abax fig. 10

Here is the number 30. By now, this must be making sense to you . If not, go back to the first lesson.

abax fig. 11

Now the tens column is full. You know that it takes 9 pebbles to fill a row. So you must know that this means that this number is 90.
Does that mean that we need another groove to represent more numbers? Not yet! We still have room for more in the ones column. So how high can you count with just two columns? Look at the next picture to find out. (If you haven't already figured it out!)

abax fig. 12

This is 99. Both columns are full. So what did our ancient Roman friends do when this happened? They cut another groove. (And boogied down?) Do you know how the next number will be represented?

abax fig. 13

This "place value" stuff was a clever invention. We got rid of the 18 pebbles which we needed to represent the number 99, and traded them in for a single pebble in the new "hundreds column" to represent 100.

abax fig. 14

With a pebble representing 100 in the hundreds column, it's almost as if we were starting from the beginning, with one pebble in the ones column. Together, they make one hundred and one (101). Now, to keep going, you keep adding to the ones column, until it is full, just like before.

abax fig. 15

Here the ones column is full, and there is still one pebble in the hundreds column, making this number 109. To continue, we will need to trade in all the pebbles in the full ones column for a one pebble in the tens column, and keep going like that.

abax fig. 16

Can you tell at a glance if this is more or less than the number represented in the last picture?
 If you practice and test yourself after you study this entire page, you will be able to tell in an instant.
 There is one in the hundreds, and one in the tens, so the number is going to be at least 110, which is larger then the 109 in the last picture. Actually, with the pebbles in the ones column, the number in this picture is 116.

abax fig. 17

Now we are getting up there in the higher numbers. But it should be no problem to recognize this number as 299. Are you starting to see how you can visualize numbers by getting used to an abax?

abax fig. 18

We've managed to fill all the columns. There are nine pebbles in each column, giving us a total of 999. To count to more, our ancient Roman friends would have to get a larger abax, with room for more grooves.
 Fortunately, we don't. This is as high as we are going to count with the abax, for now. The next lesson is about adding with the abax.
Before you go there, make sure that you can count easily by testing yourself.