Have
you ever thought of how much of our daily life is based on numbers? We
use it for shopping, reckoning the time, counting distances and so on.
Simple calculations seem effortless and trivial for most of our
necessities, but it was not so for ancient cultures, including the
Greeks and the Romans, who did not know the place value
notation of numbering.
Most
probably counting originated some 4,000 years ago for religious
reasons. Pagan priests needed to calculate the frequency of certain
natural phenomena for ceremonial purposes. One of the best known
examples of this period is the Stonehenge stone circle in Britain,
built by the Druids as a kind of celestial observatory in 1,800 BC
At
that time man still dwelled in caves or in the forest and lived in
groups for protection and survival.
So,
counting is telling someone how many things we have or there are, but
there exist various ways to do so. I can bring the things along with
me and show them; not very practical, though, especially if I want to
brag about my 286 cows and 122 goats I have on my farm. Or I could
carve a notch on a stick for every object I have: not very smart
either. Finding out an easy and efficient way to count was a
tremendous cultural advancement for mankind. It is the discovery of
the power of symbols: things that stand for other things. One notch,
or a line in the sand, for every object means that there is only one
symbol that needs be replicated as many times as there are objects to
represent. What if I want to communicate that in my village live 350
people? Definitely too many lines in the sand to draw or, still worse,
too many notches to carve! It would be a good idea to employ a higher
number of symbols, this would shorten the length of the numbers. So
another great conceptual leap forward was the invention of the
place-value notational system of multiples of 10.
This
system made use of the additive law: The sequence of the symbols meant that they had to be
added (ex. XVI = X+V+I = 10+5+1 = 16). However, writing numbers and
performing calculations was a real pain in the neck. Even very easy
arithmetic operations, like 15,786
x 7,812, was only in the power of the skilled mathematicians. About
2000 years ago, though, when the Romans were still hobbling with their
awkward number representations, people in India wrote the numbers two
and three
They
also invented a very ingenious way to perform calculations. They made parallel ditches in
the sand and placed pebbles in them. Starting from the right, each
pebble in the first ditch counted as 1, in the second ditch as 10, in
the third as 100.
This
was the development of the abacus
technology – whose first prototypes were invented in China about
2,500 years ago and soon spread out to India –
and
also an example of the base_10 numbering system. The abacus is the
first true precursor of the adding machines and computers which would
follow centuries later.
The
abacus not only was very handy but also spun
an extraordinary discovery. By observing how the pebbles were displayed
in the abacus they noticed that the various combinations gave
different results. But... what about the empty ditch? They decided to mark it with a
dot, that therefore counts for nought (zero). Later on, the dot blew up
into a circle, as we now know it. So the figures we write as 10
101 and 110 were originally annotated as 1.
1.1
11.
The
foundations for modern arithmetic were thus laid down and the
place-value notational system was invented. Since then, the ten symbols
we are so familiar with nowadays have become the common way of writing
numbers all over the world: a nice array of 0123456789.
But those were times of
slow communications and only in the 7th century AD the system was passed
to the Arabs. It took another 600 years for the Europeans to acquire
this knowledge from the Arabs, in fact only in the 13th century the
positional system of numbers became common in Europe.
