The Joy Of X

A Guided Tour of Math, from One to Infinity

Steven Strogatz

• Publisher: Mariner Books
• Year: 2012
• ISBN:
  9780547517667
• Topic: Matemáticas
• Subtopic: Álgebra

"Delightful... easily digestible chapters include plenty of helpful examples and illustrations. You'll never forget the Pythagorean theorem again!" — Scientific American Many people take math in high school and promptly forget much of it. But math plays a part in all of our lives all of the time, whether we know it or not. In The Joy of x, Steven Strogatz expands on his hit New York Times series to explain the big ideas of math gently and clearly, with wit, insight, and brilliant illustrations. Whether he is illuminating how often you should flip your mattress to get the maximum lifespan from it, explaining just how Google searches the internet, or determining how many people you should date before settling down, Strogatz shows how math connects to every aspect of life. Discussing pop culture, medicine, law, philosophy, art, and business, Strogatz is the math teacher you wish you'd had. Whether you aced integral calculus or aren't sure what an integer is, you'll find profound wisdom and persistent delight in The Joy of x.

Notes

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1. From Fish to Infinity

[>] Sesame Street: The video Sesame Street: 123 Count with Me (1997) is available for purchase online in either VHS or DVD format.

[>] numbers . . . have lives of their own: For a passionate presentation of the idea that numbers have lives of their own and the notion that mathematics can be viewed as a form of art, see P. Lockhart, A Mathematician’s Lament (Bellevue Literary Press, 2009).

[>] “the unreasonable effectiveness of mathematics”: The essay that introduced this now-famous phrase is E. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences,” Communications in Pure and Applied Mathematics, Vol. 13, No. 1 (February 1960), pp. 1–14. An online version is available at http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html.

For further reflections on these ideas and on the related question of whether math was invented or discovered, see M. Livio, Is God a Mathematician? (Simon and Schuster, 2009), and R. W. Hamming, “The unreasonable effectiveness of mathematics,” American Mathematical Monthly, Vol. 87, No. 2 (February 1980), available online at http://www-lmmb.ncifcrf.gov/~toms/Hamming.unreasonable.html.

2. Rock Groups

[>] The playful side of arithmetic: As I hope to make clear, this chapter owes much to two books—one a polemic, the other a novel, both of them brilliant: P. Lockhart, A Mathematician’s Lament (Bellevue Literary Press, 2009), which inspired the rock metaphor and some of the examples used here; and Y. Ogawa, The Housekeeper and the Professor (Picador, 2009).

[>] a child’s curiosity: For young readers who like exploring numbers and the patterns they make, see H. M. Enzensberger, The Number Devil (Holt Paperbacks, 2000).

[>] hallmark of an elegant proof: Delightful but more advanced examples of visualization in mathematics are presented in R. B. Nelsen, Proofs without Words (Mathematical Association of America, 1997).

3. The Enemy of My Enemy

[>] “Yeah, yeah”: For more of Sidney Morgenbesser’s witticisms and academic one-liners, see the sampling at Language Log (August 5, 2004), “If P, so why not Q?” online at http://itre.cis.upenn.edu/%7Emyl/languagelog/archives/001314.html.

[>] relationship triangles: Balance theory was first proposed by the social psychologist Fritz Heider and has since been developed and applied by social network theorists, political scientists, anthropologists, mathematicians, and physicists. For the original formulation, see F. Heider, “Attitudes and cognitive organization,” Journal of Psychology, Vol. 21 (1946), pp. 107–112, and F. Heider, The Psychology of Interpersonal Relations (John Wiley and Sons, 1958). For a review of balance theory from a social network perspective, see S. Wasserman and K. Faust, Social Network Analysis (Cambridge University Press, 1994), chapter 6.

[>] polarized states are the only states as stable as nirvana: The theorem that a balanced state in a fully connected network must be either a single nirvana of all friends or two mutually antagonistic factions was first proven in D. Cartwright and F. Harary, “Structural balance: A generalization of Heider’s theory,” Psychological Review, Vol. 63 (1956), pp. 277–293. A very readable version of that proof, and a gentle introduction to the mathematics of balance theory, has been given by two of my colleagues at Cornell: D. Easley and J. Kleinberg, Networks, Crowds, and Markets (Cambridge University Press, 2010).

In much of the early work on balance theory, a triangle of three mutual enemies (and hence three negative sides) was considered unbalanced. I assumed this implicitly when quoting the results about nirvana and the two-bloc state being the only configurations of a fully connected network in which all triangles are balanced. However, some researchers have challenged this assumption and have explored the implications of treating a triangle of three negatives as balanced. For more on this and other generalizations of balance theory, see the books by Wasserman and Faust and by Easley and Kleinberg cited above.

[>] World War I: The example and graphical depiction of the shifting alliances before World War I are from T. Antal, P. L. Krapivsky, and S. Redner, “Social balance on networks: The dynamics of friendship and enmity,” Physica D, Vol. 224 (2006), pp. 130–136, available online at http://arxiv.org/abs/physics/0605183. This paper, written by three statistical physicists, is notable for recasting balance theory in a dynamic framework, thus extending it beyond the earlier static approaches. For the historical details of the European alliances, see W. L. Langer, European Alliances and Alignments, 1871–1890, 2nd edition (Knopf, 1956), and B. E. Schmitt, Triple Alliance and Triple Entente (Henry Holt and Company, 1934).

