The Abacus: A History

How did people keep track of numbers before pen and paper were widely available?
How does addition and subtraction work if you don't have a handy written form for your numbers?
Say you can't read or write, but you can count - how do you add, subtract, multiply, or divide large numbers?
The answer to all these questions is . . . the abacus!

What is an abacus?
An abacus is a device used for addition and subtraction, and the related operations of multiplication and division. It does not require the use of pen and paper, and it's good for any base number system. There are two basic forms for the abacus: a specially marked flat surface used with counters (counting table), or a frame with beads strung on wires (bead frame).

Timeline of the Abacus

Chinese Abacus *************************************

Japanese Abacus ***************************

Aztec Abacus *******

Counting Table ******************************************

600 BC 500 BC 0 900 AD 1600 AD 1826 AD 1999 AD

The Counting Table

Although counting tables are obsolete now, they have over 2000 years of documented use. Their earliest form was that of a simple stone slab with incised parallel lines. The lines served to mark the place values. The earliest known counting table is from the island of Salamis in Greece. The latest surviving counting table is from the end of the 16th century and is in a museum at Strasbourg (Pullan 53). The only real difference between the early and later forms is the direction of the place value lines: horizontal or vertical.

The medieval European form is the most well documented, in part because the English Exchequer only stopped using the counting table for tallying tax payments and the like in 1826 (Pullan, 27).

Here is a diagram of the counting table, as used in medieval England:

-----------------+----------------- Billion
|
-----------------+-----------------
|
-----------------+-----------------
|
-----------------+--O-------------- Million
|
-----------------+--O-O-O----------
|
-----------------+--O-O------------
| O
-----------------+--O-O------------ Thousand
| O
-----------------+--O--------------
|
-----------------+-----------------
| O
-----------------+--O-O-O-O-------- Unit

Medieval counting table with counters reading 1,327,609 (Moon 26)

These tables were used with counters to represent various values. Small stones, called calculi, were used with counting tables in Greece and Rome, and stamped metal counters, rather like coins, were used with counting tables in Europe and England (Pullan 18). Although the idea of doing sums with pencil and paper became widespread around 1600 in Europe, and the counting table became completely obsolete in the first quarter of the 18th century, our word 'counter' as used for 'a flat surface upon which business is transacted' is a direct survival of the medieval counting table.

The Bead Frame Abacus

The bead frame form is what most people think of when they hear the word, "abacus". There are 3 main forms of abacus in use today; the Chinese, the Japanese, and the Russian. All are composed of a rectangular frame with beads on vertical wires. The number of wires and beads vary, and there may or may not be a horizontal divider in the frame.

It cannot be proven, but the Chinese are often credited with the invention of the abacus. The abacus was a great invention in ancient China and has been called by some Western writers "the earliest calculating machine in the world." The abacus has a long history behind it. It was already mentioned in a book of the Eastern Han Dynasty, namely Supplementary Notes on the Art of Figures written by Xu Yue about the year 190 A. D. Its popularization occurred at the latest during the Song Dynasty (960-1127), when Zhang Zeduan painted his Riverside Scenes at Qingming Festival. In this famous long scroll, an abacus is clearly seen lying beside an account book and doctor's prescriptions on the counter of an apothecary's (Feibao).

The most common Chinese abacus has 13 vertical wires, with 7 beads on each wire. The wires and beads are in a rectangular frame. There is a horizontal divider within the frame so the 7 beads on each wire are separated into 2 beads above the divider (the heaven beads) and 5 beads below the divider (the earth beads).

Diagram of a typical Chinese abacus.

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O || Heaven beads, each worth 5
|| O O O O O O O O O | O O O ||
|| | | | | | | | | | O | | | ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | O | O ||
|| O O O O O O O O O O O O O || Earth beads, each worth 1
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O | O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

This abacus is showing the value 5402. Moving beads towards the horizontal divider adds that value, moving beads away clears the value.

The Chinese abacus was brought into Japan around the 17th century (Rentchz). It was studied by the Japanese mathematician Seki Kowa (1640 - 1708) and many refinements were made to the Chinese abacus, including removing one bead on each wire above and below the horizontal dividing bar. The transformation of the Chinese abacus into the modern Japanese form was completed by 1920 (Kojima 25). This modern form has 4 beads below the horizontal divider, and only one bead ...

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This abacus is showing the value 5402. Moving beads towards the horizontal divider adds that value, moving beads away clears the value.

The Russian abacus is similar in form to the Chinese or Japanese abacus, and was probably brought to Russia from China. The Russian form is set up to do calculations in rubles and kopeks. It has no horizontal divider, but some of the beads on each wire are a different color, to help as place-keepers (Pullan 100). Older Russian abaci have some additional columns for quarter kopeck as well as quarter ruble values (Leipala). If you go to Russia today, you will still see the abacus used. Ed Oswalt made this observation when he went to Russia in 1997: "The same store . . . where you can buy a Pentium computer, computes your bill using an abacus."

