Abacus

The Chinese abacus has 13 wires on a wooden frame, with a wooden divider not quite down the middle of all the wires. Each wire has seven beads, five on one side of the division (the "earth" beads, worth one each) and two on the other (the "heaven" beads, each of which is worth five.)

The modern Japanese abacus has only one bead above the divider and four below, and usually has 21 wires.

To represent a number on the abacus, beads are moved toward the divider. The rightmost wire represents the ones, the next one over the tens; the third from the right stands for hundreds, and so on. For example, here's a diagram of a Chinese abacus showing 54,927:

heaven beads
-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |	
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O | | |
| O O O O O O O O O O O O O |
| 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

Note that the beads furthest away from the divider don't need to be used to represent base 10 numbers on the Chinese abacus; that's why they were able to be removed in the Japanese abacus. The extra beads help, though, when you are doing math. Also, you can use the Chinese abacus to represent (and do calculations with) numbers other than base 10. E.g., you can do binary calculations by working just with the heaven beads. You can also slide the decimal point around, and do arithmetic with fractions if you so desire.

Addition

To add numbers with an abacus, enter the first number. Suppose we wanted to calculate 54,927 + 18. Start with the bigger number:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |	
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O | | |
| O O O O O O O O O O O O O |
| O O O O O O O O O | | O O |
| O O O O O O O O O O O O O |
-----------------------------

Then go column by column, adding the second number to the first. It's best to work from the right to the left. There aren't enough beads in the ones column to add 8 to 54,927, so instead add one to the tens ...

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |	
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O | |
| O O O O O O O O O O O | O |
| O O O O O O O O O | | O O |
| O O O O O O O O O O O O O |
-----------------------------

... and take away two from the ones:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |	
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O | O |
| O O O O O O O O O | | O O |
| O O O O O O O O O O O O O |
-----------------------------

Now, add one to the tens column (the "1" in "18"):

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |	
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O | | | O |
| O O O O O O O O O O O O O |
-----------------------------

... and we are left with 54,945.

Subtraction

Subtraction is just like addition, but in reverse. To calculate 54,945 - 290, start with 54,945:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |	
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O | | | O |
| O O O O O O O O O O O O O |
-----------------------------

Subtracting 0 is a no-op; to subtract 90, first subtract 100 ...

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |	
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O | O O |
| O O O O O O O O O | O | O |
| O O O O O O O O O O O O O |
-----------------------------

Then add 10:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | | | |	
| | | | | | | | | O | O O O |
-----------------------------
| | | | | | | | | | O O | | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O | O O |
| O O O O O O O O O | O O O |
| O O O O O O O O O O O O O |
-----------------------------

Now, subtract 200 to get the answer (54,655)

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | | | |	
| | | | | | | | | O | O O O |
-----------------------------
| | | | | | | | | | O O | | |
| O O O O O O O O O O | O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O | O O O |
| O O O O O O O O O O O O O |
-----------------------------

Multiplication

To perform multiplication with an abacus, basically you perform addition repeatedly. In order to keep track of how many times you have added the number, you can use one of the unused leftmost columns of beads. This works well with smaller numbers. If you want to multiply numbers that have multiple digits, then memorize your multiplication table and do it like you would on paper. E.g., 37 times 23 would be 21 + 90 + 140 + 600.

Division

Division can be performed as repeated subtraction, using one of the leftmost columns to keep track of how many times the subtraction has been performed. You have to decide before you start how many significant digits will be allowed in your answer. For example, to compute 19/17 to the nearest hundredth, declare the ones column to be third from the left, with tenths and hundredths in the right two columns. Enter 19:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O O O | O O |	
| | | | | | | | | | | O | | |
-----------------------------
| | | | | | | | | | O O | | |
| O O O O O O O O O | O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O 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 17 and increment the far left column as a counter:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |	
| | | | | | | | | | | | | | |
-----------------------------
| O | | | | | | | | | O | | |
| | O O O O O O O O O O O O |
| O O O O O O O O O O | O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
-----------------------------

