Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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1. Card Trick
1. Card Trick
Pages 3-18
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From page 3... ...
Take an ordinary deck of 52 cards, lying on a table, all four sides of the deck squared away. Now, with a finger slide the topmost card forward without moving any of the others.
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From page 4... ...
Well, it's halfway along the unit, which is altogether one and a half cards long; so it's three-quarters of a card length from the leading edge of the top card (see Figure 1-2~. The combined overhang is, therefore, three-quarters of a card length.
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From page 5... ...
I'm not going to (lo the arithmetic; you can trust me, or work it out as I slid for the first three cards. Total overhang with four cards: one-half plus one-quarter plus one-sixth plus one-eighth, altogether one and onetwenty-fourth card lengths (see Figure 1-5~.
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From page 6... ...
To help pay my way through college I used to spend summer vacations as a laborer on construction sites, work that was not heavily unionize(1 at the time in England. The (lay after I found out about this thing with the cards I was left on my own to do some clean-up work in an indoor area where hundreds of large, square, fibrous ceiling tiles were stacked.
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From page 7... ...
With 52 cards the total overhang was 1 1 1 1 1 1 1 1 1 _+_+_+_+ + + + +~+2 4 6 8 10 12 14 16 102 Since all the denominators are even, I can take out one-half as a factor and rewrite this as 1 1 1 1 1 1 1 1 1 _ 1+-+-+-+_+_+_+_+~+ 2 2 3 4 5 6 7 8 51 If there were a hundred cards, the total overhang would be 16 1 1 1 1 1 1 1 1 _ 1+-+-+-+_+_+_+_+~+ 2 ~ 2 3 4 5 6 7 8 99 With a trillion cards it would be 16 1 1 1 1 1 1 1 _ 1+-+-+-+-+-+_+_+. 2 2 3 4 5 6 7 8 999999999999 J That's a lot of arithmetic; but mathematicians have shortcuts for this kind of thing, and I can tell you with confidence that the total overhang with a hundred cards is a tad less than 2.58868875882, while for a trillion cards it is a wee bit more than 14.10411839041479.
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From page 8... ...
Fourteen card lengths is more than four feet, with standard playing cards. The second surprise, when you start thinking about it, is that the numbers aren't bigger.
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From page 9... ...
Leonhar(1 Euler, one of the half-(lozen greatest mathematicians who ever lived, used them all the time with very fruitful results. However, the proper mathematical term of art is: The harmonic series is divergent.
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From page 10... ...
Neither Mengoli nor the Bernoullis seem to have been aware of d'Oresme's fourteenth-century proof, one of the barely known masterpieces of medieval mathematics. D'Oresme's proof remains the most straightforward and elegant of all the proofs, though, and is the one usually given in textbooks today.
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From page 11... ...
. and you have moved to the right a total distance of 1 1 1 1 1 1 1+-+-+-+ + + 2 4 8 16 32 64 which is, as you can see, 163.
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From page 12... ...
The harmonic series has its charms, and it stands at the center of the topic this book addresses the Riemann Hypothesis. Generally speaking, however, mathematicians are more interested in convergent series than divergent ones.
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From page 13... ...
Now, suppose that instead of starting out with a ruler marked in halves, quarters, eighths, sixteenths, and so on, I have a ruler marked in thirds, ninths, twenty-sevenths, eighty-firsts, and so on. In other words, instead of halves, halves of halves, halves of halves of halves .
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From page 14... ...
That is, 1+ - + - +—+—+—+ + +~. = 13 9 27 81 243 729 Expression 1-3 2187 2 And of course, I can do the alternating movement with this new ruler, too: right one inch, left a third, right a ninth, left a twenty-seventh, and so on (see Figure 1-12~.
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From page 15... ...
When Leonhar(1 Euler of whom I shall write much more later published the first great textbook of analysis in 1748, he called it Introductio in anal/ysin infinitorum: "Introduction to the Analysis of the Infinite." The notions of the infinite and the infinitesimal created serious problems in math during the early nineteenth century, though, and eventually they were swept away altogether in a great reform. Modern analysis does not admit these concepts.
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From page 16... ...
Consi(ler the following sequence of numbers: i, 3' 57' i2' 249, 979' i69, 508 ~ i938953 ~ 236s, .... Each fraction is built from the one before by a simple rule: add top and bottom to get new bottom, add top and twice bottom to get new top.
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From page 17... ...
· Algebra The use of abstract symbols to represent mathematical objects (numbers, lines, matrices, transformations) , and the stu(ly of the rules for combining those symbols.
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From page 18... ...
It includes set theory, for example, created by Georg Cantor in 1874, an(1 "foundations," which another George, the Englishman George Boole, split off from classical logic in 1854, and in which the logical underpinnings of all mathematical ideas are studied. The traditional categories have also been enlarged to include big new tonics ~eom~7 ~7 1 ~7 etry to inclu(le topology, algebra to take in game theory, an(1 so on.
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