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