We are left with 2; you can't subtract 17 from 2 again. But, we could imagine 2 is 20 by using the tenths column. Do this, and instead of using the leftmost column as your counter, use the next one over:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |	
| | | | | | | | | | | | | | |
-----------------------------
| O O | | | | | | | | | O | |
| | | O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O | O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
-----------------------------

We are left with 3 in the tenths column; you can't subtract 17 from 3, so scoot over again, and imagine the hundredths are the ones and the tenths are the tens. Subtract 17, and use the THIRD column at the left as your counter:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |	
| | | | | | | | | | | | | | |
-----------------------------
| O O O | | | | | | | | O O |
| | | | O O O O O O O O | O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
-----------------------------

We are left with 13, and you can't subtract 17 from 13, but 13 is more than half of 17, so we can probably round up the last digit of our answer. Reading the counters on the left, we come up with an answer of 1.11, rounded up to 1.12.

The analogy to normal long division with pencil and paper should be evident.

Roman Hand Abacus

The Roman hand abacus predated the Chinese and Japanese versions as described in the Wikipedia Abacus page. Other examples include this at the the London Science Museum, and probably the best example at the Online Museum.

This latter version clearly shows the seven leftmost columns for integer calculations in a decimal place value layout, from ones, then tens, hundreds and all the way up to millions. Each bead in the lower slot counted as one of that power while the bead in the top slot denotes five of that power. This clearly shows that the Romans were conversant with a decimally based place value system. The separation into values of one in the bottom slots and five times that value in the upper slots allowed for very easy translation of values from the Roman written numbers with their I and V for 1 and 5, X and L for 10 and 50, C and D for 100 and 500 and so on for the other powers, onto the abacus itself. Calculations could then be performed on the abacus and the result easily translated back to the Roman written format. It is unfortunate that they did not make the mental leap to a decimally based written place value format.

The second column from the right has 5 beads in the lower slot and 1 in the top. This column was a duodecimal based form of fractions, where each bead in the lower section was worth 1/12 and the bead in the top had a value of 6/12. This fitted with the usual Roman usage of duodecimal fractions for which they had names for 11/12, 10/12 down to 1/12 (and also a name for 1/8 which was one and a half twelfths), and duodecimal subdivisions of 1/12 down to 1/2304 (1/16 x 1/144).

The first column was used for fractions of 1/12. In some versions there was a single slot with three markings and in others three separate slots with the same markings, one to each slot. The top slot was for 1/2 of 1/12 and the middle slot for 1/4 of 1/12, while most current documents state the lower slot to be for thirds of 1/12. This is supported by the arrangement of beads with one bead in the top and middle slots each, and two beads in the lower slot.

There is however an alternative suggestion that seems to provide greater sense and regularity which claims the beads in the lower slot represent twelfths of a twelfth (1/12 x 1/12). This is supported by two strong pieces of evidence, that of the logical progression of values allowed only if the value of the beads in the lower slot are twelfths of a twelfth, and by the evidence from the tables of Gottfried Friedlein that states the symbol that resembles a digit 2 on the abacus denotes a value of 1/72 or 1/6 x 1/12. This can only be true if each bead has a value of 1/12 of 1/12 and the two together sum to 2/12 for a result of 2/12 or 1/6 x 1/12. This is described in full detail in the Wikipedia entry for the Roman Abacus.

The abacus is a mechanical aid for counting and doing arithmetic. It is wrong to think of the abacus as a calculating device like an electronic calculator or a slide rule, because is not a machine and it performs no calculations itself. It is really just a tally device that stores the results of a series of small computations made mentally by the user.

The concept

An abacus has a number of columns that represent place values in a number system. For a decimal system, if a particular column may represents ones, the column to its left represents tens, the column to its right represents tenths, and so on. Each column has beads that slide up and down on a thin rod and is divided into an upper part and a lower part by a bar. The positions of those beads represent a numerical value for that column.

The modern abacus

The abacus is not an obsolete antique; it is still being used actively in China, Japan, and in other countries that have been strongly influenced by Chinese culture. It is not unusual to see an abacus being used at counters in shops, with a cash register or computer in hand's reach. Elementary school children in those countries learn the abacus as part of their basic math curriculum.