An interesting form of the abacus was found during archeological excavations in Central America. This abacus dates to around 900 AD and is constructed from maize kernels threaded on a string, all contained within a wooden frame (Fernandes). Grado states that an abacus of this form would have 7 beads by 13 columns.

Math on the Counting Table

Simple mathematical calculations on either the abacus or the counting table are pretty much the same as pen and paper sums, but instead of using abstract representations of the quantity, you use counters or beads to represent a quantity, and position to represent value. If you were to add up two sums of money by sorting each pile out by denomination, then adding the pennies of the first pile to the pennies of the second pile, the nickels of the first to the nickels of the second, and so on; and also normalizing as you go (5 pennies are replaced by one nickel, 2 nickels by one dime, etc.) you would get a good feel for addition on the counting table.

Subtraction on the counting table depends upon the idea that you can represent larger sums of money in more than one way: 10 cents as one dime, or as 2 nickels, or as 10 pennies, or even as 1 nickel and 5 pennies! Of course, the pennies would be represented by counters on the penny line, and nickels by counters on the nickel line, etc. For subtraction, you have to make sure that the number your are subtracting from (minuend) has no fewer counters in each unit than the number you are subtracting (subtrahend) (Moon 28).

has a diagram of subtraction on the counting table.

Division can be done on the counting table as though you were doing standard long division. However, that is not how it was done in Medieval and Renaissance Europe. Modern long division on the counting table is a difficult operation, even though it is only repeated subtraction. It is a long, tedious process made a little easier if you can factor first and then divide. It is useful to keep a tally of how many subtractions have been done. Multiplication on the counting table, at least in medieval Europe, was NOT done as we do today - it used the processes of mediation and duplation to find the product of two numbers. If you did not know what 13 x 6 was, this is how you would find the answer:

Make two columns beginning with the numbers you want to multiply. Repeatedly halve the values of the first column (toss out remainders) and repeatedly double the values of the second column. For each even value in the first column discard the matching value in the second column. Add the remaining second column numbers up for your answer (Moon 54).

13----6

6----12
3----24
1----48
discard 12 from the second column, leaving 6, 24, and 48.
6+24+48=78 or 13x6=78

Math on the Bead Abacus

To do simple arithmetic on the bead abacus, you first must determine what base you want to use, and if you want to have fractional values available. Dave Bernazzani gives this very concise explanation of a base 10 numbering system for the abacus:

One of the columns is chosen to be the "Units" column and can be any column of your choosing (if you are working with only whole numbers, simply use the rightmost column). If you are working out calculations for monetary sums - reserve the rightmost 2 columns for the decimals and make the 3rd column the "Units" column. Each column to the left of the units column is worth 10 times more than its right-hand neighbor. Each column to the right is worth 0.1 times its left-hand neighbor. So, for example, if you choose the third column to be the "Units" column then you would have the following values:
First Column: Hundredths
Second Column: Tenths
Third Column: Units (Ones)
Fourth Column: Tens Fifth
Column: Hundreds
Sixth Column: Thousands Etc.
(Bernazzani).

An everyday use for the abacus is adding numbers together as you would in a shop. The basic idea of addition is to enter one number into the abacus and then add the second number, column by column, to the values that are already in each column. Although you can work from left to right or right to left, it is more efficient to work right to left (move fewer beads). The only tricky part of addition is when you do not have enough unused beads in that column to represent the value you want to add. Subtraction is just the reverse of addition, with the same necessity of sometimes subtracting 10 and adding 1 in order to take away 9 (Dilson 43ff). and show the basis of addition and subtraction operations.

Simple multiplication (i.e. 5 x 7) is done as repeated addition. It is useful to keep a running tally of how many times you have added 7. The unused columns on the left hand side of the abacus can be used for this. Two digit or higher multiplication is done just as with pencil and paper. It requires memorizing the times tables or having a second abacus to do the simple multiplication. The units column is done first. The results of the multiplication of the unit of the bottom number and the tens of the top number are stored IN THE SECOND column. Keep in mind the process of pen and paper multiplication, and the abacus presents no real difficulty. Division is done as repeated subtraction - again keeping a tally of how many times you have subtracted (Dilson 121ff). Figures 4 and 5 illustrate these 2 processes.

If you would like a little challenge, try doing calculations in other number systems. The Chinese abacus is already useful for base 2 and base 16 calculations. Each rod on the Chinese abacus has 16 possible values because that was the unit of weight in China (Accent Interactive). Because of this, you can also use the Chinese abacus for hexadecimal calculation. And if you ignore the earth beads, you can use the heaven beads for binary calculations.

The best way to learn about the abacus is by doing. Get a stack of pennies or buttons, clear a spot on the kitchen table and try out some sums! And as you are busily calculating away, keep in mind that you are continuing a two thousand year tradition that is still alive and well today.

