One to Nine
The Inner Life of NumbersBook - 2008
From the critics
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Hopeless to select best quotes because of the vast variety of interests. Here is a small sample:
Indian mathematicians have a self-deprecating joke: ‘India contributed zero to mathematics.’ The joke plays on the unnatural language of zero. Contributing ‘zero’ is not the same as contributing zero: ‘zero’ is something! This joke is also the basis of a simpler approach to the logic of numbers. Take Zero to be the empty set, the set containing nothing. Then One is the set containing just one thing: namely Zero. Then, Zero and One give two things…
That NOT (NOT A) is the same as A also requires this peculiar precision. Alan Turing suggested explaining it by saying, ‘It’s like crossing the road. You cross it, and then you cross it again, and you’re back where you started.’ But in real life, we often can’t get no such logical satisfaction. Two wrongs make no rights, ...
Tie a pair of shoes to each other with (suitably long) shoe-laces. Now rotate one shoe completely round, so that the laces get twisted too. Rotate again, in the same direction, so the laces are doubly twisted. You will find that you can untwist the laces by passing the shoe between them, without rotating it at all. Only a double rotation can be undone like this.
Explain why a mirror exchanges right and left, but not up and down.
It is a weak but also a profoundly weird force, with an extraordinary property: it is asymmetric. The weak force knows left from right just as definitely as does a helix of DNA. This was only shown unequivocally by Chien-Shiung Wu in 1956. Her experiment, though it has never become a popular icon of science, marks a major step in the understanding of space and its Two-ness.
The first rule is unsurprising, but the second is highly unobvious: (a, b) + (c, d) = (a + c, b + d) (a, b) × (c, d) = (a × c − b × d, a × d + b × c). The natural numbers 1, 2, 3 … can be identified with the pairs (1, 0), (2, 0), (3, 0) … and for these, the new addition and multiplication coincides with what we had before. The same goes for the fractions and decimals: 1/2 is (1/2, 0) and π is (π, 0). It goes for negative numbers too: so (−1, 0) is the same as −1. But something new happens when the second element in the pair is used. Following the rules, (0, 1) × (0, 1) = (−1, 0). The pair (0, 1) behaves as a square root of minus one.
The K for a thousand, the third power of ten, is convenient because it is close to 1024, the tenth power of two. Less obvious is that the approximation 1024 ≈ 1000 is related to the problem of music, and this gives a starting point for both the harmonies and the cussedness of numbers.
In practice, people do co-operate in many ways, despite the lack of individual benefit. ... But co-operation is fragile, as gun control and gang revenge questions indicate.
The same applies to another famous problem Poincaré attacked, also featuring the number Three. This is the three-body uproblem.
… it is difficult to recover the excitement of the seventeenth century, when the solar system became the test-bed for mathematical prediction. The periodic return of Halley’s comet, whose successful prediction was one great early achievement, is a reminder of that original drama of the skies.
Electric force becomes stronger and stronger as two charges approach. Colour force is the other way round. It acts like a spring, becoming stronger as two colour charges separate, and diminishing to nothing as they approach.
It brings together the most abstract mathematics and the most sophisticated high-energy experiments. Fortunately, mathematics comes particularly cheap.
Everyone knows that two and two make four. Not everyone sees that two and two make four in several different ways: 2 + 2 = 4, 2 × 2 = 4, 2^2 = 4. In short, four is the square of two. Musically, a beat of four, in two twos, gives common time. This rhythm is the first to have an internal structure, a double duality, a micro-drama to every bar. Hidden under the surface, this beat will be found in the numbers. Squares are special. The periodic table of the chemical elements shows that the elements are built up from certain magic numbers: 2, 8, 18, 32. These numbers all come from squares: 2 × 1, 2 × 4, 2 × 9, 2 × 16, although the usual layout of the table fails to make this plain.
It is common to applaud ‘win-win situations’ which arise from games which are not zero-sum. Unfortunately, turning now from optimism to pessimism, they can also be lose-lose situations. If you are a billionaire, you can probably make the rules to suit yourself, but those on the receiving end may be faced with less convenient constraints. The classic example is that of ‘prisoner’s dilemma’, a situation which could well be imagined in a Caribbean setting. Suppose two captured prisoners are interrogated separately. Both have a choice: to inform against the other, or not. They are both best off if both refuse to inform. But the direst outcome for a prisoner is when, having refused to inform, he finds himself incriminated by the other. The only way to avoid this worst scenario is to inform; the same logic applies to both and so both will incriminate each other. Neither can choose the strategy that would be to their collective advantage.
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Notes, Updates, Links and Answers to Problems ... All the answers to the problems are on these pages (see bottom hyperlinks:)
An example which is not too hard to understand/follow:
75025 and 46368 give an example of where the Euclidean algorithm makes the slowest progress, because they are Fibonacci numbers. The quotients are always 1 so the reduction is very slow.
75025 = 1 × 46368 + 28657 reducing to (46368, 28657)
46368 = 1 × 28657 + 17711 reducing to (28657, 17711)
28657 = 1 × 17711 + 10946 reducing to (17711, 10946)
...and so on through all the Fibonacci numbers to
13 = 1 × 8 + 5 reducing to (8, 5)
8 = 1 × 5 + 3 reducing to (5, 3)
5 = 1 × 3 + 2 reducing to (3, 2)
3 = 1 × 2 + 1 reducing to (2, 1).
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