Using an Abacus/Basic concepts



When humans gathered in groups large enough for barter or trade operations to acquire some importance, the need arose for basic accounting which in turn required being able to count up to high numbers, perform basic arithmetic operations and keep a permanent record of transactions. Thus both arithmetic and writing seem to have a common origin in this need.

As for the basic arithmetic operations, “these seem to have been carried out universally using an abacus of some kind” and perhaps the first historical testimony of its use is found in the proto-cuneiform character: SANGA, which appeared as part of the signature of Sumerian scribes on clay tablets some 5000 years ago and which the Assyriologists identify with such a device.

What is an abacus?
An abacus is a tool or instrument in which numbers are physically represented in a way that allows them to be manipulated to mechanically simulate arithmetic operations.

In an abacus, the numbers are represented by "counters" or "tokens" (pebbles, seeds, shells, coins and the like, rods, etc.) to which a numerical value is assigned. The counters do not have to be all identical or have the same assigned value. To represent a number we arrange together the necessary counters on a table or any suitable surface in a similar way to how we would take a series of coins to reach a certain amount of money; it is the same process.

Addition is simulated by gathering the sets of counters representing the two addends, while subtraction is simulated by removing from the set of counters representing the minuend a set of counters representing the subtrahend. Consider the simplest case in which we only use identical counters with an assigned value of one.



In the image above we have arranged four counters of value one to represent the number 4 (left-a), after attaching another three counters (left-b) representing the number 3 we have a representation of the number 7 (left-c); that is, the sum 4 + 3. Similarly, if we start from the representation of the number 7 (right-a) and remove a set of counters that represent the number 4 (right-b), what remains on the table is three or the result of the subtraction: 7-4 ( right-c).

Please note that to perform the above operations it is not necessary to know anything about the addition or subtraction tables, in particular you do not need to know that 4 + 3 = 7 or 7 - 4 = 3, you only need to know how to manipulate the counters; on the contrary, it is the abacus that will allow you to "discover" that the result of 4 + 3 is 7 and that of 7 - 4 is 3! This is an essential point about the use of the abacus that we will return to in the chapter Addition and Subtraction.

It is commonly considered that in Arithmetic there are four fundamental operations: Addition, Subtraction, Multiplication and Division, and that any other calculation (e.g. obtaining a square root) ultimately reduces to a sequence of these four fundamental operations. But multiplication can be considered as a repeated addition in the same way that a division can be considered as a repeated subtraction, so that any arithmetic calculation is ultimately reduced to a sequence of addition and subtraction (and, going further from a modern perspective, addition and subtraction are just two aspects of the same additive composition law of numbers). Therefore, with an abacus, any arithmetic calculation can be performed in principle. But this would be extremely difficult or perhaps impossible without a few refinements to our rudimentary abacus.

With the abacus used above (only identical counters with assigned value one), it is evident that if we start working with progressively larger numbers, our table (abacus) will be cluttered with counters, making their use and interpretation impractical. We need a way to reduce the number of physical objects, counters, to manipulate and keep it within some limits that are comfortable for us. There are a couple of solutions:


 * Using physically different counters with different assigned values. It is the most primitive system, already used by the Sumerians more than 5000 years ago ... and still used today since the use of coins and banknotes of different nominal values in any current monetary system corresponds perfectly with this concept of abacus.
 * Define spatial regions in our table (abacus) so that a counter represents one value or another according to the region it occupies.



Let's look at an example. In the figure above we have added 7 + 7 (a and b) with our primitive abacus, and 14, the result, is shown as a cluttered table full of counters (c). We can replace some of these counters with a physically different one that has a higher value assigned, for example 10 (the replacement value). With this, the state of our abacus is easier to interpret (d), it has been simplified as 10 1-counters have been replaced by only one 10-counter.

Alternatively, we can consider the abacus divided into two spatial regions and use identical counters to which we will associate one value or another according to the region in which we place it. At (e) in the figure above, the abacus has been divided into two regions, left and right separated by the double vertical line. If we assign a value of one to the counters located on the right and 10 to the ones located on the left, the number 14 would be represented as illustrated. This way of proceeding is preferable to the previous one since we can repeat the process, defining as many regions as we need with the replacement values that suit us, allowing us to represent arbitrarily large numbers with counters of a single type, for example, in (f) we have depicted 114 using three regions and two replacement values of 10; we only needed 6 counters. We are witnessing here the birth of Positional notation.

Before continuing, it is convenient to indicate that there are two main types of abacos:


 * Free-counters or Table Abacus: The counters are independent and are normally kept apart in a box or bag and are placed or removed from the table as needed. It is the most primitive type and the one that we have considered here so far.
 * Fixed-Bead Abacus: The counters, called beads in this context, are always present, integrated into a frame and can be slid from an inactive position to an active one along grooves, rails, strings, wires or rods. This is the most sophisticated, portable, compact type that allows a faster calculation and, as we will see, the eastern abacus to which this book is dedicated is of this type.

Now we can mention the Russian Abacus (Schoty), the Iranian Abacus (Chortkeh) and the School Abacus as examples of fixed bead abacos that conform to what we have explained so far. Both consist of a wooden frame with horizontally arranged wires along which ten beads are strung in the case of the Russian abacus and nine in the Iranian abacus. The beads can slide from an inactive position (right) to an activated position (left) and each wire represents one of the regions mentioned above, with a replacement value of 10, so that a bead in each of the wires has a associated value ten times higher than that of the beads on the immediately lower wire.



