308 1309 1310 1311 131 1313 1314 1315 1316 1317 1318 1319 130 131 13 133 134 135 136 137 138 139 1330 1331 133 1333 1334 1335 1336 1337 1338 1339 1340 1341 134 1343 Check Point I Solution: Subtracting these two equations gives: 10000000x = 387198.198198... 10000000x 1000x = 387198.198... 387.198... 9999000x = 3871811 3871811 x = 9999000 A check on a calculator shows this to be accurate. Write 0.468346834 as a rational number. Check the endnote for answer. 35 So, this concludes this portion of our examination of rational numbers. There are actually many holes that we ve left unfilled, but you could take a whole course on the topic, so we have to stop somewhere. Right? Irrational Numbers At this point, we might ask the question: If the decimal does not terminate or does not repeat, then what are you left with? In this case, we have what are called irrational numbers. That is, if a real number is not rational, then it is irrational. Irrational numbers thus have the property that they never terminate and they never repeat. This leaves us with the reality that irrational numbers have decimal expansions that go on forever and never establish any pattern. While that may seem a bit sad or scary, it s actually pretty convenient because it allows us to work with and recognize numbers such as and π, both of which are irrational. Furthermore, if you lay all the rational numbers (which include all the integers) out on the real number line, you will find that even though there are infinitely many of them, they do not fill up the number line. The irrational numbers fill up all those holes so that we have complete density on the familiar number line. Another way to say this is that the real number line (which is just a picture of the set of real numbers) is completely made up of both rational and irrational numbers. Because of the definition of irrational numbers and its dependence on what rational numbers are, irrational numbers are a little harder to work with. Thus, if you have a number and you want to show that it s irrational, then you essentially have to show that it never terminates and it never repeats. This is much more subtle and difficult than showing you can produce a terminating or repeating decimal expansion. In order to see just how tricky this can get, let s take the classic MAT107 Chapter 4, Lawrence Morales, 001; Page 41

1344 example of and show that it is irrational. That is, let s show that it is NOT rational (i.e. does 1345 not terminate or repeat). 1346 1347 Proof by Contradiction 1348 1349 To show that is not rational, we will employ a method of proof that is very common in 1350 mathematics. It is called proof by contradiction, and its strategy is exposed in its name. The idea 1351 behind such a proof is to assume the opposite of what you want to actually prove. With that 135 assumption, you then explore what such a statement would imply logically, moving along step 1353 by step. Each step must be a logical deduction of the previous one(s). You continue to do this 1354 looking for a statement or result that is either blatantly/obviously false or which contradicts 1355 something you definitely know to be true or have assumed to be true (for good reasons) in your 1356 problem. When you get such a false, contradictory statement, then the only conclusion you can 1357 reasonably reach is that one of your steps or your original opposite assumption is false. But since 1358 all the intermediate steps follow strict rules of logic, only your opposite assumption can be the 1359 culprit and is therefore false. 1360 1361 This method is not natural or intuitive to follow. Budding mathematicians generally need lots of 136 time and practice before the art of writing proofs with this method becomes a comfortable tool 1363 with which to work. But we are going to use the method here to prove that is not rational 1364 1365 (i.e. irrational) so you can see how the process works. 1366 First however, here are some facts to think about and recognize before we get started: 1367 Any even number can be written as k, where k is an integer. 1368 Any odd number can be written as k + 1, again where k is an integer. 1369 The product of two even numbers is even. (Can you prove it?) 1370 The product of two odd numbers is odd. (Can your prove it?) 1371 The product of an even number and an odd number is even. (What about this one!) 137 1373 Proof: 1374 1375 Show that is irrational. 1376 We start by assuming that the opposite is true. That is, we will assume that 1377 is rational and can therefore can be written in the form a, where this b fraction is in lowest terms, a and b are integers, and b is not zero. 1378 1379 1380 1381 138 1383 1384 1385 The assumption that the fraction is in lowest terms is key to the argument. If it s not in lowest terms, then reduce the fraction so that it is. a Logical Step 1: =. b This is simply a result of assuming that is rational. MAT107 Chapter 4, Lawrence Morales, 001; Page 4

