The Abax (Lesson 1

Notes to teachers and parents

The Abax (ancient counting board)
(continued from the introduction to the abacus)

New !
Make your own Abax
from things around the house

As you can tell from the picture, the ancient counting board was a board with grooves in it, in which you could place pebbles or other counters to represent numbers.

The advantage of using a counting board was that you could do basic operations (subtraction, addition, division and multiplication) with it. You may be wondering why people didn't just do it on paper. You must consider that in most societies thousands of years ago, there was no such thing as paper.

The abax was first used over 2000 years ago, when the only society that had papyrus (the forerunner of the paper we use today) was Egypt. And only the richest people (future mummy-types) could afford such a luxury. Besides that, have you ever seen Egyptian numbers? Hereare two examples:

Doing operations with number like these was a major pain. They were made much easier by the use of a counting board. The name "counting board" tells only half the story. You can also calculate with an abax. It is sort of primitive, but considering that it was the basis for the abacus, which is still used today in some places, it is a good start.

The board illustrated in these pages is probably the kind that was used in ancient Rome. Often they would have the Roman numerals I, X, C, etc. beneath the ones, tens, hundreds, etc. columns.

The main thing about the counting that makes it convenient, is that you can represent large numbers with a single pebble. It works with the place system. That means that where the pebble is placed is just as important as how many pebbles are used.

Before the place system, mankind pretty much used one-to-one correspondence. With that system, you needed one pebble or mark on paper for each thing you are representing. So if you had six sheep, and wanted to show someone how many you had, you needed to show six other things. Six pebbles, six fingers, six marks in the sand. Like the tally system. You know that system. Here is the number 6 in the tally system:

Well, that was swell for small numbers. But how would you like to have to represent the number 265 with pebbles, tally-marks or fingers?! Not too convenient.

Of course, one-to-one correspondence was not the only way to represent numbers before the counting board. We have already seen the Egyptian numbers, which came way before the counting board. It also used 9 symbols to represent ones, tens, hundreds, etc. But it depended on different symbols to represent each power of ten.

In other words, there was one symbol for 1 (those are the symbols that look like tally marks in the picture of Egyptian numerals, above), and if you used that symbol nine times, it meant nine ones.

Then they had a different symbol for the tens (which were the things that look like arches), and they could use up to nine of those symbols. The spirals stood for hundreds, and the lotuses (that is the weird looking symbol in the picture - the one which looks like a deranged Pac-Man) stood for thousands.

Maybe you noticed that those symbols are not "in order." That is because, even though there were some general rules, most Egyptian scribes didn't follow them. They weren't necessary; the Egyptian system was not a "positional" (or "place") system. Different symbols (not places) represented "ones", "tens", "hundreds," etc. It didn't matter where you put the symbols, they always stood for the same thing.

let's get to the first actual lesson -
counting with the abax