One such point is easily obtained by drawing a line perpendicular to d through F and bisecting the segment from d to F (see figure 5.16). Figure 5.16 The construction of a parabola. More points of the parabola can be constructed as follows. Take any point, A, on the directrix, d, and draw the perpendicular bisector of the segment AF, which we denote by a. Then all points on a are equidistant to A and F. Finally, we construct a line through A perpendicular to d, intersecting a at point A'. Then A' will be a point of the parabola, since it has equal distance to F and d. By construction, the distance between A' and d is precisely equal to the distance between A' and A. Moreover, since A' lies on a, it is equidistant to F and A and therefore to F and d. To get points of the parabola on the other side of FP, we could continue this process, or we could reflect point A' about the axis of symmetry FP, whose reflected point is denoted by A”. Now we may pick other points, B and C, on d and construct B' and B” as well as C' and C”, respectively (see figure 5.16). Repeating this procedure for all (or many) points on d, we would get all (or many) points of the parabola. Note that the bisectors of line segments can be created in paper by folding one end of the segment onto the other. For this reason, tangents of parabolas are created by folding the points of the
directrix onto the focus. This is the reason why parabolas are the basic curves in the intrinsic geometry of paper folding, origami. Much more on this fascinating topic can be discovered in the book Geometric Origami by Robert Geretschläger (Shipley, UK: Arbelos Publishing, 2008). Let's go back to our example of the satellite dish. The three-dimensional shape of a satellite dish emerges if we rotate the parabola (see figure 5.17) around its axis of symmetry (see figure 5.18). The technical term for the surface generated by rotating a parabola around its symmetry axis is paraboloid. The reason for using this shape for satellite dishes is a wonderful geometric property of paraboloids. Imagine a mirror in the shape of a paraboloid (the reflecting layer being inside the paraboloid) and light rays hitting the mirror parallel to the paraboloid's axis of symmetry (that is, the perpendicular line from the focal point to the directrix). Then all the light would be reflected into the focal point, where the intensity will, thus, be very high, meaning that a parabolic mirror acts as an amplifier for light. But light is actually an electromagnetic wave, and so are television signals from satellites. Light rays are just a way to represent the direction in which the wave is traveling. A parabolic antenna amplifies the incoming signals by reflecting them into the focal point where a device is mounted that converts the signal to an electrical current. Figure 5.17 The parabola defined by its directrix and its focal point.
Figure 5.18 A paraboloid is obtained by rotating the parabola around its symmetry axis. Before exploring the reflective property of the parabola, we have to recall the law of reflection, which states that the angle of incidence is equal to the angle of reflection measured from the normal. The normal is perpendicular to the surface, that is, perpendicular to the plane tangent to the surface (see figure 5.18).
Figure 5.19 Light rays coming in parallel to the symmetry axis of the parabola converge at the focal point. Now let us consider a ray hitting the parabola at point P at an angle α with the tangent to the parabola, as shown in figure 5.19. We extend the ray to the directrix and denote the point of intersection by Q. Moreover, we draw the tangent to the parabola through P. Then PQ = PF by the definition of a parabola, and PR is the altitude of the isosceles triangle QPF. The angle FPR is therefore congruent to angle QPR. But , since vertical angles are congruent. Hence, PF indeed represents the direction of the ray reflected from the parabolic mirror. Since P was an arbitrary point on the parabola, this must be true for all points on the parabola, meaning that a beam of electromagnetic waves coming in parallel to the symmetry axis will converge at the focal point, F. Finally, since the law of reflection is symmetrical, the opposite is also true. Light emitted by a source placed at the focal point will be focused outward along the direction of the symmetry axis. This is exactly how flashlights or car headlights work. See figure 5.20, which shows why a car's headlights emit such a strong light when only a relatively weak light is at the focal point.
Figure 5.20 As shown in figure 5.20, when light rays emitted by a source at the focal point are reflected inside the parabolic headlight, they will all leave the headlight parallel to its axis of symmetry, resulting in a strong concentration of light in this direction. We have now come to the end of the book with the hope that you have had a chance to experience mathematics as a subject that not only is exciting in and of itself, but also has many applications of the material that is taught in school and unfortunately too often omitted. Many of these “omissions” can help explain lots of the things we take for granted without explanation. Furthermore, it is our contention that many more people will grow to love mathematics if it is taught in a way that exhibits some of the motivating ideas we presented here. We hope that you have noticed how, from the earliest grades, these topics could have been inserted in the curriculum and would have motivated students to the point that mathematics could have
been the favorite subject of the majority of students, rather than the reverse. When teachers begin to realize that teaching mathematics is not merely providing a series of techniques and skills but also an opportunity to exhibit the many wonders that the subject harbors, and the many applications that help explain our everyday lives, they will back off from merely “teaching to the test.” Perhaps, one day, having been a weak mathematics student in school will no longer be a “badge of honor” among the general adult population. Let's go forward and share some of these “curriculum omissions” with the current student population so that we can correct this universal deficit in the future!
The authors wish to acknowledge superb support services received from the publisher Prometheus Books, led by their editor in chief, Steven L. Mitchell, and his truly dedicated production coordinator, Catherine RobertsAbel. We wish to also thank Senior Editor Jade Zora Scibilia for her highly meticulous editing and clever suggestions to make the presentation as intelligible as possible. Thanks is also due to Editorial Assistant Hanna Etu, and the typesetter, Bruce Carle. The cover design exhibits the talents of Nicole Sommer-Lecht. We're also very pleased with the indexing by Laura Shelley. Each of the authors has many people to thank for their patience and support throughout this book-development process. In particular, Dr. Christian Spreitzer wants to thank Katharina Brazda for inspiring discussions, which resulted in some very creative contributions.