1386 1387 1388 1389 1390 1391 139 1393 1394 1395 1396 1397 1398 1399 1400 1401 140 1403 1404 1405 1406 1407 1408 1409 1410 1411 141 1413 1414 1415 1416 1417 1418 1419 140 141 14 a Logical Step : = b This logically follows from squaring both sides of the previous step. Logical Step 3: b = a This follows from multiplying both sides of the previous step by b. Logical Step 4: a must be an even number. This statement follows from the previous step because any number that can be written in the form k, where k is an integer, is an even number. Since b is an integer, then b is of the form k, and so b is even. Therefore, a, being equal to b, is also even. Logical Step 5: a must be an even number. This follows from the only way that two (non-zero) numbers can multiply by each other to get an even number is if both of them are even. (Odd times odd is odd. But even times even is even.) So, if a is even then, since it s just the product of a times a, then a must be even. Logical Step 6: a can be written as a = k This comes from the basic definition of what it means to be even. Logical Step 7: k = b This statement comes from substituting our previous result, a = k, into the result from Logical Step 3. a = b ( k) = b 4k = b k = b Logical Step 8: b is an even number. Since b = k, it is written in the form c, so it is even. Logical Step 9: b is an even number. Since b is even, b is even. See the logic of Logical Step 5. Think About It Prove that even times even equals even and odd times odd equals odd. What about odd times even? At this point, notice that we have concluded that both a (Step 5) and b (Step 9) are even. MAT107 Chapter 4, Lawrence Morales, 001; Page 43

143 This means that a can be reduced. An even number divided by an even b 144 number, since they both have a factor of in them, can be reduced. 145 146 BUT 147 Way back in at the beginning of this process we said that the fraction a b was 148 already in lowest terms and therefore reduced. This result contradicts that, so 149 1430 1431 143 1433 1434 1435 1436 1437 1438 1439 1440 1441 144 1443 1444 1445 1446 1447 1448 1449 1450 1451 145 1453 1454 1455 1456 1457 1458 1459 1460 1461 something is wrong. Since each of the logical steps that we walked through are perfectly legitimate, the ONLY thing that could be wrong is our assumption that is rational. If it s not rational, then it s irrational. This concludes the proof. This result, as we ve seen, was a major discovery by the Pythagoreans, and created a major problem in their philosophical outlook of the world. No one knows exactly how the Pythagoreans proved this fact, and there are many possibilities, but their discovery (and other interests) played a crucial role in establishing the field of number theory. Other numbers can be shown to be irrational with similar arguments that depend on this method of proof, but we will not explore them in great detail here. But it is important to recognize that there are an infinite number of irrational numbers and some of them are very familiar to us. For example, 3 is irrational. The same goes for 6. Unfortunately, these proofs by contradiction can get increasingly complex so we want to be able to find and identify irrational numbers with other methods. (If you re interested in seeing a proof that 3 is irrational, you can read a proof in the endnotes. 36 ) Generating Hoards of Irrational Numbers FACT If a is any irrational number and r is any rational number except for zero, then all of the following are irrational: a r 1 a + r, a r, r a, ar,,, a, r a a This latest fact (which we will not prove here) says that if you combine an irrational number and a rational number (except maybe for zero) together with addition, subtraction, multiplication, or division, you will end up with an irrational number. This allows us to generate a whole bunch of irrational numbers without even trying. For example, since we know that is irrational, then we automatically know that all of the following are irrational: + 3, 5, 1 +, 4 3 MAT107 Chapter 4, Lawrence Morales, 001; Page 44

