The Number Mysteries: A Mathematical Odyssey through Everyday Life (MacSci)

by du Sautoy, Marcus

  Figure 1.11

  But they didn’t think of 0 as a number in its own right. For them, it was just a symbol used in the place-value system to denote the absence of certain powers of 60. Mathematics would have to wait another 2,700 years until the seventh century AD, when the Indians introduced and investigated the properties of 0 as a number.

  As well as developing a sophisticated way of writing numbers, the Babylonians are responsible for discovering the first method of solving quadratic equations—something every child is now taught at school. They also had the first inklings of Pythagoras’s theorem about right-angled triangles. But there is no evidence that the Babylonians appreciated the beauty of prime numbers.

  How to Count to 60 with Your Hands

  We see many leftovers of the Babylonian base 60 today. There are 60 seconds in a minute, 60 minutes in an hour, 360 = 6 × 60 degrees in a circle. There is evidence that the Babylonians used their fingers to count to 60, in a quite sophisticated way.

  Each finger is made up of three bones. Not counting the thumb, there are four fingers on each hand, so with the other hand, you can point to any one of 12 different bones. The left hand is used to count to 12. The four fingers on the right hand are then used to keep track of how many lots of 12 you’ve counted. In total, you can count up to five lots of 12 (four lots of 12 on the right hand plus one lot of 12 counted on the left hand), so you can count up to 60.

  For example, to indicate the prime number 29, you need to point to two lots of 12 on the right hand and then up to the fifth bone on the left hand.

  Figure 1.12

  Which prime is this?

  Figure 1.13

  The Mesoamerican culture of the Maya was at its height from AD 200 to 900 and extended from southern Mexico through Guatemala to El Salvador. The Maya had a sophisticated number system developed to facilitate the advanced astronomical calculations that they made, and this is how they would have written the number 17. In contrast to the Egyptians and Babylonians, the Maya worked with a base–20 system. They used a dot for 1, two dots for 2, three dots for 3, and four dots for 4. Just like a prisoner chalking off days on a prison wall, once they got to 5, instead of writing five dots, they would simply put a line through the four dots. A line, therefore, corresponds to 5.

  It is interesting that the system works on the principle that our brains can quickly distinguish small quantities: we can tell the difference between one, two, three, and four things—but beyond that, it gets progressively harder. Once the Maya had counted to 19—three lines with four dots on top—they created a new column in which to count the number of 20s. The next column should have denoted the number of 400s (20 × 20), but bizarrely, it represents how many 360s (20 × 18) there are. This strange choice is connected with the cycles of the Mayan calendar. One cycle consists of 18 months of 20 days. (That’s only 360 days. To make the year 365 days in length, they added an extra month of five “bad days,” which were regarded as very unlucky.)

  Interestingly, like the Babylonians, the Maya used a special symbol to denote the absence of certain powers of 20. Each place in their number system was associated with a god, and it was thought disrespectful to the god not to be given anything to hold, so a picture of a shell was used to denote nothing. The creation of this symbol to represent nothing was prompted by superstitious considerations as much as mathematical ones. Like the Babylonians, the Maya still did not consider 0 to be a number in its own right.

  The Maya needed a number system to count very big numbers because their astronomical calculations spanned huge cycles of time. One cycle of time is measured by the so-called long count, which started on August 11, 3114 BC. It uses five placeholders and goes up to 20 × 20 × 20 × 18 × 20 days. That’s a total of 7,890 years. Therefore, a significant date in the Mayan calendar will be December 21, 2012, when the Mayan date will turn to 13.0.0.0.0. Like kids in the back of a car waiting for the odometer to click over, Guatemalans are getting very excited by this forthcoming event—though some doom-mongers claim that it is the date of the end of the world.

  Which prime is this?

  Figure 1.14

  Although these are letters rather than numbers, this is how to write the number 13 in Hebrew. In the Jewish tradition of gematria, the letters in the Hebrew alphabet all have a numerical value. Here, gimel is the third letter in the alphabet, and yodh is the tenth. So this combination of letters represents the number 13. The table opposite documents the numerical values of all the letters.

  People who are versed in Kabbalah enjoy playing games with the numerical values of different words and seeing their interrelation. For example, my first name has the numerical value

  mem resh kaph vav samekh

  40 + 200 + 20 + 6 + 60 = 326,

  which has the same numerical value as “man of fame” . . . or, alternatively, “asses.” One explanation for 666 being the number of the beast is that it corresponds to the numerical value of Nero, who was one of the most evil Roman emperors.

  You can calculate the value of your name by adding up the numerical values in Table 1.1. To find other words that have the same numerical value as your name, visit http://billheidrick.com/works/hgemat .htm or use your smartphone to scan this code.

  Table 1.1

  Although primes were not significant in Hebrew culture, related numbers were. Pick a number and look at all the numbers that divide into it (excluding the number itself) without leaving a remainder. If when you add up all these divisors you get the number you started with, then the number is called a perfect number. The first perfect number is 6. Apart from the number 6, the numbers that divide it are 1, 2, and 3. Add these together—1 + 2 + 3—and you get 6 again. The next perfect number is 28. The divisors of 28 are 1, 2, 4, 7, and 14, which add up to 28. According to the Jewish religion, the world was constructed in 6 days, and the lunar month used by the Jewish calendar was 28 days. This led to a belief in Jewish culture that perfect numbers had special significance.

  The mathematical and religious properties of these perfect numbers were also picked up by Christian commentators. St. Augustine (354–430) wrote in his famous text City of God that “six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect.”

