**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

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

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 *¡¥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 *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

*Adapted from
¡¥Mathematics Minus Fear¡¦ by *

*Published in 2006 by
Marion Boyars Publishing Ltd*