146 1463 1464 1465 1466 1467 1468 1469 1470 1471 147 1473 1474 1475 1476 1477 1478 1479 1480 1481 148 1483 1484 1485 1486 1487 1488 1489 1490 1491 149 1493 1494 1495 1496 1497 1498 1499 1500 1501 150 1503 1504 Unfortunately, this does not tell us anything about other numbers like + 5 since these two numbers do not satisfy the condition that one be rational and the other irrational. They re both irrational so Fact will not say anything about + 5. So, we see that there are many irrational numbers and many ways to generate even more. An endless supply exists for anyone in need of such numbers. We still have to consider the question of how to identify a number as being irrational without having to do a lengthy proof. To do this for certain kinds of numbers, we turn to a brief study of polynomial equations. Polynomial Equations By using polynomial equations we can develop a common method for establishing the irrationality of a large class of numbers. To do this, we focus not on the numbers themselves but on relatively simple algebraic equations that have these numbers as solutions. Example 13 Solution Find an equation that has 3 as a solution. To do this, we proceed as follows: Let x = 3 Square both sides to get: x = 3 Subtract 3 from both sides so that the equation is equal to 0: x 3 = 0 Thus, x 3 = 0, has as a solution x = 3 if we travel backwards in the logic of this problem. Similarly, other numbers that look like they could be irrational satisfy certain equations: 5 satisfies x 5 = 0 7 satisfies x 7 = 0 3 3 9 satisfies x 9 = 0 5 5 93 satisfies x 93 = 0 MAT107 Chapter 4, Lawrence Morales, 001; Page 45

1505 1506 1507 1508 1509 1510 1511 151 1513 1514 1515 1516 1517 1518 1519 150 151 15 153 154 155 156 157 158 159 1530 1531 153 1533 1534 1535 1536 1537 1538 1539 1540 If you re unsure of any of these, you re encouraged to solve the equations to verify that the stated solutions are indeed valid. All of these equations are called polynomials and are the focus of this section. We start by looking at one of the simplest of the polynomials, the quadratic polynomial. This is an expression of the form ax + bx + c, where a, b, and c are called the coefficients. The quadratic should be familiar to you by now as you have used the quadratic formula to solve them many times before and have even developed methods for graphing the parabolic curves that are associated with these equations. A cubic polynomial, or a polynomial of degree three, looks like 3 ax + bx + cx + d. Cubic polynomials may not be as familiar to you as they are generally not seen much until you study precalculus. We can extend this idea by defining a polynomial of degree n (where n is a positive integer) by stating that it has the form: c n n 1 n x + cn a x + + c1x +... x 0, with c n not zero. The numbers cn, cn 1,..., c, c1, c0 are called the coefficients of the polynomial. Example 14 Solution Check Point J Identify the degree, n, and the coefficients, c n, of the polynomial: 5 4 3 4x + x 5x + 7x 3 = 0 The degree of the polynomial is n = 5 since 5 is the highest power of x that is present. The coefficients are as follows: c = 4 c c c c c 5 4 3 1 0 = = 5 = 0 = 7 = 3 Note that c = 0 because there is no x term, and so we can pretend that the 3 term 0x is hiding between the 5x and the + 7x terms. Identify the degree, n, and the coefficients, c n, of the polynomial: 6 4 8x 5x + 6x 8x + 3 MAT107 Chapter 4, Lawrence Morales, 001; Page 46

1541 154 1543 1544 1545 1546 1547 1548 1549 1550 1551 155 1553 Solution Check the endnotes for the answer. 37 Eventually, we will use polynomial equations to determine whether or not certain numbers are irrational or not, but before we do that, we first need to make sure we know how to check if a given number is a solution to a polynomial. This is relatively easy, since all it entails is substituting the number into the polynomial and checking to see if we get a true statement. The only hard part is making sure that we don't make any errors in the process. An example is probably helpful. Example 15 Is a solution of the polynomial 10 3 6 x + x + x = 0? 38 5 1554 1555 1556 Solution To check this, we substitute 5 everywhere we see an x. 1557 1558 1559 10x 3 + 6x + x = 10 + 6 + 5 5 8 4 = 10 + 6 + 15 5 80 4 = + + 15 5 5 80 10 50 = + + 15 15 15 50 50 = 15 15 = 0 Since we get a true statement, we can say that 5 is a solution to the 1560 polynomial. 3 5 5 50 15 1561 156 1563 1564 In the previous example, if we had ended with a false statement, then our conclusion would have been that 5 was not a solution to the given equation. 1565 1566 1567 1568 Check Point K 1 Is a solution to the polynomial 4 3 x x 5 x + 9 = 0? MAT107 Chapter 4, Lawrence Morales, 001; Page 47

