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