The Number Mysteries: A Mathematical Odyssey through Everyday Life. Marcus Sautoy du

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СКАЧАТЬ the cicadas discovered, knowing your maths is the key to survival in this world. Any teacher who is having trouble motivating their mathematics class might find some of the gory deaths in Cube a great piece of propaganda for getting them to learn their primes.

      Why do science fiction writers like primes?

      When science fiction writers want to get their aliens to communicate with Earth, they have a problem. Do they assume that their aliens are really clever and have picked up the local language, or that they’ve invented some clever Babelfish-style translator that does the interpreting for them? Or do they just assume that everyone in the universe speaks English?

      One solution that a number of authors have gone for is that mathematics is the only truly universal language, and the first words that anyone should speak in this language are its building blocks—the primes. In Carl Sagan’s novel Contact, Ellie Arroway, who works for SETI, the Search for Extra-Terrestrial Intelligence, picks up a signal which she realizes is not just background noise but a series of pulses. She guesses that they are binary representations of numbers. As she converts them into decimal, she suddenly spots a pattern: 59, 61, 67, 71 … all prime numbers. Sure enough, as the signal continues, it cycles through all the primes up to 907. This can’t be random, she concludes. Someone is saying hello.

      Many mathematicians believe that even if there is a different biology, a different chemistry, even a different physics on the other side of the universe, the mathematics will be the same. Anyone sitting on a planet orbiting Vega reading a maths book about primes will still consider 59 and 61 to be prime numbers because, as the famous Cambridge mathematician G.H. Hardy put it, these numbers are prime ‘not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way’.

      The primes may be numbers that are shared across the universe, but it is still interesting to wonder whether stories similar to those I’ve related are being told on other worlds. The way we have studied these numbers over the millennia has led us to discover important truths about them. And at each step on the way to discovering these truths we can see the mark of a particular cultural perspective, the mathematical motifs of that period in history. Could other cultures across the universe have developed different perspectives, giving them access to theorems we have yet to discover?

      Carl Sagan wasn’t the first and won’t be the last to suggest using the primes as a way of communicating. Prime numbers have even been used by NASA in their attempts to make contact with extra-terrestrial intelligence. In 1974 the Arecibo radio telescope in Puerto Rico broadcast a message towards the globular star cluster M13, chosen for its huge number of stars so as to increase the chance that the message might fall on intelligent ears.

      The message consisted of a series of 0s and 1s which could be arranged to form a black and white pixelated picture. The reconstructed image depicted the numbers from 1 to 10 in binary, a sketch of the structure of DNA, a representation of our solar system and a picture of the Arecibo radio telescope itself. Considering that there were only 1,679 pixels, the picture is not very detailed. But the choice of 1,679 was deliberate because it contained the clue to setting out the pixels. 1,679=23×73, so there are only two ways to arrange the pixels in a rectangle to make up the picture. 23 rows of 73 columns produces a jumbled mess, but arrange them the other way as 73 rows of 23 columns and you get the result shown right. The star cluster M13 is 25,000 light years away, so we’re still waiting for a reply. Don’t expect a response for another 50,000 years!

      FIGURE 1.05 The message broadcast by the Arecibo telescope towards the star cluster M13.

      Although the primes are universal, the way we write them has varied greatly throughout the history of mathematics, and is very culture-specific—as our whistle-stop tour of the planet will now illustrate.

      Which prime is this?

      FIGURE 1.06

      Some of the first mathematics in history was done in Ancient Egypt, and this is how they wrote the number 200,201. As early as 6000BC people were abandoning nomadic life to settle along the river Nile. As Egyptian society became more sophisticated, the need grew for numbers to record taxes, measure land and construct pyramids. Just as for their language, the Egyptians used hieroglyphs to write numbers. They had already developed a number system based on powers of 10, like the decimal system we use today. (The choice comes not from any special mathematical significance of the number, but from the anatomical fact that we have ten fingers.) But they had yet to invent the place-value system, which is a way of writing numbers so that the position of each digit corresponds to the power of 10 that the digit is counting. For example, the 2s in 222 all have different values according to their different positions. Instead, the Egyptians needed to create new symbols for each new power of 10:

      FIGURE 1.07 Ancient Egyptian symbols for powers of 10. 10 is a stylized heel bone, 100 a coil of rope and 1,000 a lotus plant.

      200,201 can be written quite economically in this way, but just try writing the prime 9,999,991 in hieroglyphs: you would need 55 symbols. Although the Egyptians did not realize the importance of the primes, they did develop some sophisticated maths, including—not surprisingly—the formula for the volume of a pyramid and a concept of fractions. But their notation for numbers was not very sophisticated, unlike the one used by their neighbours, the Babylonians.

      Which prime is this?

      FIGURE 1.08

      This is how the Ancient Babylonians wrote the number 71. Like the Egyptian empire, the Babylonian empire was focused around a major river, the Euphrates. From 1800BC the Babylonians controlled much of modern Iraq, Iran and Syria. To expand and run their empire they became masters of managing and manipulating numbers. Records were kept on clay tablets, and scribes would use a wooden stick or stylus to make marks in the wet clay, which would then be dried. The tip of the stylus was wedge-shaped, or cuneiform—the name by which the Babylonian script is now known.

      Around 2000BC the Babylonians became one of the first cultures to use the idea of a place-value number system. But instead of using powers of 10 like the Egyptians, the Babylonians developed a number system which worked in base 60. They had different combinations of symbols for all the numbers from 1 to 59, and when they reached 60 they started a new ‘sixties’ column to the left and recorded one lot of 60, in the same way that in the decimal system we place a 1 in the ‘tens’ column when the units column passes 9. So the prime number shown above consists of one lot of 60 together with the symbol for 11, making 71. The symbols for the numbers up to 59 do have some hidden appeal to the decimal system because the numbers from 1 to 9 are represented by horizontal lines, but then 10 is represented by the symbol (Figure 1.09).

      FIGURE 1.09

      The choice of base 60 is much more mathematically justified than the decimal system. It is a highly divisible number which makes it very powerful for doing calculations. For example, if I have 60 beans, I can divide them up in a multitude of different ways:

      60=30×2=20×3=15×4=12×5=10×6

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