1569 1570 1571 157 1573 1574 Solution Check Point L See endnote for answer. 39 1 Is a solution to the polynomial 15 3 3 x x + 9 x 1 = 0? 3 1575 1576 1577 1578 1579 1580 1581 158 1583 1584 1585 1586 1587 1588 1589 1590 1591 159 1593 Solution See endnote for answer. 40 We are now ready to link polynomials to the question of whether or not a number is rational or irrational. The Rational Roots Theorem A root of a polynomial equation is the same thing as a solution of the polynomial set equal to 0. The two words are used interchangeably. The following theorem/fact gives us what we need to make the connection between polynomials and rational or irrational numbers. Theorem Given any polynomial with integer coefficients, such as: c n n 1 1 n x + cn a x +... + c x + c1x + c0 1594 a If this equation does have a rational root, say, where this fraction is in b 1595 lowest terms, then a is a divisor of c 0 and b is a divisor of c n. 1596 1597 This theorem basically tells us what the candidates are for possible rational solutions to a 1598 polynomial equation. If we can identify what they are, then we can use that information to 1599 determine whether or not a number is irrational or not. Let's first see how this theorem works. 1600 1601 Example 16 160 What are the possible rational roots of the following equation? 41 1603 1604 1605 3 x 9x + 10x 3 = 0 1606 Solution a 1607 If there is a rational root to this equation, we will call it. By the Rational b Roots Theorem, a must be a divisor of 3. The divisors of 3 are +1, 1, +3, 1608 1609 1610 and 3. Also, b must be a divisor of +. The divisors of + are +1, 1, +, and. Here is where it gets interesting. Any rational root of this equation is MAT107 Chapter 4, Lawrence Morales, 001; Page 48

1611 a going to look like, so we now have to combine all of these possibilities b 161 with each other to get all possible rational roots. 1613 Taking a = + 1 with all possible values for b gives: 1614 1615 1616 1617 1618 1619 160 161 16 163 164 165 166 167 168 1, 1 1, 1 1, 1 Taking a = 1 with all possible values for b gives: 1 1 1 1,,, 1 1 Taking a = 3 and a = 3 with all possible combinations for b gives: 3, 1 3, 1 3, 3, 3 3 3 3,,, 1 1 A careful examination of this list shows that many of them are repeats, so if we list only the unique possibilities from these lists, we have the following as all possible rational roots of the given polynomial: 1 1 3 3 1, + 1,, +, 3, + 3,, + If we want to know which of these work, we have to painstakingly substitute each of them in to determine if each one works or not. I'll let the reader verify 1 169 that 1,, and 3 all work. (If you have and know how to use a graphing 1630 calculator, this process is a little bit easier.) 1631 163 FACT 3 1633 1634 1635 1636 1637 1638 A polynomial of degree n has at most n roots, whether they are rational or not. This fact is useful because it gives you some idea of when to stop looking for rational roots. In Example 16 above, the polynomial was degree 3 so that means it has at most 3 roots. Once we 1 1639 had determined that 1,, and 3 worked, we could have stopped because we know that there could be no more. By this Fact, a degree-4 polynomial has at most 4 roots. It is possible that it 1640 1641 164 1643 1644 1645 1646 could have fewer than that, but it will certainly not have any more than 4. Example 17 3 What are the rational roots of the equation: 3x 10x 9x + 4 MAT107 Chapter 4, Lawrence Morales, 001; Page 49

