The Number Mysteries: A Mathematical Odyssey through Everyday Life. Marcus Sautoy du
Чтение книги онлайн.
Читать онлайн книгу The Number Mysteries: A Mathematical Odyssey through Everyday Life - Marcus Sautoy du страница 7
Название: The Number Mysteries: A Mathematical Odyssey through Everyday Life
Автор: Marcus Sautoy du
Издательство: HarperCollins
Жанр: Математика
isbn: 9780007362561
isbn:
Intriguingly, there are primes hidden behind these perfect numbers. Each perfect number corresponds to a special sort of prime number called a Mersenne prime (more of which later in the chapter). To date, we know only 47 perfect numbers. The biggest has 25,956,377 digits. Perfect numbers which are even are always of the form 2n–1(2n–1). And whenever 2n–1(2n–1) is perfect, then 2n–1 will be a prime number, and conversely. We don’t yet know whether there can be odd perfect numbers.
Which prime is this?
FIGURE 1.15
You might think that this is the prime number 5; it certainly looks like 2+3. However, the
This traditional Chinese form of writing numbers did not use a place-value system, but instead had symbols for the different powers of 10. An alternative system of representing numbers by bamboo sticks did use a place-value system and evolved from the abacus, on which when you reached ten you would start a new column.
Here are the numbers from 1 to 9 in bamboo sticks:
FIGURE 1.16
To avoid confusion, in every other column (namely the 10s, 1000s, 100,000s, …) they turned the numbers round and laid the bamboo sticks vertically:
FIGURE 1.17
The Ancient Chinese even had a concept of negative number, which they represented by different-coloured bamboo sticks. The use of black and red ink in Western accounting is thought to have originated from the Chinese practice of using red and black sticks, although intriguingly the Chinese used black sticks for negative numbers.
The Chinese were probably one of the first cultures to single out the primes as important numbers. They believed that each number had its own gender—even numbers were female and odd numbers male. They realized that some odd numbers were rather special. For example, if you have 15 stones, there is a way to arrange them into a nice-looking rectangle, in three rows of five. But if you have 17 stones you can’t make a neat array: all you can do is line them up in a straight line. For the Chinese, the primes were therefore the really macho numbers. The odd numbers, which aren’t prime, though they were male, were somehow rather effeminate.
This Ancient Chinese perspective homed in on the essential property of being prime, because the number of stones in a pile is prime if there is no way to arrange them into a nice rectangle.
We’ve seen how the Egyptians used pictures of frogs to depict numbers, the Maya drew dots and dashes, the Babylonians made wedges in clay, the Chinese arranged sticks, and in Hebrew culture letters of the alphabet stood for numbers. Although the Chinese were probably the first to single out the primes as important numbers, it was another culture that made the first inroads into uncovering the mysteries of these enigmatic numbers: the Ancient Greeks.
How the Greeks used sieves to cook up the primes
Here’s a systematic way discovered by the Ancient Greeks which is very effective at finding small primes. The task is to find an efficient method that will knock out all the non-primes. Write down the numbers from 1 to 100. Start by striking out number 1. (As I have mentioned, though the Greeks believed 1 to be prime, in the twenty-first century we no longer consider it to be.) Move to the next number, 2. This is the first prime. Now strike out every second number after 2. This effectively knocks out everything in the 2 times table, eliminating all the even numbers except for 2. Mathematicians like to joke that 2 is the odd prime because it’s the only even prime … but perhaps humour isn’t a mathematician’s strong point.
FIGURE 1.18 Strike out every second number after 2.
Now take the lowest number which hasn’t been struck out, in this case 3, and systematically knock out everything in the 3 times table:
FIGURE 1.19 Now strike out every third number after 3.
Because 4 has already been knocked out, we move next to the number, 5, and strike out every fifth number on from 5. We keep repeating this process, going back to the lowest number n that hasn’t yet been eliminated, and then strike out all the numbers n places ahead of it:
FIGURE 1.20 Finally we are left with the primes from 1 to 100.
The beautiful thing about this process it that it is very mechanical—it doesn’t require much thought to implement. For example, is 91 a prime? With this method you don’t have to think. 91 would have been struck out when you knocked out every 7th number on from 7 because 91=7×13.91 often catches people out because we tend not to learn our 7 times table up to 13.
This systematic process is a good example of an algorithm, a method of solving a problem by applying a specified set of instructions—which is basically what a computer program is. This particular algorithm was discovered two millennia ago in one of the hotbeds of mathematical activity at the time: Alexandria, in present-day Egypt. Back then, Alexandria was one of the outposts of the great Greek empire and boasted one of the finest libraries in the world. It was during the third century BC that the librarian Eratosthenes came up with this early computer program for finding primes.
It is called the sieve of Eratosthenes, because each time you knock out a group of non-primes it is as if you are using a sieve, setting the gaps between the wires of the sieve according to each new prime you move on to. First you use a sieve where the wires are 2 apart. Then 3 apart. Then 5 apart. And so on. The only trouble is that the method soon becomes rather inefficient if you try to use it to find bigger and bigger primes.
As well as sieving for primes and looking after the hundreds of thousands of papyrus and vellum scrolls in the library, Eratosthenes also calculated the circumference of the Earth and the distance of the Earth to the Sun and the Moon. The Sun he calculated to be 804,000,000 stadia from the Earth—although his unit of measurement perhaps makes judging the accuracy a little difficult. What size stadium are we meant to use: Wembley, or something smaller, like Loftus Road?
In addition to measuring the solar system, Eratosthenes charted the course of the Nile and gave the first correct explanation for why it kept flooding: heavy rains at the river’s distant sources in Ethiopia. He even wrote poetry. Despite all this activity, his friends gave him the nickname Beta—because he never really excelled at anything. It is said that he starved himself to death after going blind in old age.
You can use your snakes and ladders board on the cover to put the Sieve of Eratosthenes into operation. Take a pile of pasta and place pieces on each of СКАЧАТЬ