Written Number Systems

The history of written arithmetic is long and complicated, and it took thousands of years for societies to develop to the point where they could tackle complex number problems. Nobody should ever suggest that it is easy.

When Christ was born, the most common method of recording numbers was the Greek system. The Greeks took their alphabet, and assigned a letter to each number from one to nine, to each ten from ten to ninety, and to each hundred from one hundred to nine hundred. Sadly, they ran out of letters in their alphabet, so they brought an old letter back, and borrowed a couple from the Phoenicians to make up the numbers.

So, using their system, a was one, b was two, q was nine, i was ten, k was twenty, r was one hundred and s was two-hundred - just to name a few examples. By combining the letters, they could produce any number up to 999. For example, they would write 112 as rtb, 229 as sk q and 354 as tnoi. They were perfectly capable of dealing with larger numbers. They invented symbols for a thousand (/) and ten thousand (M). If they wanted to express two thousand, they wrote / b, and if they wanted to express twenty thousand, they wrote M with b above it:

b

M

This probably sounds very sensible, but unfortunately (or fortunately, depending on how you see it) it makes calculations very difficult. We only have to deal with combinations of ten different symbols, where the Greeks had to deal with twenty-seven. For us, adding twenty and thirty is essentially the same as adding two and three, but this is not true for the Greek system. There is no connection between the symbol for two and the symbol for twenty.

As a result, the Greeks tended to use abaci to do their sums, rather than use written methods. In fact, they were quite snooty about calculations in general. Philosophers and mathematicians were much more interested in number patterns than working out sums, because they felt that such patterns were behind the very fabric of the universe. The Greeks got so excited by this sort of thing that they became downright religious about numbers.

So, Greek numbers are excellent if you want to identify your neighbour as the Antichrist, but not so useful if you want to work out the price of beans. With the rise of the Roman Empire, Roman numerals gradually took their place. The Romans had symbols for one (I), five (V), ten (X), fifty (L), one hundred (C), five hundred (D) and one thousand (M). These were much more useful for addition and subtraction, but they were very unwieldy for more complicated sums. And it took a long time to write large numbers. For example, when the BBC wanted to represent the year 1988 on its title page, this is what it had to write: MCMLXXXVIII. So you can see, as arbitrary as it sounds, the system that you use to write down your numbers can actually have quite a significant effect on the kinds of maths that you can do … or at least the kinds of maths that you can do easily.

Even so, Roman numerals stuck around for a long time. They were still used in some mathematics books in the 16th century, and they only really disappeared when printed books became widespread. Meanwhile, since they were so cumbersome for complex calculations, merchants stuck to using the abacus and waited for something better to turn up.

Whilst the Europeans were wallowing in the Dark Ages, other parts of the world were making impressive advances in their number systems. It all started in India, when, in about the 7th century, the Hindus carne up with a decimal place-value system. They realized that they could-write numbers far more economically, if the position of a digit influenced its value. So, they invented the 'units column', 'the tens column', 'the hundreds column' and so on. As a result of this invention, the Hindus could represent any number they liked, using just nine symbols. So, where the BBC took eleven symbols to write just one number, the Hindus only took four: 1988. The one represents 'one thousand', the nine represents 1 nine hundred', the first eight represents 'eight tens', and the second eight represents 'eight units'.

However, it wasn't all plain sailing for the Hindus with their new invention. Their main problem was how to show that a column was empty. Initially, they just left a gap, but this caused trouble, because it was very difficult to say whether a space in a number represented one gap, or two gaps, or three gaps. And yet the size of the space had a big influence on the size of the number. For example, 2 space 3, could be two-hundred-and-three, two thousand-and-three, or twenty-thousand-and-three, depending on how many gaps you thought it represented.

So finally, at some time during the 9th century, they came up with the symbol '0' to represent an empty column. From India, the decimal place-value system spread to the Middle East. Copies still exist of a book by a man called al-Khwarizmi (approximately 790-840) with the title of ‘A Book on Addition and Subtraction after the method of the Indians.’ In this book, al-Khwarizmi gives methods for addition, subtraction, multiplication and division. Later Arab mathematicians even began to experiment with extending the decimal place-value system to deal with fractions.