1647 1648 1649 1650 1651 165 1653 1654 1655 1656 Solution By the Rational Roots Theorem, we know that a must divide 4, so a could be ± 1, ±, or ± 4. Also, b must divide 3, so b could be ± 1 or ± 3. Hence the combinations for b a are as follows: ± 1, 1 ± 1, 3 ±, 1 ±, 3 4 ±, 1 Generally, it is easy to check if integers are solutions first, so we by checking 1, 1,,, 4, and 4, we find that 1 and 4 are solutions. (The details are left out here.) This means there is at most one more solution to this polynomial and good old-fashioned brute force will reveal that 3 1 is the remaining root. ± 4 3 1657 1658 1659 1660 1661 166 1663 1664 1665 1666 1667 1668 1669 1670 1671 167 1673 1674 1675 1676 1677 1678 1679 1680 1681 168 Check Point M Solution Find the rational roots of the polynomial x + 9x 5 = 0 by using the Rational Roots Theorem. See the endnote for the answer. 4 Using Rational Roots Theorem to Show Irrationality We now move to the punch line for all of this work. We want to be able to determine if a given number is irrational or not by using the Rational Roots Theorem. We saw earlier that is irrational, but we used a long, intricate proof by contradiction to do it. We now have the tools to give a much more compact proof. Example 18 Solution Use the Rational Roots Theorem to show that We first find a polynomial that has bill: 43 x is irrational. as a solution. The following fits the = 0 From the Rational Roots Theorem, we know that any rational root, b a, for this 1683 equation will have to meet the requirements that a is a divisor of -, so a could 1684 1685 1686 be 1, -1,, or -; and b is a divisor of 1 and is thus either -1 or 1. So, the list of possible roots for this equation is as follows: MAT107 Chapter 4, Lawrence Morales, 001; Page 50

1687 1688 1689 1690 1691 169 1693 1694 1695 1696 1697 1698 1699 1700 1701 170 1703 1704 1705 1706 1707 1708 1709 1710 1711 171 1713 1714 1715 1716 1717 1718 1719 170 171 17 173 174 175 176 177 178 179 1, 1,, There are no other possible candidates for solutions that are rational. Since is not on this list, then could not be rational and is therefore irrational! Note that it is not necessary to do any substitution here. We are only interested in generating a list of all of the possible rational roots for this polynomial and then comparing that to the number we are examining. Example 19 Solution Example 0 Solution Use the Rational Roots Theorem to show that 8 is irrational. The polynomial that has 8 as a solution is x 8 = 0. By the Rational Roots Theorem, the list of possible rational roots for this equation is: +1, 1, +,, +4, 4, +8, -8 Since 8 is not on this list, 8 is not rational and is therefore irrational. Use the Rational Roots Theorem to show that 3 6 is irrational. We start by letting x = 3 6. Cubing both sides gives: x 3 = 6 Setting this equal to 0 gives: 3 x 6 = 0 This polynomial (by Rational Roots Theorem) has the following list of possible rational roots: +1, 1, +,, +3, 3, +6, 6 Since 3 6 is not on this list, 3 6 is not rational and is therefore irrational. An alternative proof by contradiction for this last example would be much more lengthy and much more difficult to follow. So, even though the Rational Roots Theorem takes a little time to understand and master, once we do that we have a very handy way of proving that a given number is irrational. As we mentioned earlier, this method will not work for all irrational numbers. In fact, it's only suited for numbers n x. Think About It Why can't the Rational Roots Theorem be used to prove that π is irrational? MAT107 Chapter 4, Lawrence Morales, 001; Page 51

1730 1731 173 1733 1734 1735 1736 1737 1738 1739 1740 1741 174 1743 1744 1745 1746 1747 1748 1749 1750 1751 175 1753 1754 1755 1756 1757 1758 1759 1760 1761 176 1763 1764 1765 1766 1767 1768 1769 1770 However, you will not be able to show that π is irrational using the Rational Roots Theorem. (The proof that π is irrational was developed relatively late in the history of mathematics and requires mathematics that goes well beyond what this chapter or course provides.) Check Point N Check Point O Use the Rational Roots Theorem to show that 0 is irrational. 44 Use the Rational Roots Theorem to show that 4 5 is irrational. 45 Earlier in this chapter, we talked about how we could generate a long list of irrational numbers by adding, subtracting, multiplying, or dividing a rational number (except maybe 0) to/with/by an irrational number. (See FACT on page 44.) What about adding two irrational numbers? What does that give you? Example 1 Solution Example Solution What is added to, rational or irrational? We know that each of these two numbers are irrational. But what about their sum? It's easy to see that they add up to 0, which is rational. Therefore, it's possible for the sum of two irrational numbers to be rational. What is + 3, rational or irrational? 46 Certainly, this one is not as straightforward. To determine this number's personality, we start by letting x = + 3. Now subtract from both sides to get: x = Now square both sides carefully to get the following: 3 ( x ) = ( 3) ( x )( x ) = 3 x Rearranging this we have: x + = 3 x 1 = x MAT107 Chapter 4, Lawrence Morales, 001; Page 5