  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 on this later in the chapter). To date, we know of only 47 perfect numbers. The biggest has 25,956,377 digits. Perfect numbers that 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 vice versa. 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 here is not a plus symbol—it is in fact the Chinese character for 10. The three characters together denote two lots of 10 and three units: 23.

  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, 1,000s, 100,000s, etc.) they turned the numbers around and laid the bamboo sticks vertically:

  Figure 1.17

  The ancient Chinese even had a concept of negative numbers, which they represented by different-colored 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 th
e 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 were 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 that 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 that is very effective at finding small primes. The task is to find an efficient method that will knock out all the nonprimes. 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, which is 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 humor isn’t a mathematician’s strong point.

  Figure 1.18 Strike out every second number after 2.

  Now take the lowest number that 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 striking 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 is 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 seventh number on from 7 because 91 = 7 × 13. 91 often catches people off guard 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, which is 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 nonprimes, 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 two apart—then three apart, five apart, and so on. The only problem 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. He calculated the sun to be 804 million stadia from the earth—although his unit of measurement perhaps makes judging the accuracy a little difficult. What size stadium are we supposed to use: Wembley or something smaller, like Loftus Road, the home ground of Fulham Soccer Club in London?

  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, downloadable from the Number Mysteries website, to put the Sieve of Eratosthenes into operation. Take a pile of pasta and place pieces on each of the numbers as you knock them out. The numbers left uncovered will be the primes.

  HOW LONG WOULD IT TAKE TO WRITE A LIST OF ALL THE PRIMES?

  Anyone who would try to write down a list of all the primes would be writing forever, because there are an infinite number of these numbers. What makes us so confident that we’ll never come to the last prime, that there will always be another one waiting out there for us to add to the list? It is one of the greatest achievements of the human brain that with just a finite sequence of logical steps, we can capture infinity.

  The first person to prove that the primes go on forever was a Greek mathematician living in Alexandria, named Euclid. He was a student of Plato’s, and he also lived during the third century BC, though it appears he was about 50 years older than the librarian Eratosthenes.

  To prove that there must be an infinite number of primes, Euclid started by asking whether, on the contrary, it was possible that there were in fact a finite number of primes. This finite list of primes would have to have the property that every other number could be produced by multiplying together primes from this finite list. For example, suppose that you thought that the list of all the primes consisted of just the three numbers 2, 3, and 5. Could every number be produced by multiplying together different combinations of 2s, 3s, and 5s? Euclid concocted a way to build a number that could never be captured by these three prime numbers. He began by multiplying together his list of primes to make 30. Then—and this was his act of genius—he added 1 to this number to make 31. None of the primes on his list—2, 3, or 5—would divide into it exactly. He always got a remainder of 1.

  Euclid knew that all numbers are built by multiplying together primes, so what about 31? Since it can’t be divided by 2, 3, or 5, there had to be some other primes not on his list that created 31. In fact, 31 is a prime itself, so Euclid had created a “new” prime. You might say that this new prime number could have just been added to the list. But Euclid could have then played the same trick again. However big the table of primes, Euclid could have just multiplied the list of primes together and added 1. Each time, he could have created a number that left a remainder of 1 upon division by any of the primes on the list, so this new number had to be divisible by primes not on the list. In this way, Euclid proved that no finite list could ever contain all the primes. Therefore, there must be an infinite number of primes.

  Although Euclid could prove that the primes go on forever, there was one problem with his proof—it didn’t tell you where the primes are. You might think that his method produces a way of generating new primes. After all, when we multiplied 2, 3, and 5 together and added 1, we got 31—a new prime. But it doesn’t always work. For example, consider the list of primes 2, 3, 5, 7, 11, and 13. Multiply them all together: 30,030. Now add 1 to this number: 30,031. This number is not divisible by any of the primes from 2 to 13, because you always get a remainder of 1. However, it isn’t a prime number since it is divisible by the two primes 59 and 509, and they weren’t on our list. In fact, mathematicians still don’t know whether the process of multiplying a finite list of primes together and adding 1 will
infinitely often give you a new prime number.

  There’s a video available of my soccer team in their prime-number gear explaining why there are an infinite number of primes. Visit www.youtube.com/watch?v=0LU4nkQKIN4 or use your smartphone to scan this code.

  WHY ARE MY DAUGHTERS’ MIDDLE NAMES 41 AND 43?

  If we can’t write down the primes in one big table, then perhaps we can try to find some pattern to help us to generate the primes. Is there some clever way to look at the primes you’ve got so far and know where the next one will be?

  Here are the primes we discovered by using the Sieve of Eratosthenes on the numbers from 1 to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. The problem with the primes is that it can be really difficult to work out where the next one will be, because there don’t seem to be any patterns in the sequence that will help us to locate them. In fact, they look more like a set of lottery ticket numbers than the building blocks of mathematics. Like waiting for a bus, you can have a huge gap with no primes and then suddenly several come along in quick succession. This behavior is very characteristic of random processes, as we shall see in Chapter 3.

  Apart from 2 and 3, the closest that two prime numbers can be is two apart, like 17 and 19 or 41 and 43, since the number between each pair is always even and therefore not prime. These pairs of very-close primes are called twin primes. With my obsession for primes, my twin daughters almost ended up with the names 41 and 43. After all, if Chris Martin and Gwyneth Paltrow can call their baby Apple, and Frank Zappa can call his daughters Moon Unit and Diva Thin Muffin Pigeen, why can’t my twins be 41 and 43? My wife was not so keen on the idea, however, so these have had to remain my “secret” middle names for the girls.