These abaci have everything necessary to allow arithmetic operations with numbers expressed in decimal notation: several rods to represent successive powers of ten and 9 beads to represent the digits from 0 to 9 (for convenience, the Russian abacus has one more bead than is strictly necessary). You can try a Russian abacus model  at this link.

But we still need one last refinement to fully understand the East Asian abacus.

Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of items. We can make such a judgment if the number of objects to be counted is 4 or 5 at most; from there, we will have to invest time in counting. In the Russian and Iranian abacos we have 9 or 10 beads per rod, so the reading of the number represented may be beyond the subitization limit. This is alleviated by using beads of two different colors as illustrated in the previous images, but there are also a couple of additional techniques that not only allow us to stay within the limits of subitization but also reduce the number of beads needed in the abacus.



In the upper image (a) we have the number 18 represented in two regions (rods); one of them contains 8 counters that are above the subitization limit. To simplify the reading of the abacus we can:


 * Use a different type of counter with a replacement value of, for example, five (b).
 * Subdivide the regions or rods into two zones: one in which a counter takes the value one and the other in which it takes the value five (c,d).

In either case, we do not need to have more than four identical counters per region to be able to represent numbers in decimal notation, so we are guaranteed the fast reading of the abacus. When using 5 as the second replacement value, we are using a bi-quinary decimal notation for numbers. Examples of both solutions are counting rods and the eastern abacus.

Counting rods
Counting rods were a table type abacus or free counters abacus in which the counters were small rods of wood, bamboo, bone, etc. that were arranged on a flat surface, using a checkerboard or not. By the way, this abacus that dominated Chinese mathematics for at least 14 centuries and Japanese mathematics (Wasan) until the Meiji restoration, is probably the most versatile that has ever existed, although unfortunately it is also very slow to handle.



In the figure above (a) we use vertically arranged rods as counters of value one to represent the number 18. In (b) we use a horizontally arranged rod as a counter of value five and in (c) we use a more compact rod arrangement with alternation of orientation or not depending on whether we use a smooth table or a checkerboard (see details in Wikipedia). Digits from 1 to 9 are represented as:



Zero was represented by an empty cell on the checkerboard or by a space or other object (for example, a Go token) on the table. For example, the number 1547 would be expressed:



It is interesting to mention that this is the only abacus that is known to use the orientation of the counters to assign them one value or another; but we find a parallel to this concept, if not a precedent, many centuries before the appearance of counting rods in the Babylonian numerals used to write numbers in sexagecimal notation. Each sexagesimal digit was constituted by a series of vertical impressions of the edge of a reed stylus on the fresh clay tablet with unit value (,, , , ..., ), and impressions made with the stylus turned 45 degrees or more in an anticlockwise direction of value 10. The Decimal number 1547 is expressed in sexagesimal in the form 25:47, where "25" and "47" are two sexagesimal digits written as: and



The very appearance of these digits suggests their immediate representation in a table abacus using counting rods.

The Eastern Abacus
The second solution is the one adopted both in the Roman abacus and the abacuses that appeared in China.



While a few examples of Roman abacus such as the one in the figure are known, where the beads slide along grooves, nothing is known for certain about the origin of the eastern abacus. A confusing phrase from the Shushu Jiyi (術數紀遺) of Xu Yue (徐岳), which perhaps dates from the 2nd century, is often quoted as describing a computing device that we could identify with an abacus and that has been interpreted in different ways as in the above figure (a). In this interpretation of a first Chinese abacus as a table abacus, the central part is divided into a series of columns with two parts; the upper one would assign a value of 5 to each bead and the lower one a value of 1, while the inactive (unused) beads wait scattered above and below the central part.

It is unknown when the abacus of beads strung along rods appeared, but when this abacus replaced the use of counting rods throughout the 16th century it did not have four lower beads and one upper one like the Roman abacus (we will refer to this disposition as a 4 + 1 type abacus) but five in the lower part and two in the upper part (5 + 2 type abacus), separated by a horizontal beam. The additional beads, not necessary for the calculation with decimal numbers, were introduced for convenience to adapt the calculation algorithms that had been developed with the counting rods to the abacus. Historically, the four types of abacus described in the figure below have been used.



Symbolically, the upper and lower areas of the abacus have been designated Heaven (天, Tiān in Chinese, Ten in Japanese) and Earth (地, De in Chinese, Chi in Japanese).

In this book we will focus on the use of the 4 + 1 type abacus or modern abacus, following what we will call the modern abacus method. If you have understood the principles on which any abacus is based and you learn to use the modern abacus, you will have no difficulty in imagining how you can use any other type of abacus, at least for elementary operations of addition and subtraction. This could include, why not? the following abacus for sexagesimal calculations conjectured by Woods as the Babylonian abacus based on what we know about mathematics in Mesopotamia ... and the mistakes the scribes made!



Finally, if after achieving some experience with the modern abacus following this book you want to know about the traditional techniques and the use of the 5 + 2 abacus, you can continue with the book: Traditional Abacus and Bead Arithmetic.

Soroban Trainer
If you are interested in trying the abacus but do not have an abacus yet, you can use the JavaScript application: Soroban Trainer


 * You can run it directly from GitHub in your browser
 * or you can download it to your computer from the repository on GitHub.

It may be used as an 4+1, 5+1 or 5+2 type abacus.