1771 177 1773 1774 1775 1776 1777 1778 1779 1780 1781 178 1783 1784 1785 1786 1787 1788 1789 1790 1791 179 1793 1794 1795 Check Point P Solution Squaring both sides again we get: ( x 1) = ( x) x x 4 4 x 10x + 1 = 8x + 1 = 0 This is the equation to which we want to apply the Rational Roots Theorem. By that Theorem, the only possible rational roots would be +1 or 1. Since + 3 is clearly larger than either of these two possibilities, + 3 could not be rational and is therefore irrational. Show that 5 is irrational. Check the endnote for results. 47 This concludes our study of modern facts on irrational numbers. Although the Pythagoreans did not have these sophisticated tools and theorems, the ideas they discovered and explored left a lasting impact on the future of mathematics, especially with respect to the field we now call number theory. As time has passed, mathematicians have sought to learn more and more about the qualities, properties, and behavior of both rational and irrational numbers. Much of what they have learned has evolved into increasingly complex theories and facts about numbers and their natures that are now catalogued in the history of the subject. MAT107 Chapter 4, Lawrence Morales, 001; Page 53

1796 1797 1798 1799 1800 1801 180 Part 5: Pythagorean Geometry, A Brief Discussion The Pythagoreans are also known for some their studies of geometry, but their achievements here are more questionable. 48 Of course, the famous Pythagorean theorem can be considered a study in geometry, but beyond that, it appears that they also investigated the five regular polyhedra, which we will study more in the next chapter and are shown here 49 : 1803 1804 1805 1806 1807 1808 1809 1810 1811 181 1813 1814 1815 1816 1817 1818 1819 180 181 18 183 184 185 There is conflicting opinion on whether or not the Pythagoreans actually knew how to construct all five of these. Calinger conjectures that because the iron pyrite crystals found in southern Italy have faces that are composed of regular pentagons, the Pythagoreans became curious about figures such as these. 50 There is also some speculation that the Pythagoreans did geometric algebra, showing such facts as ( a + b) = a + ab = b by drawing appropriate pictures. We ll also save this topic for the next chapter when we look at geometric algebra. Conclusion The Pythagoreans, while known for their famous Theorem, also investigated many other ideas in mathematics. Their basic belief that numbers were the substance of the universe drove them to study mathematics in ways that allowed them to interpret the universe around them through a lens of mathematics. This is a unique approach and sets the stage for the development of number theory as well as the general attempt to interpret the world around us by investigating related mathematics. Whether we see it or not, mathematics is used all around us to fuel the technological change that is going on and to find new and better ways to view the world around us. As we have seen, the Pythagoreans and their studies branch off into many areas that they are not generally known for but for which they should be given due credit. MAT107 Chapter 4, Lawrence Morales, 001; Page 54

186 187 188 189 1830 1831 183 1833 1834 1835 1836 1837 1838 1839 1840 1841 184 1843 1844 1845 1846 1847 1848 1849 1850 1851 185 1853 1854 1855 1856 1857 1858 1859 1860 1861 186 1863 1864 1865 1866 1867 Part 6: Homework Problems Triangular Numbers Find each of the following triangular numbers: 1) T 7 ) T 18 3) T 34 4) T 100 For each of the following numbers, determine whether or not they are triangular numbers. If they are, indicate which one they are. If not, show why not. Use the quadratic formula in all calculations, and show all work neatly. 5) 595 6) 41,616 7) 3,488 8) 8,515 In around 100 C.E., Plutarch noted that if a triangular number is multiplied by 8 and then 1 is added to the result, then the resulting number would be a square number. 51 9) Use Plutarch s statement to do the following: a. Show that it is true for T. b. Draw a picture that clearly shows this fact for T geometrically. 10) Show that Plutarch s statement is true for T 5. 11) Show that Plutarch s statement is true for T 0. 1) Show that Plutarch s statement is true for any triangular number, T n. (You will need to work algebraically with variables rather than particular numbers. The algebraic expression you obtain after multiplying by 8 and adding 1 should be factorable.) Write each of the following numbers as the sum of three or fewer triangular numbers. 5 13) 56 14) 69 15) 185 16) 87 MAT107 Chapter 4, Lawrence Morales, 001; Page 55

