One
advantage the counting board had over this system, was that there was
only one "symbol" - the pebble. Depending on where it was
placed, you could represent as large a number as you wanted. If you
had 4 columns in your board, you could represent the number 1,000 with
a single pebble in the thousands (the fourth) column.
Now
that you have some Idea of the principle of the counting board, let's
get reckoning. This is going to be pretty easy, and I have made lots of
illustrations for you. Go through all of them, because even though you
will "get" the Idea within a minute or so, there are some more
principles along the way that you might like to learn.
The entire
lesson about the abax should take very little time. Soon you will be on
to the abacus. You could skip the abax and go to the abacus, but
that would be like building a house without a foundation - you could do
it, but it wouldn't be as sturdy.
One
thing should be made perfectly clear - in principle, an abax can have
grooves long enough for any number of marbles or pebbles. Usually, though,
we make room for only 9 pebbles in each column. That means every time
you reach the number 10 in any column, you must begin another column.
That is because we can only fit one digit in any column. (Once you have
a second digit, you have automatically started another column). Because
we start a new column any time we reach the number 10 in any column, we
call this system the "base ten" system. It is also called the
"decimal" system. That name comes from the Latin word "decem"
(meaning "ten"). There are other bases, and you can use an abax
for them, but you have to make some changes. We will get to that eventually.
You should
know what a "digit" means. A digit is a whole number (not a
proper fraction or decimal) from 0 to 9. You may notice that all the digits
can be represented by a single symbol. Numbers from 10 on, all need at
least two digits to write them. (In the case of 10, you have the digit
1 in the tens column, and the digit 0 in the ones column. This is called
a "two-digit" number).
abax fig. 1
Here
is the number 0. We use this column to represent the "ones."
How many ones are in this column? None, so we call it zero.
If we were using paper, we would not leave it blank, though. The
numeral for nothing is 0 ("zero"), and we always have
to write it in a column with nothing in it when we are using paper.
The reason we do that, is that if we wanted to write, let's say,
the number fifty, and you just wrote a 5, you would think it was
the number 5. You have to write a zero in the ones column to let
people know that there actually is a
ones column which is empty, and the 5 means 5 tens, and
not five ones.
With an abax, you don't need to put anything in an empty column,
because people can see that there is a column there,
whether there is something in it or not.
This is what me mean when we talk about "place value."
Depending on what place a digit is in, you know if it means ones,
tens, hundreds, etc.
So far, we have only used the ones column. There will be more.
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abax fig. 2
This
is the number 1. Since there is only one column,
you know that the single pebble stands for one "one."
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abax fig. 3
This
is the number 5.
To represent more ones, you just put more pebbles in the ones
column.
The difference between this and how you would do it on paper,
is that on paper you only need one symbol to represent up to 9
things in any column. On an abax in base ten (the base we commonly
use in real life), you need up to 9 different pebbles to represent
up to 9 things in any column.
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abax fig. 4
This
is the number 9.
Now
you see we have filled the column. What did the ancients do when
they filled a column? Simple! They cut another groove for the
next column! I did that for you in the next picture.
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abax fig. 5
This
is the number 10. You see the empty ones column,
and only one pebble in the new tens column.
How
would we write the next number? A lot of people automatically
think you just put one more in the tens column. But that would
mean there were two tens (and that would mean twenty,
not eleven!)
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abax fig. 6
This
is the correct way to represent the number 11.
You
see that there is still only one in the tens column, but there
is also one in the ones column.
To
write the number up to 19, you just continue to add pebbles in
the ones column.
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abax fig. 7
For
example, this is the number 17. There is still
only one pebble in the tens column, but there are 7 in the ones
column.
How
do you think you would represent the number 19?
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abax fig. 8
This
is the number 20. Remember how we have already
mentioned that two pebbles in the tens column would represent
the number 20?
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abax fig. 9
What
number do you think this is? How many in the tens column? There
are two in the tens column. How many in the ones column? There
are nine in the ones column. Two tens and nine ones make 29.
How
do you imagine we will make the number 30? Do you think we will
have to cut another groove, or not? Click on the link below to
continue with your abax lessons.
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Let's
go to the next lesson, counting higher with the
abax
Would
you like to learn this on your own Abax?
To order an abax and instruction booklets for more detailed speedmath
lessons, click
here.
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