The new numbers finally made their way to Europe around the 14th century. The Italian mathematician Fibonacci travelled with his merchant father to North Africa where he learnt about the Arabic number system. In 1202 he published a book, ‘Liber Abaci’, which showed the possible practical applications of the new methods. The book was well received by European academics, but the use of the decimal number system didn't become widespread until after the invention of the printing press in the 1400s. You would have thought that the long-suffering merchants there would have been grateful for a system that made calculations quicker and easier. But, on the contrary, many refused to use it, claiming that it was a satanic practice, and that the numbers themselves contained dark magic. Another disadvantage for the new numbers was that their very simplicity made forgery a threat to their use in trade. To solve the problem, enterprising merchants came up with the idea of cheques.

So, it took a long time for societies to create a written number system that would allow them to deal easily with written calculations, and we, in the West, were particularly slow to catch on. Just to rub it in, several other civilizations have invented a place-value system over the years. The Babylonians figured it out in the 19^th century BC, although they worked with a number system that was a mixture of base ten and base sixty. They had a symbol for ten, which looked a bit like ‘<’ and a symbol for one, which looked like ‘Y’.

To write any number up to sixty, they just combined the correct number of each symbol. So forty-three was ‘YYY <<<’ (although arranged slightly differently) and 55 was ‘YYYYY <<<<<’. Beyond this they used a place-value system, but because they worked in base sixty, the values of their columns were different. Each successive column had a value sixty times greater than the previous one. So, the first one was units, but the second was for 60s, the third for 3600s, and so on.

Therefore, to write the number that we would call 581, the Babylonians would not consider it, like us, as made up of units, tens and hundreds (in this case five hundreds, eight tens and one unit). They would see it as a combination of units and sixties. For them, 581 was nine 60s and 41 units [(9 x 60) + (41 x 1) = 540 + 41= 581]. Therefore they would represent this number as YYYYYYYYY <<<<Y (i.e. the symbol for nine in the 60s column, and the symbol for forty-one in the units column).

In a similar way, for the Babylonians, 3730 was not three thousands, seven hundreds, three tens and 0 units, but 1 three thousand-six-hundred, 2 sixties, and 10 units [(1 x 3600) + (2 x 60) + (10 x 1) = 3600 + 120 + 10 = 3730]. Therefore, a Babylonian would write: Y YY <.

Other systems have been created in other places over the centuries. The Chinese came up with one, the Mayans have decorated their temples with their symbols for numbers, and alternatives have been created in modem times as well. Computers and other machines use the binary place-value system to operate. In this system, the values of the 'columns' are based on powers of two. Each column has a value that is two times greater than the previous one. So the first column (reading from right to left) is for units, the second for 'twos', the third for 'fours', the fourth for 'eights', and so on.

Using this system, a computer considers the number that we write as 46 to be made up of 1 thirty-two, 0 sixteens, 1 eight, 1 four, 1 two and 0 ones:

(1 x 32) + (0 x 16) + (1 x 8) + (1 x 4) + (1 x 2) + (0 x 1)

~ 32 + 0 + 8 + 4 + 2 + 0

~ 46

So, in binary, this number becomes 1 0 1 1 1 0. And 129 is thought of as 1 one-hundred-and-twenty-eight, 0 sixty-fours, 0 thirty-twos, 0 sixteens, 0 eights, 0 fours, 0 twos and 1 one ((1 x 128) + (1 x 1) ~ 128 + 1 ~ 129). Therefore, in binary, our 129 becomes 1 0 0 0 0 0 0 1.

As you can see, there were plenty of options for how to write down numbers, and plenty of different civilizations who tried them out. But eventually, in Europe at least, the merchants admitted defeat, put away their abaci, and started scribbling sums on bits of paper. It was the beginning of written arithmetic.

Adapted from ‘Mathematics Minus Fear’ by Lawrence Potter

Published in 2006 by Marion Boyars Publishing Ltd