As we mentioned earlier, Ceva's theorem might well have been introduced to a high school class, since it merely applies similarity relationships that are an integral part of the geometry curriculum. We offer one of many proofs available to justify Ceva's theorem. It is perhaps easier to follow the proof by looking at the left-side diagram in figure App.1 and then verifying the validity of each of the statements in the right-side diagram. In any case, the statements made in the proof hold for both diagrams. Figure App.1. Consider figure App.1, for which we have on the left triangle ABC with a line (SR) containing A and parallel to BC, intersecting CP extended at S and BP extended at R. The parallel lines enable us to establish the following pairs of similar triangles:
Now by multiplying (I), (II), and (V), we obtain our desired result: This can also be written as AM ∙ BN ∙ ∙ CL = MC ∙ NA ∙ BL. A nice way to read this theorem is that the product of the alternate segments along the sides of the triangle made by the concurrent line segments (called cevians) emanating from the triangle's vertices and ending at the opposite side are equal. Yet, it is the converse of this proof that is of particular value to use here. That is, if the products of the alternate segments along the sides of the triangle are equal, then the cevians determining these points must be concurrent. We shall now prove that if the lines containing the vertices of triangle ABC intersect the opposite sides in points L, M, and N, respectively, so that , then these lines, AL, BM, and CN, are concurrent. Suppose BM and AL intersect at P. Draw PC and call its intersection with AB point N'. Now that AL, BM, and CN' are concurrent, we can use the part of Ceva's theorem proved earlier to state the following: But our hypothesis stated that . Therefore, , so that N and N′ must coincide, which thereby proves the concurrency.
For convenience, we can restate this relationship as follows: If AM ∙ BN ∙ CL = MC ∙ NA ∙ BL, then the three lines are concurrent. The ambitious reader may want to see other proofs of Ceva's theorem, which can be found in Advanced Euclidean Geometry by Alfred S. Posamentier (New York: John Wiley and Sons, 2002, pp. 27–31.
INTRODUCTION 1. Michel Chasles, Aperçu historique 2 (1875). CHAPTER 1: ARITHMETIC NOVELTIES 1. See MacTutor History of Mathematics Archive, School of Mathematics and Statistics, University of St Andrews, Scotland, “An Overview of Babylonian Mathematics,” http://wwwhistory.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html and the references therein. 2. Pi World Ranking List, http://www.pi-world-ranking-list.com/index.php? page=lists&category=pi. 3. Proper divisors are all the divisors, or factors, of the number except the number itself. For example, the proper divisors of 6 are 1, 2, and 3, but not 6. 4. If k = pq, then 2k – 1 = 2pq – 1 = (2p – 1)(2p ( q –1) + 2p (q–2) + ∙ ∙ ∙ + 1). Therefore, 2 k – 1 can only be prime when k is prime, but this does not guarantee that when k is prime, 2 k – 1 will also be prime, as can be seen from the following values of k: Note that 2,047 = 23 ⋅ 89 is not a prime and so doesn't qualify. 5. Rounded to nine decimal places. CHAPTER 2: ALGEBRAIC EXPLANATIONS OF ACCEPTED CONCEPTS 1. Kurt von Fritz, “The Discovery of Incommensurability by Hippasus of Metapontum,” Annals of Mathematics 46, no. 2, 2nd ser. (April 1945); see also chap. 6 in Lore and Science in Ancient Pythagoreanism, by Walter Burkert (Cambridge, MA: Harvard University Press, 1972).
CHAPTER 3: GEOMETRIC CURIOSITIES 1. L. Hoehn, “A Neglected Pythagorean-Like Formula,” Mathematical Gazette 84 (March 2000): 71–73. 2. Deanna Haunsperger and Stephen Kennedy, eds., The Edge of the Universe: Celebrating 10 Years of Math Horizons (Washington, DC: Mathematical Association of America, 2006), p. 231. 3. This reference to pons asinorum would appear to be wrong, since we usually consider the proof that the base angles of an isosceles triangle are congruent as the pons asinorum, or “bridge of fools.” It is clear that they meant to refer to the Pythagorean theorem. New England Journal of Education 3, no. 14 (April 1, 1876). 4. Elisha S. Loomis, The Pythagorean Proposition (Reston, VA: NCTM, 1940, 1968). 5. A book to consider is The Pythagorean Theorem: The Story of Its Power and Beauty, by A. S. Posamentier (Amherst, NY: Prometheus Books, 2010). 6. This idea was published by Cabre Moran, “Mathematics without Words,” College Mathematics Journal 34 (2003): 172. 7. For more information about Ceva's theorem and related topics, see The Secrets of Triangles, by A. S. Posamentier and I. Lehmann (Amherst, NY: Prometheus Books, 2012). 8. “Small stellated dodecahedron” was created by Robert Webb's Stella Software, http://www.software3d.com/Stella.php. 9. “Great stellated dodecahedron” was created by Robert Webb's Stella Software, http://www.software3d.com/Stella.php. 10. “Great dodecahedron” was created by Robert Webb's Stella Software, http://www.software3d.com/Stella.php. 11. “Great icosahedron” was created by Robert Webb's Stella Software, http://www.software3d.com/Stella.php. CHAPTER 4: PROBABILITY APPLIED TO EVERYDAY EXPERIENCES 1. Interested readers might want to read the book by Jason Rosenhouse, The Monte Hall Problem: The Remarkable Story of Math's Most Contemptuous Brainteaser (New York: Oxford University Press, 2009). 2. For readers interested in methods of calculating the number of possible hands of a certain type, here is a brief introduction to the procedures you can use to get these numbers. Here is how to calculate the number of three-of-a-kind hands when there are two jokers in the deck: Out of the possible hands that can be dealt from a deck of 52 cards plus two jokers, there are three types of hands that will normally be counted as three of a kind. These can include 0, 1, or 2 jokers. First of all, let us consider how many three-of-a-kind hands there are that do not include a joker. The three cards of the same kind can have any of thirteen values, from a 2 through an ace. There are four ways to choose three of these (as one of the four will not be chosen). Furthermore, the other two cards can be of any of the remaining twelve values, and there are