1868 1869 1870 1871 187 1873 1874 1875 1876 1877 1878 1879 1880 1881 188 1883 1884 1885 1886 1887 1888 1889 1890 1891 189 1893 1894 1895 1896 1897 1898 1899 1900 1901 190 1903 1904 1905 1906 1907 1908 1909 1910 1911 17) In 187, the mathematician who is best know for discovering calculus, Lebesgue, showed two things related to triangular numbers. 53 Every positive integer is the sum of a square number (possibly 0 ) and two triangular numbers. Every positive integer is the sum of two square numbers and triangular number Show that Lebesgue s findings are true for the following numbers: a. 9 b. 44 c. 81 d. 100 18) The greatest mathematician in all of antiquity, Archimedes, found that you could add up the first n square numbers with a relatively simple formula: 1 + + 3 + 4 +... + n n = ( n + 1)( n 1) + To verify this formula works in particular cases, you would physically add up the first n square numbers on the left and then plug the appropriate value of n into the formula on the right to see that it matches what you got on the left. Verify that the formula works for: a. The sum of the first three square numbers. b. The sum of the first four square numbers. c. The sum of the first five square numbers. 19) A famous Hindu mathematician, Aryabhata, showed (in around 500 C.E.) that you could add up the first n triangular numbers with the following formula: n T1 + T + T3 + T4 +... + Tn = 6 ( n + 1)( n + ) To verify this formula works in particular cases, you would physically add up the first n triangular numbers on the left and then plug the appropriate value of n into the formula on the right to see that it matches what you got on the left. For example, to add up the first three triangular numbers, you would let n = 3 and substitute into the formula. This would give you the sum of T 1 + T + T3 = 1+ 3 + 6 = 10 a. Verify that the formula works for the sum of the first 10 triangular numbers by using the formula to compute the sum AND by adding up the first 10 triangular numbers. b. We certainly don t want to have to add up the first 100 triangular numbers, but use the formula to easily find the sum. c. What is the sum of the first 1000 triangular numbers? 6 MAT107 Chapter 4, Lawrence Morales, 001; Page 56

191 1913 1914 1915 1916 1917 1918 1919 190 191 19 193 194 195 196 197 198 199 1930 1931 193 1933 1934 1935 1936 1937 1938 1939 1940 1941 194 1943 1944 1945 1946 1947 1948 1949 Pentagonal Numbers A pentagonal number is one that can be arranged in the shape of a pentagon (five sides). The first three pentagonal numbers are shown and the number of dots in a figure corresponds to the value of the pentagonal number. (The first one is trivial. ). The n th Pentagonal 3( nn 1) number is given by Pn = n+. 0) Use the formula to compute P 4 and then draw it. (Use a ruler or straightedge, please.) 1) Use the formula to compute P 5 and then draw it. (Use a ruler or straightedge, please.) ) What is the value of P 15? 3) What is the value of P 5? 4) What is the value of P 100? For each of the following numbers, determine whether or not they are pentagonal numbers. If they are, indicate which one they are. If not, show why not. Use the quadratic formula in all calculations, and show all work neatly. 5) 05 6) 590 7) 5735 8),150 Oblong Numbers An oblong number 54 gives the number of dots in a rectangular grid having one more row than it has columns. The first few oblong numbers are shown below: O 1 = O =6 O 3 =1 O 4 =0 O n. 30) Use the formula given in the previous problem to find the 100 th oblong number 31) Show algebraically and geometrically that any oblong number is the sum of two equal triangular numbers. 9) Explain why the n th oblong number is given by = n( n +1) n=1 P 1 =1 n= P =5 n=3 P 3 =1 MAT107 Chapter 4, Lawrence Morales, 001; Page 57

