# The Abacus

How to Use an Abacus

Before I begin describing how to use an abacus, I ought to specify that there really is not just one, quintessential abacus.  In general, there are three types of abaci: dust, counter, and frame.  (These categories are suggested in passing by Gardner, Smith, and Williams, but I would like to establish them formally in my final paper.)  The dust abacus was a dustboard (defined in post) with columns for place value marked, and Hindu numerals (excluding zero) written in these columns (Ifrah, 2000).  This method is only a few steps from the modern pen-and-paper method.

Counter and frame abaci, on the other hand, are more in line with what I initially considered “abaci”.  The counter abacus (also called counting board, counting table, calculating board, calculating table, line abacus, or exchequer) is composed of a table (material irrelevant) into which lines are carved or drawn.  Loose counters (or calculi, Latin for “pebbles”) of any form are manipulated on the place-value lines to perform arithmetic.  The counters in frame abaci are attached to the device, either as beads on rods or as pieces in grooves.  Arithmetic operations on frame abaci are performed using the same principles as counter abaci.  To give examples of addition and subtraction using these principles, I will employ diagrams of arithmetic on a Japanese abacus (soroban) from Moon’s treatise on the history and operation of the abacus (1978).  Note that beads above the “crossbar” are equal to five beads below the crossbar, which are worth one unit in their respective place values.  Beads touching the crossbar or touching other beads touching the crossbar are read for the number they represent.

Figure 2.13

) is performed by setting on the abacus the largest number of the two being combined.  In Figure 2.13a, the number 7248 is set up.  In order to add 3517 to 7248, first note that 7 (from the units place) = 10 – 3.  So, add 10 to 7248 (that is, add 1 in the tens place), then subtract 3 from the ones place (Figure 2.13b).  Now, we’re adding 3510 to 7255, so add 1 to the tens place (Figure 2.13c).  Then 5 to the hundreds place (Figure 2.13d), which leaves us with 7765 + 3000.  In the final step (Figure 2.13e), 3 is added to the thousands place using 3 = 10 – 7 because 3 + 7 carries.

Figure 2.14

Subtraction (Figure 2.14) is the same idea as addition, just in reverse.

Figure 2.15

In Figure 2.15, 73 is multiplied by 47.  First the partial product of 3 and 7, the two numbers in the digits place are multiplied and the product, 21, set on the abacus (Figure 2.15a).  The partial product of 7 (from 73) and 7 (from 47) is added to 21, adjusting to the fact that one seven is from the tens place by treating  as 490 (Figure 2.15b).  The third partial product, 3 (from 73) and 4 (from 47), is added to the number on the abacus in the same way; that is, skipping the ones place (Figure 2.15c).  Finally, the partial product of the 7 (from 73) and 4 (from 47) is added, ignoring both the ones and the tens places in order to correct for the fact that both multicands came from tens places (Figure 2.15d).

Division is a more complex process, especially on a soroban.  The Chinese suan-pan has a second bead above the crossbar, which simplifies things.  Nonetheless, for my sake and for the sake of time/space, I will forbear to explain the process.  For those who are still curious, division on the abacus is related to the galley method.

References

[Author’s note: This post is lacking in citations and references because I have been separated from my books.]

Gardner, M. (1970). MATHEMATICAL GAMES. Scientific American, 222(1), 124-127. Retrieved from EBSCOhost.

Ifrah, G. (2000). The universal history of numbers: From prehistory to the invention of the computer (D. Bellos, E. F. Harding, S. Wood, & I. Monk, Trans.). New York, NY: John Wiley & Sons.

Smith, D. E. (1958). History of mathematics.  New York, NY: Dover Publications. (Original work published in 1925)

Williams, M. R. (1997). A history of computing technology. [Incomplete bibliography entry.]

This is fascinating stuff! I’ve always wondered how an abacus works. The method of executing the math reminds me of Incan quipus, although they both have their nuances.

2. sehartzell says:

And I barely understand how to use my calculator!

Random Thoughts:

It must be an interesting experience to research the history of a body of knowledge. I feel a sense of disconnect when I look at all of the “progress” we’ve achieved as far as science and technology go, then consider the original thinkers who came up with ideas, abaci for example. And yet, we could still add numbers more quickly on a TI-84. Are we collectively more knowledgeable, but individually less inventive? Is it easier now that researchers have more of a knowledge base to to build off of, or are we still facing problems that will require a level of ingenuity on the level of, “hey, i’m going to invent math!”?

3. Michael says:

I think the Japanese abacus we have been keeping in my family’s house was meant for this moment! I had to get it out and follow along, and it finally all makes sense! Thank you for enlightening me and giving purpose to this ancient house decoration with hidden power! And the pictures were very helpful.

4. dwood says:

Sharon, I’m so glad you commented: I feel the same disconnect you mention, and your questions resonate with ones I ask myself frequently in the course of my research. Despite my extensive contemplation on the subject of present-day math versus that in the past, I’m afraid I have no answers for you. At the moment, I think I am inclined to believe that due to the extensive knowledge base we have accumulated over centuries, we each individually know a medium amount about a large number of subjects, whereas in the past, some people knew a great amount about a small to medium number of subjects and some knew very little about a few subjects. This results in modern-day prodigies being at something of a disadvantage compared to prodigies of the past. Furthermore, in math at least, our understanding has reached a point where it is necessary for people to work in groups and to work with computers. Group-work inspires progress by bringing more intellect in contact with each other, but it also inhibits it by the typical inefficiencies of groups. By working with technology, researchers are able to do more complicated computations than ever before in relatively short stretches of time, but I think it also enables people to proceed with their research when they do not have as thorough an understanding of future steps as might be helpful.

So…I don’t know. But it seems that I just threw some of my thoughts at you. I hope they made sense. I think your questions would make for a really great discussion topic at a Monroe get-together; I’d love to hear other people’s opinions on it, especially yours.

5. dwood says:

Thanks! And I’m glad I could enlighten you. I am curious about how the method of calculation on an abacus reminds you of Incan quipus? My research included investigating quipus and the Incan abacus, and I was not hit with any obvious similarities. I enjoy connecting these things in my head–I find it helps me understand the underlying mathematical principles–so I would love to hear about your impressions.

6. dwood says:

In my opinion, the ONLY way to learn how to perform arithmetic on an abacus is with an abacus right in front of you. I struggled from Day 1 of my research to figure out the abacus because I had to do it all in my head, and I’m still a bit fuzzy, especially multiplication and division. (To be honest, I think I will do my best to having to learn division on the abacus; I’m not convinced it’s worth trying.) I’m thrilled you had one available to you, and it’s nice to hear that I could help open its potential to you!