such choices possible. Each of these values can be represented by any of the four suits. In total, there are, therefore, such hands. Next, let us consider how many three-of-a-kind hands include exactly one joker. These hands all result from one pair, two other different cards, and the joker to complete the three of a kind together with the pair. We obtain numbers in a similar fashion to the previous case. The pair can have any of thirteen values, from a 2 through an ace, and there are ways to choose two of these from the four cards of this value. Furthermore, the other two cards can once again be of any of the remaining twelve values, and there are again such choices possible, with each of these values being represented by any of the four suits. Finally, there are also 2 possible jokers in play, and either of these can be part of the hand. In total, there are therefore, such hands. (Note that this is three times as many as there are “normal” three of a kind hands.) Finally, we have the most difficult case to consider, the case in which two of the cards in the hand are jokers. Since we have only three more cards to choose, this may seem to be easier, but this is not the case. First, we note that the three cards must be of different values, since any pair would make four of a kind possible, which would yield a more valuable hand than three of a kind. These three may not be of the same suit, since we could otherwise define the two jokers to complete a flush, which is also more valuable than three of a kind, and they may also not be within the same sequence of five values (since we could otherwise define the two jokers to complete a straight). Also, in eliminating these two possibilities, we must note that a straight flush (or royal flush) is also an option, and this is included in both, so simple subtraction of these cases will not suffice. Since these hands will have been subtracted twice, we can add their number again once (an application of the socalled inclusion-exclusion principle) to obtain the number we want. This gives us the calculation:
and therefore, the total number of three-of-a-kind hands is 54912 + 164736 + 13320 = 232968. (Note that the number 64 for the number of possible combinations leading to a straight results from allowing ace high or low, which is not always applied.) 3. See S. Gadbois, “Poker with Wild Cards—A Paradox?” Mathematics Magazine 69 (1996): 283–85. CHAPTER 5: COMMON SENSE FROM A MATHEMATICAL PERSPECTIVE 1. Florian Cajori, A History of Mathematical Notations, vol. 2 (Chicago: Open Court Publishing, 1929), pp. 182–83. 2. This triangle is formed by beginning on top with 1, then the second row has 1, 1, then the third row is obtained by placing 1s at the end and adding the two numbers in the second row (1 + 1 = 2) to get the 2. The fourth row is obtained the same way. After the end 1s are placed, the 3s are gotten from the sum of the two numbers above (to the right and left), that is, 1 + 2 = 3, and 2 + 1 = 3. 3. Image from Wikimedia Creative Commons; author: Keithonearth; licensed under CC BY-SA 3.0 via Commons, https://commons.wikimedia.org/wiki/File:Derailleur_Bicycle_Drivetrain.svg#/media/File:Derailleur_ Bicycle_Drivetrain.svg.
NUMBERS 0. See zero (0) 1 not included in list of prime numbers, 52–53 numbers composed entirely of 1s (see repunits) 1.618. See golden ratio (ϕ) 2 , 35, 36 bisection method to approximate, 83–86 and continued fractions, 86–89 irrationality of, 80–82 divisibility by, 13, 17 natural log of (.693147), 263 as only even prime number, 54 3, divisibility by, 13, 14–15 4, divisibility by, 17 5 divisibility by, 17 squaring numbers with last digit of, 22 6 as a perfect number, 62 curiosities related to, 65 proper divisors of, 293n3 7, divisibility by, 17–19 7-gon, 198 9, divisibility by, 13, 14–15 11 divisibility by, 13, 15–16 first four powers of 11 are palindromic numbers, 48 and sum of any ten consecutive Fibonacci numbers, 72 13, divisibility by, 19–20 17, divisibility by, 20–21, 62–63 28
as a perfect number, 62–63 curiosities related to, 65 72, rule of, 253–55 496 curiosities related to, 65 as a perfect number, 63 SYMBOLS AND BASIC EQUATIONS a 2 + b 2 = c 2 . See Pythagorean theorem ax2 + bx + c. See quadratic equation ℵ, 272 c 2 = a 2 + bd. See Hoehn, Larry, and Hoehn's formula ÷. See division =, 241 e. See Euler's constant (e) ϕ. See golden ratio (ϕ) > (greater than), 241 ∞. See infinity < (less than), 241 – (minus sign). See subtraction × or ·. See multiplication +. See addition √. See square root operations NAMES AND SUBJECTS Adams, John, sharing death date with Thomas Jefferson and James Monroe, 214–15 addition and friendly numbers, 45–47 reverse-and-add as way to find palindromic numbers, 49–50 sums of Fibonacci numbers, 72–73 finding by pairing numbers in an alternate harmonic series, 262–63 finding by pairing numbers in an infinite series, 261–62 four-square theorem (sums of squares of numbers), 23–24 number equal to sum of all two-digit numbers formed by, 44–45 of numbers in rows of Pascal's triangle, 226 of odd positive integers, 32–34 partial sum of the series of perfect numbers, 65 of positive integers, 30–34 of the products of consecutive Fibonacci numbers, 75 of proper divisors (see perfect numbers) “staircase” for determining sums of positive integers, 30–31 sum-of-consecutive-squares primes, 54
