Since the dawn of civilisation, humankind has needed to count things. Early farmers had to be able to keep track of their animals. They needed to count the days so they could plant the crops at the right time. Officials had to calculate taxes so that the rulers could continue to organise and protect the country.

Various ways of counting have been devised based on grouping things into twos, fives, tens and twenties. Along with these, many different ways of writing down the numbers have been invented over the years. The most popular by far is the one we now use, the Indo-Arabic system. This Entry gives some details of older number systems.

Tallies

The earliest records of numbers are tallies. A mark is made for each object being counted - the total number of marks is the same as the total number of objects. Such tallies have been found scratched on bones dating back to 20,000 years ago. More sophisticated versions allow a few different types of thing to be counted at the same time.

In Sumer (modern Iraq) about 6,000 years ago, tiny clay models were used as tallies. If a farmer had six cows and three goats, six tiny model cows and three model goats were placed inside a clay sphere, which was then sealed. This formed a record of the farmer's possessions. Some experts believe that writing itself originated as a shorthand for such possession tokens.

The British Exchequer kept records on wooden tally sticks until as recently as 1826. A few years later, it was decided to destroy all the old records by burning them. This took place on 16 October, 1834. The fire got out of hand and burnt down the Houses of Parliament.

Egyptian Numbers

The Egyptians were manic record keepers. They had an abundant supply of paper in the form of papyrus. This was made from the pith of the reeds of the papyrus plant which grew everywhere in the marshy ground of the Nile Delta. They recorded small numbers with a tally system of vertical lines:

Above nine, they introduced a new symbol, a half circle opening downwards. This represented ten. This was a big step forward, with one symbol representing a number of objects. In this system, we have:

The exact position of the symbols wasn't very important, but they were usually arranged with the tens on the left and units on the right. The system was extended with lots more symbols:

Numbers up to just less than ten million could be written down. The system was good for records, but not good for doing calculations.

Inca Numbers

On a par with Egypt is the system used in South America by the Quechua people, (normally known as the Incas). They never developed writing, so they did not write down numbers in books or inscriptions. But they invented a way of storing numbers as knots on string. Messengers memorised the messages they carried, but the numbers were held on the string. Of course, the messenger had to remember which string referred to which record, but elaborate colour coding was used.

Inca number strings are called 'quipu'. Each string holds one number, and the strings hang from a main string. Strings are different colours, and presumably the colours and the order in which they were attached to the main string served as reminders to the messenger as to what they represented.

Within each string, knots were tied to represent the number. A single knot represent 1; two knots meant 2 and so on. A decimal system was used: there was a place for units close to the main string, a place for tens a bit further down and so on. So the number two hundred and thirteen would be represented as three knots, a space, one knot, a space, then two knots.

Zeroes were represented by a larger space, but this could lead to confusion. To get around this, there were three different types of knot used: one was used for the digit 1 in the ones position only. Another was used for any number other than 1 in the units position. The third sort was used for any number in any other position. This meant that 1 could not be confused with 10, but it was still possible to confuse 11 with 101.

Babylonian Numbers

The Babylonians, who lived about 4,000 years ago in what is now Iraq, did not have an abundant supply of paper. Theirs was a much drier countryside. They kept records by making marks in clay tablets. The tablets were put out in the sun to dry. They could be re-used by adding a small amount of water. Alternatively, they could be baked in ovens to make permanent records. The marks were put into the clay using a special stick which left two types of triangular mark: a narrow, long, vertical one and a wide, horizontal one. These type of records are known as cuneiform, from the Latin for 'wedge shape'. This Entry will use vertical lines | and angle brackets < to represent these marks.

Numbers from one to nine were represented by vertical marks, like in the Egyptian system. These were grouped in threes, so for example four was represented by three vertical marks, and then below this another vertical mark. Tens were represented by the < mark. Numbers up to 59 were done in the same way as the Egyptian system, with combinations of these:

When the Babylonians reached sixty, they brought in a new principle which had not been used anywhere before: the positional system. The symbol for 60 was the same as that for 1. The first position (starting at the right) represented units. The next position to the left of this represented multiples of 60. So the number 61 was represented by a vertical mark, a small gap and then another vertical mark. The same numbers from 1 to 59 were used to represent multiples of 60 in the second position as were used for units in the first position. For really big numbers, the third position represented multiples of 3,600, which is 60 squared. This positional system could be extended as far as you liked with multiples of 60 cubed, 60 to the fourth power and so on. In this way, enormous numbers could be written down. In the following table, we've also written the numbers out in an intermediate form showing the number in each position:

The obvious problem with this was that there was no symbol for zero, so there was no way of telling 1 from 60 or 3605 from 65 and so on.

