The Number Mysteries: A Mathematical Odyssey through Everyday Life

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

FIGURE 1.10 The different ways of dividing up 60 beans.

The Babylonians came close to discovering a very important number in mathematics: zero. If you wanted to write the prime number 3,607 in cuneiform, you had a problem. This is one lot of 3,600, or 60 squared, and 7 units, but if I write that down it could easily look like one lot of 60 and 7 units—still a prime, but not the prime I want. To get around this the Babylonians introduced a little symbol to denote that there were no 60s being counted in the 60s column. So 3,607 would be written as

How to count to 60 with your hands

We see many hangovers 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. There are four fingers on each hand, so with the thumb 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 along on the left hand.

FIGURE 1.11

FIGURE 1.12

But they didn’t think of zero 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 zero 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.

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. They 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 one, two dots for two, three dots for three. Just like a prisoner chalking off the days on the prison wall, once they got to five, instead of writing five dots they would simply put a line through the four dots. A line therefore corresponds to five.

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 Mayans 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 up the year to 365 days 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 for nothing was prompted by superstitious considerations as much as mathematical ones. Like the Babylonians, the Maya still did not consider zero 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 11 August 3114BC, uses five place-holders and goes up to 20×20×20×18×20 days. That’s a total of 7,890 years. A significant date in the Mayan calendar will be 21 December 2012, when the Mayan date will turn to 13.0.0.0.0. Like kids in the back of the car waiting for the milometer 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. Table 1.01 details the numerical values of all the letters.

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

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, one of the most evil Roman emperors.

TABLE 1.01

You can calculate the value of your name by adding up the numerical values in Table 1.01. To find other words that have the same numerical value as your name, visit http://bit.ly/Heidrick or use your smartphone to scan this code.

Although primes were not significant in Hebrew culture, related numbers were. Take a number and look at all the numbers which 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 the 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 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