triangular numbers as sum of first n consecutive natural numbers, 57–58 sums and powers odd numbers and sum of power of 2 and a prime number, 66–67 sum of digits in a number raised to same power, 42–43 sums of digits raised to consecutive powers, 44 and triangular numbers, 57–59, 60–61 use of plus symbol (+), 240 Advanced Euclidean Geometry (Posamentier), 291 algebraic explanations, 77–112 comparing means, 91–94 and Descartes's rule of signs, 100–102 Diophantine equations (solving equations with two variables), 94–98 of division by zero, 78–79 of falling objects, 98–100 Fermat's method of factoring to distinguish prime and composite numbers, 89–91 and Frobenius problem of largest amount of money using only coins of specified denominations, 110–12 and generating Pythagorean triples, 105–10 Horner's method to evaluate polynomials, 102–105 and logic, 77–78 solving equations with two variables, 94–98 and square roots bisection method to approximate, 83–86 continued fractions of square roots, 86–89 irrationality of the square root of 2, 80–82 taught as a mechanical process, 113 algorithms, 10 for better approximation of square root of 2, 84 Euclidean algorithm to find greatest common divisor, 28–29 Fermat's method of factoring, 89–91 Newton-Raphson method for extracting a square root, 25–26 for quantum computers, 42 and the “square root test,” 90 for successive percentages, 251–53 Almagest (Ptolemy), 141 alternating harmonic series, 262–63 amicable numbers. See friendly numbers Analysis alquationum universalis (Raphson), 25 angles of equal size, 194, 195 impact on Pythagorean theorem of changing right angle to an angle of 60°, 159–62 negative angles, 144 sine of an angle, 142, 146 (see also sines) small angle measurement, 142–44 Apollonius of Perga, and Apollonian circles, 188–91 arbitrary numbers, 23 determining if number is a prime number, 39–40 (see also prime numbers: prime factorization) sums of arbitrary positive integers, 30 using squares to multiply arbitrary numbers, 24–25 areas, finding for geometric shapes
in circles and semicircles, 172–75 for exotic-shaped polygons (Pick's formula), 135–37 formulas for, 114, 124 in parallelograms, 114–21 of a square, 116 in triangles, 114–15, 124–27 going beyond Heron's formula, 125–26 Heronian triangles, 127–31 for isosceles triangles, 131–34 using a grid to calculate areas, 115–21 volume and surface areas of a sphere and cylinder, 192–94 Aristotle, 266 Arithmetica (Diophantus), 23, 95 arithmetic mean, 92, 93, 94, 230, 232, 254 arithmetic operations, 13–75 arithmetic sequence of numbers, 91–92 comparing large numbers, 26–27 mathematical symbols, 239–41 triangular numbers resulting from arithmetic series, 58 See also addition; division; integers; multiplication; numbers; square root operations; subtraction “Ask Marilyn” column, 217 atoms, 39 atoms of numbers, 41 and the periodic table of elements, 37 averages. See means Ayn new Kunstlich Buech (Schreyber), 240 Babylonian number system, 25 basketball tournament elimination problem, 244–46 Bayes, Thomas, and Bayes's rule, 220 Behende und hüpsche Rechenung auff allen Kauffmanschafft (Widmann), 240 Benford, Frank Albert, Jr., and Benford's law applying, 211–12 and discrete uniform distribution, 209–11 Bernoulli, Johann, 240–41 Bertrand, Joseph, and Bertrand's box, 218–21 B-fever and the false positive paradox, 221–24 bicycle, mathematics involving, 273–78 birthday phenomenon, 212–15 chart of probability of birth date match, 214 boundary points in a polygon, 136 Brahmagupta, 126 “bridge of fools.” See pons asinorum Brownian motion and random walks, 232 Cantor, Georg, 256 on comparing sizes of different sets even if infinite, 266–73 card games. See poker game cardinality of a set (collection) (ℵ), 272 Cartesian plane, 142
centerpoint of a quadrilateral, 121, 122–24 centroid of a triangle, 121 Ceva, Giovanni, and Ceva's theorem, 176–77 justification of, 289–91 chainrings. See gears and ratios on a bicycle, mathematics involving Champernowne, D. G., and Champernowne's constant, 36, 36–37 Charlie, the bicycle rider problem, 276 chords, 145 chord function, 140–41 in an isosceles triangle, 140–41 Circle, The: A Mathematical Exploration Beyond the Line (Posamentier and Geretschläger), 192 circles Apollonian circles, 188–91 area of, 172 rectilinear figure of equal area to a circle, 194 and bicycles, 273–78 circumcircles, 146 concurrency in, 177, 179 constructing, 131, 191, 279 cyclic quadrilateral, 125–26 diameters of two circles intersecting, 188 escribed circles, 179–82 and inscribed squares (finding area of square), 116–17 and inscribed triangles and sines, 145–46 intersecting lines meeting, 137–40 partitioning of, 260 and points, 186–91, 260 Gergonne point, 177 Miquel point, 178–79 semicircles (lunes and the right triangle), 172–75 and sines of small angles, 142–44, 145–46 triangle circumscribing a circle, 177 unit circle trigonometry, 142, 143 circumference (π times diameter), 275 in a cylinder, 193–94 Clavis Mathematicae [The Key to Mathematics] (Oughtred), 240 coins and the Bertrand's box problem, 218–21 coin tosses and Pascal's triangle, 225–26 finding largest amount of money (Frobenius problem), 110–12 flipping coins and probability, 207–208 flipping in the dark, 78 compasses used in geometric constructions, 191–92 composite numbers, 39, 53, 55 determining if number is a prime number or composite, 39–40 Fermat's method of factoring, 89–91 “square root test,” 89–91 divisibility by, 21 and Fibonacci numbers, 72 compound interest, problem involving, 253–55
concurrency, 176–82 Ceva's theorem, 176–77 justification of, 289–91 constants Champernowne's constant, 36–37 Euler's constant (e), 34, 80 See also pi (π) continued fractions of square roots, 86–89 convex polygon, determining, 195–96, 203 Conway, John, 36 cosines, 142 definition of evolving from right triangle trigonometry, 143 law of cosines, 162 Coss (Rudolff), 240 counterintuitive situations in mathematics birthday phenomenon, 212–15 and the concept of infinity, 263–65 false positive paradox, 221–24 finding unexpected patterns, 258–61 Grand Hotel paradox, 264 Monty Hall problem, 215–17 toothpick arrangement problem, 242–43 See also problems and problem solving counting the uncountable, 266–73 cubes (geometric shape), 200, 202 Euler formula for counting sides, faces, and vertices of polyhedra, 169 curve, parabola as a remarkable, 278–84 cylinder and sphere, 192–94 death dates, 214–15 decimal numeral system, 17 nonterminating decimals, 34–37 repeating decimals, 34–35 decomposition of prime, method for, 28, 29, 38, 39–40 Desboves, A., 256 Descartes, René, and rule of signs, 100–102 diagonal counting scheme, 269–72 Diophantus of Alexandria, 23, 95 Diophantine equations, 94–98 directrix of a parabola, 279, 280–81, 283 discounts, problem involving, 251–52 discrete uniform distribution and Benford's law, 209–11 division divisibility rules for division by 2, 13, 17 division by 3, 9, or 11, 13, 14–16 division by 4, 17 division by 5, 17 division by 7, 17–19 division by 13, 19–20