Greek Alphabetic Numbers

The Greeks were great mathematicians, but their number system left a lot to be desired. They had two different ways of writing down numbers. One, called the 'acrophonic' system, was used for day-to-day numbers by the people and was similar to Roman Numerals described later in this Entry. The other was the alphabetic system. They used capital Greek letters for the numbers 1 to 9, another set of Greek letters for the numbers 10, 20 ... 90 and a third set for 100, 200 ... 900. They needed 27 separate symbols for this, but there were only 24 letters in the Greek alphabet, so they revived three ancient letters which were no longer used in writing.

The numbers 7, 73, 123 and 500 were therefore written: Ζ, ΟΓ, ΡΚΓ, and Φ. Sometimes, they would put a line over the symbols to indicate that they were being used as numerals rather than as normal letters.

Above 999, they ran out of symbols, so things started to get complicated. A superscript iota, the symbol for 10, meant multiply by 1,000. A big M stood for 10,000. A number written above an M meant that the number was multiplied by 10,000. Since these big numbers were not used in everyday life, it was only the mathematicians who knew how to write them down, and they didn't even agree on a system among themselves.

The Greeks still use these ancient numbers in inscriptions, in the sort of places that the Western World might use Roman numerals. In the picture of Constantine XI (the Eleventh), the last Byzantine Emperor, you might just be able to make out that his Greek title to the right of the statue is Konstantinos ΙΑ. Similarly, Queen Elizabeth the Second of the United Kingdom is known as Elizabeth Β in Greece.

Roman Numerals

The Romans used the letters I, V, X, L, C, D and M to represent 1, 5, 10, 50, 100, 500 and 1,000. These were repeated as often as necessary to make bigger numbers. To make the number 1998, you would write MDCCCCLXXXXVIII.

Sometimes the Romans wrote the symbol for a small number before the symbol for a big number, for example, IX, and this indicated that the small value was to be taken away from the big value. The example IX therefore meant 1 less than 10, which is 9. This is known as the subtractive principle, but the Romans didn't use it much. If you look at Roman inscriptions, you'll see that they generally wrote their numbers out in full rather than using this subtractive shorthand.

Roman numerals continued to be used in Europe throughout the Middle Ages, and as time went on, the subtractive principle became more popular. It was no longer acceptable to have four copies of the same symbol in a number as it was difficult to read: IX is much clearer than VIIII.

People often ask how mediæval merchants could do calculations using Roman numerals. The answer is that they didn't. They used calculating devices such as the abacus or the chequerboard accounting table¹, and only used the Roman numerals for recording the final result.

The Italian mathematician Leonardo 'Fibonacci' of Pisa² (about 1170–1250) popularised the Indo-Arabic numbers we now use in his book Liber Abaci, but it was many centuries before their use was commonplace, and Roman numerals were still used for monumental inscriptions up to the middle of the 20th Century.

Indo-Arabic (Standard Western) Numbers

The system we now use for writing down numbers is known as the Indo-Arabic number system. It was invented in India and came into Europe through Arabia, hence the name. Europeans have spread the system throughout the world.

The system uses the same position system as the Babylonian numbers - the value of a symbol is not fixed but depends on the position of the symbol in the number. In the Indo-Arabic system, a base of ten rather than sixty is used, so 11 represents ten plus one rather than sixty plus one as it did in the Babylonian system. The Indo-Arabic system also introduced something which had not been seen (at least in Europe) before: the number zero. This allows it to be clear where digits have been left out, so that 402 means something different from 42.

Only ten digits are needed, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The digits 1, 2 and 3 started out as a single line, two horizontal lines and three horizontal lines. Written with an ink pen, it was easier not to lift the pen between writing the horizontal lines, so the curved sections of the 2 and 3 came into being. It's not clear where the digits 4 to 9 came from; even in the earliest inscriptions they appear to be arbitrary symbols and not graphic depictions of a number of lines.