4. Commuting

[>] revisit multiplication from scratch: Keith Devlin has written a provocative series of essays about the nature of multiplication: what it is, what it is not, and why certain ways of thinking about it are more valuable and reliable than others. He argues in favor of thinking of multiplication as scaling, not repeated addition, and shows that the two concepts are very different in real-world settings where units are involved. See his January 2011 blog post “What exactly is multiplication?” at http://www.maa.org/devlin/devlin_01_11.html, as well as three earlier posts from 2008: “It ain’t no repeated addition” (http://www.maa.org/devlin/devlin_06_08.html); “It’s still not repeated addition” (http://www.maa.org/devlin/devlin_0708_08.html); and “Multiplication and those pesky British spellings” (http://www.maa.org/devlin/devlin_09_08.html). These essays generated a lot of discussion in the blogosphere, especially among schoolteachers. If you’re short on time, I’d recommend reading the one from 2011 first.

[>] shopping for a new pair of jeans: For the jeans example, the order in which the tax and discount are applied may not matter to you—in both scenarios you end up paying $43.20—but it makes a big difference to the government and the store! In the clerk’s scenario (where you pay tax based on the original price), you would pay $4 in tax; in your scenario, only $3.20. So how can the final price come out the same? It’s because in the clerk’s scenario the store gets to keep $39.20, whereas in yours it would keep $40. I’m not sure what the law requires, and it may vary from place to place, but the rational thing would be for the government to charge sales tax based on the actual payment the store receives. Only your scenario satisfies this criterion. For further discussion, see http://www.facebook.com/TeachersofMathematics/posts/166897663338316.

[>] financial decisions: For heated online arguments about the relative merits of a Roth 401(k) versus a traditional one, and whether the commutative law has anything to do with these issues, see the Finance Buff, “Commutative law of multiplication” (http://thefinancebuff.com/commutative-law-of-multiplication.html), and the Simple Dollar, “The new Roth 401(k) versus the traditional 401(k): Which is the better route?” (http://www.thesimpledollar.com/2007/06/20/the-new-roth-401k-versus-the-traditional-401k-which-is-the-better-route/).

[>] attending MIT and killing himself didn’t commute: This story about Murray Gell-Mann is recounted in G. Johnson, Strange Beauty (Knopf, 1999), p. 55. In Gell-Mann’s own words, he was offered admission to the “dreaded” Massachusetts Institute of Technology at the same time as he was “contemplating suicide, as befits someone rejected from the Ivy League. It occurred to me however (and it is an interesting example of non-commutation of operators) that I could try M.I.T. first and kill myself later, while the reverse order of events was impossible.” This excerpt appears in H. Fritzsch, Murray Gell-Mann: Selected Papers (World Scientific, 2009), p. 298.

[>] development of quantum mechanics: For an account of how Heisenberg and Dirac discovered the role of non-commuting variables in quantum mechanics, see G. Farmelo, The Strangest Man (Basic Books, 2009), pp. 85–87.

5. Division and Its Discontents

[>] My Left Foot: A clip of the scene where young Christy struggles valiantly to answer the question “What’s twenty-five percent of a quarter?” is available online at http://www.tcm.com/mediaroom/video/223343/My-Left-Foot-Movie-Clip-25-Percent-of-a-Quarter.html.

[>] Verizon Wireless: George Vaccaro’s blog (http://verizonmath.blogspot.com/) provides the exasperating details of his encounters with Verizon. The transcript of his conversation with customer service is available at http://verizonmath.blogspot.com/2006/12/transcription-jt.html. The audio recording is at http://imgs.xkcd.com/verizon_billing.mp3.

[>] you’re forced to conclude that 1 must equal .9999 . . . : For readers who may still find it hard to accept that 1 = .9999 . . . , the argument that eventually convinced me was this: they must be equal, because there’s no room for any other decimal to fit between them. (Whereas if two decimals are unequal, their average is between them, as are infinitely many other decimals.)

[>] almost all decimals are irrational: The amazing properties of irrational numbers are discussed at a higher mathematical level on the MathWorld page “Irrational Number,” http://mathworld.wolfram.com/IrrationalNumber.html. The sense in which the digits of irrational numbers are random is clarified at http://mathworld.wolfram.com/NormalNumber.html.

6. Location, Location, Location

[>] Ezra Cornell’s statue: For more about Cornell, including his role in Western Union and the early days of the telegraph, see P. Dorf, The Builder: A Biography of Ezra Cornell (Macmillan, 1952); W. P. Marshall, Ezra Cornell (Kessinger Publishing, 2006); and http://rmc.library.cornell.edu/ezra/index.html, an online exhibition in honor of Cornell’s 200th birthday.

[>] systems for writing numbers: Ancient number systems and the origins of the decimal place-value system are discussed in V. J. Katz, A History of Mathematics, 2nd edition (Addison Wesley Longman, 1998), and in C. B. Boyer and U. C. Merzbach, A History of Mathematics, 3rd edition (Wiley, 2011). For a chattier account, see C. Seife, Zero (Viking, 2000), chapter 1.

[>] Roman numerals: Mark Chu-Carroll clarifies some of the peculiar features of Roman numerals and arithmetic in this blog post: http://scienceblogs.com/goodmath/2006/08/roman_numerals_and_arithmetic.php.

[>] Babylonians: A fascinating exhibition of Babylonian math is described by N. Wade, “An exhibition that gets to the (square) root of Sumerian math,” New York Times (November 22, 2010), online at http://www.nytimes.com/2010/11/23/science/23babylon.html, accompanied by a slide show at http://www.nytimes.com/slideshow/2010/11/18/science/20101123-babylon.html.

[>] nothing to do with human appendages: This may well be an overstatement. You can count to twelve on one hand by using your thumb to indicate each of the three little finger bones (phalanges) on the other four fingers. Then you can use all five fingers on your other hand to keep track of how many sets of twelve you’ve counted. The base 60 system used by the Sumerians may have originated in this way. For more on this hypothesis and other speculations about the origins of the base 60 system, see G. Ifrah, The Universal History of Numbers (Wiley, 2000), chapter 9.

7. The Joy of ...