division by 17, 20–21 finding greatest common divisor (Euclidean algorithm), 28–29 prime numbers, 13, 17–21 divisors of friendly numbers, 45 and Fibonacci numbers, 74 proper divisors, 293n3 sum of proper divisors (see perfect numbers) resulting in a nonterminating decimal, 34 synthetic division, 105 use of symbols to indicate, 241 DNA molecules and Champernowne's constant, 37 dodecahedron, 200, 203 stellated dodecahedron, 203–204 Een sonderlinghe boeck in dye edel conste Arithmetica (Hoecke), 240 election fraud, finding evidence for, 212 Elements [Stoicheia] (Euclid), 39, 131–34 encryption systems and prime factorization, 41–42 equalities, finding by taking certain numbers to powers of 1, 2, 3, 4, 5, 6, and 7, 67–68 equals, symbol for, (=), 241 equilateral triangles and concurrency, 176 and creation of a great dodecahedron, 204 and creation of a star, 198 and definition of triangular numbers, 56 as an example of a regular polygon, 194 points inside, 9 and Pythagorean theorem extension of Pythagorean theorem, 159–62 sharing a side with a square, 157–58, 159 Euclidean algorithm to find greatest common divisor, 28–29 Euclid of Alexandria, 131, 147 fundamental theorem of arithmetic (Euclid's theorem on prime numbers), 38–42 Euler, Leonhard, 95, 168, 255 Euler's constant (e) as an irrational number, 35, 80 nonterminating decimals, 34, 80 formula for counting sides, faces, and vertices of polyhedra, 168–72 method for finding integral solutions, 95–97 even numbers 2 as only even prime number, 54 comparing size of set of to set of natural numbers, 263, 265 Goldbach's conjecture on even numbers greater than 2, 255–56 using squares to multiply two even numbers, 24–25 expected value in probability theory, 230 extremes, use of in problem solving, 246–47 Fabulous Fibonacci Numbers, The (Posamentier and Lehmann), 75 faces of polyhedra, counting, 168–72 factorization
Fermat's method of factoring, 89–91 of prime numbers, 40–42, 46 falling squares or objects, 98–100 false positive paradox, 221–24 Fermat, Pierre de finding friendly numbers, 46 method of factoring, 89–91 and theory of probability, 207, 208, 233 Fibonacci (Leonardo of Pisa) and Fibonacci numbers, 69 and Benford's law, 211 Fibonacci numbers, 69–75 known triangular numbers, 60 in Pascal's triangle, 228 ratios of consecutive Fibonacci numbers, 70 Fibonacci Association, 71 Fibonacci Quarterly, The, 71 Fillmore, Millard, having same death date as William Taft, 214 four-dimensional figures, 165–66 four-square theorem of Lagrange, 24 fractions constructing fractions for nonterminating decimals, 34–35 continued fractions of square roots, 86–89 infinite continued fractions, 87 friendly numbers, 45–46 Frobenius, Ferdinand Georg and Frobenius problem, 110–12 fundamental theorem of arithmetic (Euclid's theorem on prime numbers), 38–42 gambling. See poker game games and probability. See probability theory Garfield, James A., and proof of the Pythagorean theorem, 147–49 use of pons asinorum, 147, 294n3 Gauss, Carl Friedrich, 30 gcd. see greatest common divisor (gcd) gears and ratios on a bicycle, mathematics involving, 273–78 generalizations, mistaken, 66–69 Geometria del Compasso (Mascheroni), 192 geometric explanations, 113–205 center of a quadrilateral, 121–24 compasses used in geometric constructions, 191–92 concurrency, 176–82 decomposing complex shapes, 137 geometric proof of the irrationality of the square root of 2, 82–83 intersecting lines meeting a circle, 137–40 lunes and the right triangle, 172–75 point and circle relationships, 186–91 and Pythagorean theorem (see also Pythagorean theorem) extending into three dimensions, 163–68 extending to other polygons, 153–62 proofs of, 147–53 “rubber sheet geometry” (topology), 170–72
sides, faces, and vertices of polyhedra, 168–72 similarity and the golden ratio, 182–86 stars, creation of, 194–200 taught as a logic process, 113 See also areas, finding for geometric shapes; solid geometry; three-dimensional space geometric mean, 92, 93–94 Geometric Origami (Geretschläger), 280 geometric sequence, 91 Geretschläger, Robert, 280 Gergonne, Joseph Diaz, and Gergonne point, 177 Goldbach, Christian, and Goldbach's conjectures, 255–58 golden ratio (ϕ) and Fibonacci numbers, 69–71 and similarity, 182–86 Grammateus, Henricus. See Schreyber, Heinrich Grandi, Luigi Guido, 261 greater than, symbol for, (>), 241 greatest common divisor (gcd), 28–29 Guinness Book of World Records, 217 Harding, Warren G., having same birthday as James Polk, 213 harmonic means, 92–93, 94 harmonic series, alternating, 262–63 Harriot, Thomas, 241 Helfgott, Harald, 257 hens laying eggs problem, 249–50 heptagon, 197 Heron of Alexandria Heronian triangles, 127–31 Heron's formula for area, 124–25 hexagons, 194, 195 hexahedron, 202 Hilbert, David, 264 Hipparchus, 140, 141 Hippasus of Metapontum, 80 Hjelmslev, Johannes, 192 Hoecke, Gillis van der, 240 Hoehn, Larry, and Hoehn's formula, 132–34 Horner, William George, and method to evaluate polynomials, 102–105 hypotenuse and Pythagorean theorem, 105–106, 117, 152, 154–55, 157, 159, 164, 172 and semicircles, 173 and sine ratio, 143, 145–46 icosahedron, 200 inclusion-exclusion principle, 234n3 inequality symbols, (<) and (>), 241 infinite continued fractions, 86–87. See also square root operations: square root of 2 infinity comparing sizes of different infinite sets, 266–73