The most common configuration for a modern abacus has between 10 and 30 or so columns. Each column has four beads in the lower part and one bead in the upper part. An older configuration that has five beads below and two beads above can still be seen, but with rapidly diminishing frequency. The rods and beads that form the columns are held together by a rectangular frame made of wood or plastic. The thin bar separating the upper and lower parts (which are called 'heaven' and ' earth' in Japanese) has dots on every third column that can serve as the decimal points or as commas do in written numerals to show which place values the columns represent and make the result easier to read. Size varies, especially with the number of columns, but most modern abacuses are small enough to hold in one hand and operated with the other.

Evolution

The abacus evolved from the earliest forms of tallying or counting. According to the American Heritage Dictionary, the word abacus is rooted in the Hebrew term abaq, which referred to 'sand used for drawing and counting'. Later, there was the classical Greek abax, and then the Latin abacus, which was absorbed into Middle English and handed down to us without change.

Sand was eventually replaced by pebbles as a counting medium, and people began placing the pebbles in columns for counting. Later improvements include the use of grooved boards to hold the pebbles or beads. It is difficult to date the counting board, because they were made of wood or other non-durable materials, but the oldest surviving artifact is the Salamis tablet, which was used in Babylonia some three centuries BC. The Salamis tablet is a marble slab inscribed with eleven vertical lines that enclose 10 columns. A horizontal line crosses through the vertical lines about half way from top to bottom. This is the basic configuration of the abacus, but without the beads.

The hand abacus appeared sometime in the later Roman or Greek era before 500 AD. That abacus had five columns, wth four beads on the bottom and one on the top. In the Middle Ages, the abacus used in Europe had a horizontal orientation of rows rather than columns. Later, a ten-bead abacus came into use, particularly in Russia.

In China, the abacus appears in paintings dated in the Song Dynasty, and by the time of the Ming Dynasty, the abacus was the common way of doing arithmetic. During that period, Chinese merchants spread the abacus throughout the Far East. An abacus made of corn kernels strung on strings was also used by the Aztecs around 900 to 1000 AD.

The Chinese abacus featured five beads below the bar and two above. That 5/2 configuration was suited to performing hexadecimal calculations, which was handy because of the base 16 system of weights used in the earlier days. That has now been replaced almost entirely by the 4/1 configuration, which is the most efficient for decimal system arithmetic. The abacus can also be used for the duodecimal or any other number system by simply changing the number of beads in a column.

In education

Aside from its practical value, the abacus offers important benefits as a tool for introducing the concepts of number, counting, and arithmetic operations in early education. Rather than doing the work and simply providing an answer for the student as a calculator does, the abacus makes it easier for them to build up mental computation habits a step at a time. Using an abacus also reinforces the concepts with with multi-sensory visual and physical feedback as students manipulate the beads. Proficient users can sometimes be seen calculating by doing the physical motions without even holding an abacus. Studies have shown that children who learn on the abacus can do mental calculations earlier and faster than students who learn with paper and pencil.

---

The Abacus

Ab"a*cus (#), n.>; E. pl. Abacuses ; L. pl. Abaci (#). [L. abacus, abax, Gr. ]

1.

A table or tray strewn with sand, anciently used for drawing, calculating, etc.

[Obs.]

2.

A calculating table or frame; an instrument for performing arithmetical calculations by balls sliding on wires, or counters in grooves, the lowest line representing units, the second line, tens, etc. It is still employed in China.

3. Arch. (a)

The uppermost member or division of the capital of a column, immediately under the architrave. See Column.

(b)

A tablet, panel, or compartment in ornamented or mosaic work.

4.

A board, tray, or table, divided into perforated compartments, for holding cups, bottles, or the like; a kind of cupboard, buffet, or sideboard.

Abacus harmonicus Mus., an ancient diagram showing the structure and disposition of the keys of an instrument.

Crabb.

 

© Webster 1913.