Figures

Figure 1: Subtraction of 7 from 15 on the counting table

Step One

------------------+---------------- Thousand
|
------------------+------------------
|
--------------O---+------------------
O | O
------------------+--O-O------------- Unit
value of 15 value of 7

Step Two

------------------+---------------- Thousand
|
------------------+------------------
|
------------------+------------------
O O | O
--------O-O-O-O-O-+--O-O------------- Unit
value of 15 value of 7

Step Three

------------------+---------------- Thousand
|
------------------+------------------
|
------------------+------------------
O O | O
--------O-O-O-O-O-+--O-O------------- Unit

Remove counters that cancel each other out

Step Four

------------------+---------------- Thousand
|
------------------+------------------
|
------------------+------------------
O |
------------O-O-O-+------------------ Unit

Final answer is 8

Figure 2: Adding 9 to 8 on the Chinese abacus

Step One

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | | | O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Begin with 8 in the units column

Step Two

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | | O O ||
|| O O O O O O O O O O O | O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Add 10 to the tens column

Step Three

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | | O O ||
|| O O O O O O O O O O O | O ||
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Subtract 1 from the units, giving answer of 17

Figure 3: Subtracting 9 from 27 on the Chinese abacus

Step One

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | | O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O | | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

enter value of 27

Step Two

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | | O O ||
|| O O O O O O O O O O O | O ||
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Subtract value of 10

Step Three

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | | O O ||
|| O O O O O O O O O O O | O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Add value of 1 for answer of 18

Figure 4: Multiplying 68 by 7 on the Chinese Abacus

Step One

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O | | ||
|| | | | | | | | | | | | O O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | | | O ||
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

8 x 7 so 56, so put 6 in units column and 5 in tens column

Step Two

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O | | ||
|| | | | | | | | | | | | O O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | O O O ||
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O | O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O | O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

7 x 6 is 42, so add 2 in tens column and 4 in hundreds column for final product of 476

Figure 5: Division of 15 by 5 on the Chinese abacus

Step One

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| | | | | | | | | | | | O | ||
|| O O O O O O O O O O O | O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Put 5 in units column and 10 in tens column

Step Two

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| | | | | | | | | | | | | | ||
|-------------------------------------------|
|-------------------------------------------|
|| O | | | | | | | | | | O | ||
|| | O O O O O O O O O O | O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Take 5 away, and add a marker bead on far left

Step Three

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O | ||
|| O O O O O O O O O O O O O ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| O | | | | | | | | | | | | ||
|| | O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Convert 10 to 5+5

Step Four

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O | ||
|| | | | | | | | | | | | | O ||
|-------------------------------------------|
|-------------------------------------------|
|| O | | | | | | | | | | | | ||
|| O O O O O O O O O O O O O ||
|| | O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Take 5 away, and add a marker bead on far left

Step four

+-------------------------------------------+
||-----------------------------------------||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| | | | | | | | | | | | | | ||
|-------------------------------------------|
|-------------------------------------------|
|| O | | | | | | | | | | | | ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| | O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
|| O O O O O O O O O O O O O ||
||-----------------------------------------||
+-------------------------------------------+

Take 5 away, and add a marker bead on far left for a remainder of 0 and a tally of 3

References

Accent Interactive. "Chinese Abacus"
http://www.accent.co.uk/chinabacus.html: 28 March 1999

Bernazzani, Dave, "Chinese Abacus and Japanese Soroban",
Dave Bernazzani's Home Page http://www.tiac.net/users/dber/abacus.htm: 28 March 1999.

Dilson, Jesse. The Abacus: a Pocket Computer.
New York: St. Martin's Press, 1968.

Feibao, Du. "Abacus, the Earliest Calculating Machine in the World", The China Experience http://www.chinavista.com/experience/abacus/abacus.html: 27 March 1999.

Fernandes, Luis. "Introduction", The Abacus, the Art of Calculating with Beads http://www.ee.ryerson.ca:8080/~elf/abacus: 28 March 1999

Grado, Victor M. "Nepohualtzitzin, A Mesoamerican Abacus" http://www.ironhorse.com/~nagual/abacus: 28 March 1999

Kojima, Takashi. The Japanese Abacus: Its Use and Theory. Tokyo: Charles E. Tuttle, 1954.

Leipala, Timo. "Russian Abacus (Schoty)", http://www.dotpoint.com/xnumber/pic_abacus3.htm: 2 April 1999

Moon, Parry. The Abacus: Its history; its design; its possibilities in the modern world. New York: Gordon and Breach Science, 1971.

Pullan, J.M. The History of the Abacus. London: Books That Matter, 1968.

Oswalt, Ed. "Ridiculous Observations In Russia" http://www.jao.com/ace/ruslist.html: 28 March 1999

Rentzsch H., G. Ottenbacher. "History of Computing" http://wwwstall.rz.fhtesslingen.de/studentisches/Computer_Geschichte/grp1/seite2.html: 28 March 1999