cardinality of a set (collection), 272 concept of, 263–65 Grand Hotel paradox, 264 infinite numbers of prime numbers, 55 infinity conundrum in pairing of consecutive members of a series, 261–63 symbol used for, (∞), 264 Zeno's paradox, 264 integers decomposition of integers greater than 1, 38–40 and Diophantine equations, 95, 97 dividing integers and nonterminating decimals, 34–35 and Fermat's method of factoring, 89–91 four-square theorem of Lagrange, 23–24 and friendly numbers, 47 and Heronian triangles, 127, 128–29 and lattice polygons, 136 and n-pointed stars, 199 and palindromic numbers, 49 and perfect numbers, 63 prime decomposition method and gcd, 28–29 and prime factorization, 41–42 and Pythagorean triples, 106, 107 and random walks, 229 and triangular numbers, 56, 60, 61 See also non-negative; numbers; positive integers integer sides, list of in Heronian triangles, 129–30 integral solutions, Euler's method for finding, 95–97 intercept theorem, 137–38 interior points in a polygon, 137 intersections constructing intersections, 191 diameters of two circles intersecting, 188 intersecting circles in Apollonian circles, 189–90 intersecting lines intercepted by a pair of parallel lines, 137–38 intersecting lines meeting a circle, 137–40 irrational numbers irrationality of the square root of 2, 80–82 and nonterminating decimals, 35, 83 patterns in digits of decimal approximations, 36 and square roots, 86–89 square root when number is not a square number, 35 See also Euler's constant (e); pi (π); square root operations: square root of 2 isosceles triangles area of, 131–34 as an example in trigonometry, 140–41 and right triangles, 132, 133 Jefferson, Thomas, sharing death date with John Adams and James Monroe, 214–15 Kepler, Johannes, 202
and Kepler-Poinsot solids, 203 Kumar, Suresh, 36 Lagrange, Joseph-Louis, 24 large numbers, comparing, 26–27 lattice points, 136, 137 lattice polygon, 136–37 law of cosines, 162 law of reflection and the parabola, 283–84 law of sines, 146 Leibniz, Gottfried Wilhelm, 240–41 Leonardo of Pisa. See Fibonacci and Fibonacci numbers less than, symbol for, (<), 241 Let's Make a Deal (TV show) and Monty Hall problem, 215–17 Liber Abaci [Book of Calculation] (Fibonacci), 69 lines constructing, 191, 192 intersecting lines intercepted by a pair of parallel lines, 137–38 intersecting lines meeting a circle, 137–40 radical axis or power line, 188 logic and algebra, 77–78 Loomis, Elisha S., 151, 152 lunes and the right triangle, 172–75 Mascheroni, Lorenzo, and Mascheroni constructions, 192 mathematical conjectures. See Goldbach, Christian, and Goldbach's conjectures Mathematical Gazette, The, 132 mathematical symbols, 239–41, 264, 272 means arithmetic mean, 92, 93, 94, 230, 232, 254 averages in the random walk, 230, 232 average value and rule of 72, 254 comparing means, 91–94 geometric mean, 92, 93–94 harmonic means, 92–93, 94 Method of Fluxions (Newton), 25 minus symbol (–), 240 Miquel, August, and Miquel point, 178–79, 179 mistaken generalizations, 66–69 Mohr, George, 192 Monroe, James, sharing death date with Thomas Jefferson and John Adams, 214–15 Monty Hall problem, 215–17 multiplication product of Fibonacci numbers, 74, 75 and squares, 23, 25 as combination of squares of sums and differences, 22 symbols used for, (×) or (·), 240–41 natural numbers comparing size of set of to set of all integers, 266
comparing size of set of to set of even numbers, 263, 265 as an infinite set, 263, 265, 266, 268–69, 270, 271, 272–73 in Pascal's triangle, 227 square root of as irrational numbers, 35 and triangular numbers, 57–58 “Neglected Pythagorean-like Formula, A” (Hoehn), 132 New England Journal of Education, 147 Newton, Isaac, and Newton-Raphson method for extracting a square root, 25, 25–26 New York Times (newspaper), 217 non-negative integers, 33, 110, 111 integral solutions, 95 numbers, 93 nonprime numbers. See composite numbers nonterminating decimals, 34–37 and irrational numbers, 35, 83 lack of periodic repeating patterns, 86–87 repeating decimals, 34–35 numbers arbitrary numbers, 23 determining if number is a prime number, 39–40 sums of arbitrary positive integers, 30 using squares to multiply arbitrary numbers, 24–25 arithmetic sequence of numbers, 91–92 Babylonian number system, 25 comparing large numbers, 26–27 friendly numbers, 45–46 pentagonal numbers found in Pascal's triangle, 228 real numbers as uncountable, 270, 272 See also composite numbers; even numbers; Fibonacci and Fibonacci numbers; fractions; integers; irrational numbers; natural numbers; odd numbers: palindromic numbers; perfect numbers; prime numbers: prime factorization; rational numbers; ratios; square numbers; triangular numbers obelus (÷), 241 octahedron, 200 odd numbers appearance in perfect and triangular numbers, 65 and sum of power of 2 and a prime number, 66–67 sums of odd positive integers, 32–34 odds, determining. See probability theory Oliveira e Silva, Tomás, 256 orbits, eccentricity of, 140 organized thinking, use of in problem solving, 247–50 Oughtred, William, 240 palindromic numbers, 47–52 prime numbers as, 53–54 repunits, 50–52 triangular numbers that are palindromic, 59
parabola, 278–84 construction of, 280 definition of, 279 paraboloid, 281, 282 Parade (magazine), 217 parallelograms, 118 area of, 114–15 diagonals of, 122–24 See also quadrilaterals Pascal, Blaise Pascal's triangle, 224–28, 233, 297n2 unexpected pattern in, 259–60 and theory of probability, 207, 208, 233 patterns in falling squares, 98–100 finding unexpected patterns, 42–45, 258–61 found in Pascal's triangle, 224–28, 259–60, 297n2 making mistaken generalizations based on, 66–69 in nonterminating decimals, 35–36, 83, 86–87 in palindromic numbers, 50–52 in perfect numbers, 65 in perfect squares, 32–34 for primitive Pythagorean triples, 108–10 in triangular numbers, 61–62 Pearson, Karl, and random walks, 228–32 pentagonal numbers found in Pascal's triangle, 228 pentagons and the golden ratio, 184–86 pentagrams, 194–99 as faces of a stellated dodecahedron, 203 percentages, problems involving, 251–53 perfect numbers, 62–65 end in 6 or 28, 65 Euclid's method of generating and table of, 63–65 partial sum of the series, 65 perfect squares, 26, 32, 33, 34, 89, 90, 99 perimeter equaling area in some Heronian triangles, 131 periodic table of elements, 37–38 pi (π), 175, 275 decimal places computed to, 36 as an irrational number, 35, 80, 172 as a nonterminating decimal, 34, 35–36, 80 Pick's theorem, 135–37 pineapples and pinecones, number of spirals on, 71 Pipping, N., 256 Platonic solids and star polyhedra, 200–204 plus symbol (+), 240 Poinsot, Louis, and Kepler-Poinsot solids, 203 points, 186 boundary points in a polygon, 136 and circles, 186–91