The Indo-Arabic system is very practical and allows us to write down some very large numbers, such as 12,079,595,520 which is the number of bytes of storage on the author's mobile phone.

In the Arabic-speaking world, a slight variation of the Indo-Arabic system is used. Instead of the familiar digits of 0-9, different more cursive-looking digits are used. The zero looks like a dot, the two like a backwards 7, and the five looks like a zero, for example. Nevertheless, despite the different appearance, this is still the same number system.

Mayan Numbers

The Indo-Arabic system of numbers could be seen as the end of the line, since it is the one we use and it is very practical, without the shortcomings of the other systems described in this Entry. Mention should be made, however, of one other method of writing numbers, since it independently had the innovation of a symbol for zero and got a lot of press in the last few years. This is the Mayan number system.

The Maya civilisation existed in Central America from about 2000 BC, reaching its peak in the 3rd to 9th Centuries AD. The Maya developed an elaborate calendar which they used for classifying days as lucky or unlucky. Sacrifices to the gods and blood-letting rituals³ had to be done on the right sort of day. They invented a method of writing down numbers which they used for recording dates.

The Maya used two different calendars simultaneously. The first, called the 'Calendar Round', had no year number but had two different numbers and two different names for each day in a pattern that repeated every 52 years, so could be used to identify a day uniquely within recent time. For more permanent, long-term dates, they used the 'Long Count' which was the number of days since a mythical creation date in the distant past.

Small numbers up to 19 were written using dots and horizontal lines. You could have up to four dots and up to three lines. The dots represented 1 and the lines 5. A shell symbol represented zero:

For numbers above 19, a positional system was used; a number from the table above was used in each position. The first position represented units, the second position represented multiples of 20. The positions were arranged vertically, with the first position at the bottom, the second position above it, and so on. Thus, for example, 392 is 19×20 + 12 so it would be written with 12 in the first position and 19 in the second position:

The Maya were studied in the 16th Century by the Catholic Bishop Diego de Landa Calderon. Landa is a controversial figure. He burned many Mayan books and hundreds of wooden statues of their gods, considering them blasphemous, but was interested in learning about the Maya and their culture. Many of the facts we know about the Maya come from Landa's records, such as the correlation between the Mayan calendar and our own. He totally misinterpreted the Mayan writing system, but his attempts to write down a 'Mayan alphabet' were crucial in the modern decipherment of the writing system.

Landa recorded that the Maya merchants, particularly those dealing in cacao, used a purely base-twenty system for writing numbers. In this, the third position was multiples of 20 squared (400), the fourth position was multiples of 20 cubed (8000) and so on. In this system for example, our number one million is 6×20⁴ + 5×20³ so it would be written as the five numbers 6,5,0,0,0 stacked up with the six on top.

Dates were a different matter. The Maya were aware of the exact length of the year but instead chose to treat the year as 360 days for counting purposes. This short year was called a 'tun'. They modified their number system when writing dates so that the third position represented multiples of 360 rather than 400. Subsequent positions such as the fourth, fifth, etc were modified accordingly so that the fourth position represented 20×360, the fifth was 20×20×360 and so on. This meant that you could read off the number of 'tuns' (short years) by just ignoring the lowest two positions. Because the Maya recorded dates for everything in their history, all known examples of Mayan numbers are in this modified date format - there are no examples of the pure base-20 system used by the merchants.

Dates were reckoned from a mythical creation date which corresponds to 11 August, 3114 BC in our⁴ calendar. The current date was given as a number of days since the creation date, using the special date number system described above. So for example, 1 January, 900 AD, a date during the golden age of Mayan civilisation, would be 1,465,496 days since the start of the calendar. This is equal to 10×144,000 + 3×7,200 + 10×360 + 14×20 + 16, so it can be represented by the five numbers (10,3,10,14,16). These were written using dots and lines and stacked up with the 10 at the top and the 16 at the bottom.

The Mayan date (12,19,17,19,19) was followed by the date (13,0,0,0,0). This is like the digits on a car's odometer clocking over. This date in our calendar corresponded to 21 December, 2012. It was predicted that it would be an extremely unlucky day by the Mayan reckoning, featuring such unpleasant events as the End of the World. As it happened, it was fairly uneventful. Perhaps the gods got bored waiting for someone to engage in ritual blood-letting, and decided to leave the World in peace.