1950 1951 195 1953 1954 1955 1956 1957 1958 1959 3) Show algebraically and geometrically that O n n = n. 33) Show algebraically that On + n = Tn. (To find an expression for T n, you will need to plug n into the general formula for a triangular number.) Non-Standard Figurative Numbers 34) Consider the following sequence of figures. D n is the total number of dots in the figure. n=1 n= n=3 n=4 1960 1961 196 1963 1964 1965 1966 1967 1968 1969 1970 1971 197 1973 1974 1975 1976 1977 1978 1979 1980 1981 198 1983 1984 1985 D n = 8 D n = 13 D n = 19 D n = 6 a. Find a general formula for D n which can be used to find the number of dots in the n th figure. Explain in words or show how you got your formula. (Hint: Combine the formulas for two known types of figurative numbers.). Show that your formula works for n = 3 and 4. b. Use your formula to find the number of dots in the 50 th figure. c. One of these figures has 818 dots. Which one is it? Use the formula from part (a) to and the quadratic formula to answer this. Show all algebraic steps. Other Proofs of the Pythagorean Theorem 35) Many civilizations had proofs of the Pythagorean theorem, including the Chinese. The attached picture is the Chinese hsuan-thu. The square that is in a diamond orientation is what we re most interested in. Notice that it is made up of four triangles surrounding a small center square in the middle of the diamond. (The triangles can be a bit hard to see, but look hard enough and you should see four of them arranged in such a way that they form and surround a small square in the middle.) Label the shorter legs of the triangles with lengths a, and the longer legs with length b. Label the hypotenuse of each with length c. a. Carefully cut the inner square/diamond into four triangles and one small square and then rearrange them to prove visually that a + b = c. When you are convinced you have a correct diagram, glue them together on a sheet and add any labels you think help MAT107 Chapter 4, Lawrence Morales, 001; Page 58

1986 1987 1988 1989 1990 1991 199 1993 1994 1995 1996 1997 1998 1999 000 001 00 003 004 005 006 007 008 009 010 011 01 013 014 015 016 017 018 019 00 01 0 03 04 05 06 07 08 09 demonstrate the proof. Hints: You don t need the four outer triangles; the shape you get may not be a square; make copies of the diagram before cutting in case you need another copy. There s also copies on the web site. HINT: You will find this same drawing and a related proof inside this chapter that should get you going with dissecting the pieces properly. b. Find the following WITHOUT assigning particular values to a, b, and c. i. The area of the diamond, in terms of c. ii. The area of all four inner triangles combined, in terms of a and b only. iii. The area of the small square, in terms of a and b only. c. Use your answers from part (a) and part (b) to algebraically prove the Pythagorean Theorem. Your proof should only use the variables a, b, and c. Briefly explain how your rearranged drawing finishes the proof. 36) In 1876, James Garfield, who would eventually be the 0th president of the United States, published a proof of the Pythagorean Theorem. Research what it was and present it. To receive credit you must include a neat drawing similar to his (use a ruler) as well as a proof that goes along with the drawing. The proof should be in your own words and reflect your own understanding of the problem. Your write-up, logic, and organization will be the main basis of your score. Specifically state where you got your information from or no credit will be given. (No printed web sites will be accepted it must be your written summary that you present.) 37) Research and present, in your own style, one proof of the Pythagorean Theorem that we have not talked about in this chapter. Your write-up, logic, and organization will be the main basis of your score. Specifically state where you got your information from or no credit will be given. (No printed web sites will be accepted it must be your written summary that you present.) 38) Consider the figure shown where triangle ABC is a right triangle (with right angle at C). Triangle BAD is also a right angle (with right angle at A). 55 a. Explain why triangles ABC and DBA are similar to each other. D B b. Use the fact that triangle ABC is C b similar to triangle DBA to show that ac AD =. b c. ac a Use the fact that AD = and other geometry as necessary to show that DC =. b b d. Prove that a + b = c by relating the area of triangle ABD to the areas of triangles ABC and ACD. a A c MAT107 Chapter 4, Lawrence Morales, 001; Page 59