focal points in Apollonian circle, 189, 190, 191 Gergonne point, 177 in a lattice polygon, 136–37 in a line, 186, 192 Miquel point, 179 in a parabola, 279–81 in a plane, 191 points of tangency, 177, 187, 188 power of the point, 187–88 in a star, 195, 204 and triangles, 9 poker game hierarchy of winning hands, 233–34 changing when wild cards are in game, 236–37 method to calculate possible hands, 295–97n2 wild-card paradox, 233–37 Polignac, Alphonse de, 67 Polk, James K., having same birthday as Warren Harding, 213 polygons decomposing into triangles, 131 determining if it is convex, 195–96 extending Pythagorean theorem to polygons placed on the legs of a right triangle, 153–56 as faces of Platonic solids, 201 finding areas in (Pick's theorem), 135–37 lattice polygon, 136–37 regular polygons, 194 (see also hexagons; squares [geometric shape]; triangles) pentagram as, 194–99 polyhedra, 168–72 distorting polyhedron to a plane, 170–72 Platonic solids and star polyhedra, 200–204 polyline star, 194–99 polynomials Horner's method to evaluate, 102–105 second-degree polynomials and Descartes's rule of signs, 100–101 pons asinorum, 147, 294n3 positive integers and Benford's law, 210–11 and Champernowne's constant, 36–37 counting the uncountable, 266 finding positive Fibonacci numbers, 74 and Frobenius problem, 111 and greatest common divisor (gcd), 28–29 and irrationality of square root of 2, 80–81, 82 and Pythagorean triples, 106, 107 sums of odd positive integers, 32–34 of positive integers, 30–31 sums of positive integers, 30–34 sums of odd positive integers, 32–34 Potenz des Puncts [power of the point] (Steiner), 187–88
power line, 188 power of the point, 187–88 powers of 2 odd numbers and sum of power of 2 and a prime number, 66–67 in Pascal's triangle, 226 of 9 and triangular numbers, 61 equalities found by taking certain numbers to powers of 1, 2, 3, 4, 5, 6, and 7, 67–68 first four powers of 11 are palindromic numbers, 48 sum of digits in a number each taken to the same power, 42–43 sums of digits raised to consecutive powers, 44 and triangular numbers, 61 See also square numbers prime numbers, 38, 52–55, 293n4 determining if number is a prime number or composite, 39–40 Fermat's method of factoring, 89–91 “square root test,” 89, 91 divisibility by, 17–21 Euclid's use of to find perfect numbers, 63 Fibonacci numbers as relatively prime, 72 finding prime factors in friendly numbers, 46–47 Goldbach's conjecture on even numbers greater than 2, 255–56 on odd numbers greater than 5, 257 infinite numbers of, 39, 55 odd numbers and sum of power of 2 and a prime number, 66–67 and palindromic numbers, 53–54 prime decomposition method, 28, 29, 38, 39–40 prime factorization, 40–42, 46 relatively prime numbers, 21, 72, 108, 110, 199 used in Frobenius problem, 111 reversible prime numbers, 53 sum-of-consecutive-squares primes, 54 twin primes, 54 probability theory, 207–37 Benford's law and patterns of distribution, 209–12 Bertrand's box and finding gold coins, 218–21 Bayes's rule applied to, 220–21 birthday phenomenon, 212–15 conditional probabilities, 223 determining odds, 210 early history of, 207–208 expected value, 230 false positive paradox, 221–24 Monty Hall problem, 215–17 Pascal's triangle, 224–28 poker wild-card paradox, 233–37 method to calculate possible hands, 233–37, 295–97n2 random walks, 228–32 problems and problem solving
basketball tournament elimination problem, 244–46 birthday phenomenon, 212–15 Charlie, the bicycle rider problem, 276 counting number of triangles in a diagram, 247–49 different ways to look at a problem, 244–46 efforts to confirm Goldbach's conjectures, 255–58 evaluating discounts problem, 251–52 extremes used in problem solving, 246–47 Frobenius problem, 110–12 Grand Hotel paradox, 264 hens laying eggs problem, 249–50 Monty Hall problem, 215–17 organized thinking used in problem solving, 247–50 with percentages, 251–53 reconstruction used in problem solving, 247–50 rule of 72, 253–55 toothpick arrangement problem, 242–43 wine bottles problem, 246–47 See also counterintuitive situations in mathematics proof by contradiction, 80 Ptolemy, 141 pyramid and Euler formula for counting sides, faces, and vertices of polyhedra, 169 Pythagoras, 147, 150 figures showing his proof of the Pythagorean theorem, 151–52 Pythagorean Proposition, The (Loomis), 151 Pythagoreans on irrationality of the square root of 2, 80 Pythagorean theorem, 56, 113 applied to Hoehn's formula for isosceles triangles, 131–34 applied to lunes and the right triangle, 172–75 applied to points and circles, 187–88 extending Pythagorean theorem into three dimensions, 166–68 two-dimensional plane used to develop three-dimensional figures, 163–66 and Fibonacci numbers, 71–72 finding Pythagorean triples that satisfy the equation, 105–10 impact of changing right angle to an angle of 60°, 159–62 proofs of, 131 extended to polygons placed on legs of a right triangle, 153–59 figures showing proof by Pythagoras, 151–52 James Garfield's, 147, 148–49 numbers of, 147, 151, 153 shortest proof, 152–53 Xuan Tu diagram, 149–50 Pythagorean Theorem, The: The Story of Its Power and Beauty (Posamentier), 110, 168 Pythagorean triples, 71–72, 105–10 primitive Pythagorean triples, 108–10 quadratic equation, 186 Descartes's rule of signs, 100–101 and falling squares, 98–100 quadrilaterals
extending Heron's formula to area of cyclic quadrilaterals, 126 finding centroid and centerpoint of, 121–24 See also parallelograms; rectangles; squares (geometric shape) quantum computers, 42 radical axis, 188, 189, 190 radix [root] as origin of square root symbol, 240 Rahn, Johann, 241 random walks, 228–32 Raphson, Joseph, 25 rational numbers, 80 countability of, 268, 270 in Heronian triangles, 128 ratios of consecutive Fibonacci numbers, 69–71 gearing ratios in bicycles, 273, 276, 278 intercept theorem, 137–38 real numbers, 186, 231, 266, 273 as uncountable, 265, 270–71, 272 reciprocals and harmonic means, 92 reconstruction, use of in problem solving, 247–49 Recorde, Robert, 240, 241 rectangles and the golden ratio, 182–84 rectangular solid and the Pythagorean theorem, 166–68 See also quadrilaterals reflection, law of and the parabola, 283–84 regular polygons. See polygons relatively prime numbers, 21, 72, 108, 110, 199 used in Frobenius problem, 111 repunits, 50–52 reversible prime numbers, 53 rhombus and concurrency, 180–81 Richstein, Jörg, 256 right triangles area of, 127–31 Heronian right triangles, 127, 131 and isosceles triangles, 132, 133 and lunes, 172–75 and parallelograms, 114–15 and proof of the Pythagorean theorem, 151 shortest proof, 152–53 two-dimensional plane used to develop three-dimensional figures, 163–66 use of congruent right triangles as part of Garfield's proof, 148–49 in proofs going beyond the Pythagorean theorem, 154–55, 157, 158, 159, 163, 164 in solid geometry, 166, 167 and Pythagorean triples, 105–106 and square inscribed in a circle, 116, 117 in trigonometry, 143, 146 Rudolff, Christoff, 240
rule of 72, 253–55 rule of signs, 100–101 satellite dish. See parabola Schreyber, Heinrich (Henricus Grammateus), 240 semicircles, area of, 173–75 sets, comparing sizes of different sets even if infinite, 266–73 Shor, Peter Williston, 42 sides of polyhedra, counting, 168–72 similarity and the golden ratio, 182–86 sines definition of evolving from right triangle trigonometry, 143 law of sines, 146 sine function not requiring right-angled triangles, 145–46 of small angles, 142–44 solid geometry Kepler-Poinsot solids, 203 Platonic solids and star polyhedra, 200–204 and the Pythagorean theorem, 163–66 relationships of sides, faces, and vertices of polyhedra, 168–72 sphere and cylinder, 192–94 stellated dodecahedron, 203–204 sphere and cylinder, 192–94 spirals, number of found on pineapples and pinecones, 71 sprocket wheels. See gears and ratios on a bicycle square numbers, 56 Descartes's rule of signs, 100–101 falling squares, 98–100 and Fibonacci numbers, 73, 74, 75 and palindromic numbers squared, 50–51 perfect squares, 26, 33, 34, 89, 90, 99 prime numbers as sum of two consecutive squares, 54 squaring numbers quickly, 21–23 numbers with a last digit of 5, 22 and sums, 23–24 and triangular numbers, 59 (see also triangular numbers) using squares to multiply arbitrary numbers, 24–25 See also powers square root operations bisection method to approximate, 83–86 continued fractions of square roots, 86–89 extracting a square root, 25–26 of natural numbers and irrational numbers, 35 square root of 2 ( ), 35, 36 bisection method to approximate and continued fractions, 86–89 irrationality of, 80–82 “square root test,” 89–91 squares of natural numbers and irrational numbers, 35 symbol for square root (√), 240
squares (geometric shape) as an example of a regular polygon, 194 inscribed in circles, 116–17 square inscribed in another square and proof of the Pythagorean theorem, 151–52 See also quadrilaterals “staircase” for determining sums of positive integers, 30–31 stamps, recombinations of (a two-variable equation), 95–96 stars creating pentagram and other versions, 194–99 Platonic solids and star polyhedra, 200–204 Steiner, Jacob, 187 stella octangula (“eight-pointed star”), 201–202 stellated dodecahedron, 203–204 stock markets and random walks, 229 Stoicheia (Euclid). See Elements [Stoicheia] (Euclid) subtraction difference of squares and Fibonacci numbers, 73, 74 use of minus symbol (–), 240 sums. See addition Sylvester, James Joseph, 111 symbols, mathematical, 239–41, 264, 272 Taft, William H., having same death date as Millard Fillmore, 214 tangents, point of tangency, 177, 187, 188 Tennenbaum, Stanley, 81 tetrahedron, 200 and Euler formula for counting sides, faces, and vertices of polyhedra, 169–70 and making of a stella octangula, 202 and the Pythagorean theorem, 166–68 Teutsche Algebra (Rahn), 241 theory of probability. See probability theory three-dimensional space extending Pythagorean theorem into, 166–68 two-dimensional plane used to develop three-dimensional figures, 163–66 See also solid geometry Tierney, John, 217 toothpick arrangement problem, 242–43 topology, 170–72 trapezoids, isosceles, diagonal of, 134 triangles area of, 124, 131–34 Heron's formula for, 114–15, 124–31 using triangles to find area of other shapes, 114–16 centroid of a triangle, 121 and concurrency Ceva's theorem, 176–77, 289–91 extending concept from triangles to circles, 177–81 counting number of triangles in a diagram, 247–49 Heronian triangles, 127–31 impact on Pythagorean theorem of changing right angle to an angle of 60°, 159–62
Pascal's triangle, 224–28, 233, 297n2 unexpected pattern in, 259–60 and points, 9 Gergonne point, 177 Miquel point, 179 polygons composed of decomposed triangles, 131 See also equilateral triangles; isosceles triangles; right triangles triangular numbers, 56–62, 65 equilateral triangles and definition of, 56 found in Pascal's triangle, 227 and fourth power of an integer, 61 as product of three consecutive numbers, 60 as sum of first n consecutive natural numbers, 57–58 that are palindromic, 59 trigonometry origins of, 140–42 unit circle trigonometry, 142–43 See also chords; cosines; sines; tangents twin primes, 54 uncountable, real numbers as, 270 unit circle trigonometry, 142–43 unsolved problems. See Goldbach, Christian, and Goldbach's conjectures vertices of a pentagon, 196–98 of polyhedra, 168–72 of a stellated dodecahedron, 203 vos Savant, Marilyn, 217 walk, random, 228–32 Wallis, John, 264 Whetstone of Witte, The (Recorde), 240 Widmann, Johannes, 240 wild-card paradox, 233–37 method to calculate possible hands, 295–97n2 wine bottles problem, 246–47 Xuan Tu and diagram that proves Pythagorean theorem, 149–50 Zeno's paradox, 264 zero (0) division by zero, 78–79 first person to compute, 126
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