How to Use an Abacus

Monica Abacus.png

Pictured: The Answer.

Abacuses (yes, that’s all right, and so is “abaci”) have been used in various forms for thousands of years. The full history of early counting machines and a complete guide to everything you can do on an abacus is way beyond the scope of a single blog post. Today I’ll give you a quick overview of different types of abacus and a few tricks you can do with one, if that’s your chosen method of showing off at parties.

The earliest evidence of abacus-like devices comes from Mesopotamia in about 2700 BC, where we know they divided numbers into columns based on orders of magnitude (ones, tens, hundreds, and so on for those of us working in base ten). This early form was probably a board covered with sand or dust in which lines were drawn, hence the word abacus, from the Latin form of Greek abax, probably from a Semitic word for “dust.”

Speaking of Greek, the Egyptians, Persians, and Greeks were all probably using some form of abacus by the last several centuries BC. Based on our knowledge of finds like the Salamis tablet, at least the Greek ones were counting tablets rather than abacuses as we think of them. These tablets used grooves or lines and removable counters instead of beads fixed on wires; the Roman version was similar and continued to be used in medieval Europe and as recently as the nineteenth century.

Salaminische_Tafel_Salamis_Tablet_nach_Wilhelm_Kubitschek_Numismatische_Zeitschrift_Bd_31_Wien_1899_p._394_ff

The Salamis tablet, c. 300 BC. Photo by Wilhelm Kubitschek, public domain courtesy of Wikimedia Commons.

I mentioned beads and wires as a contrast to the Greek and Roman tablets because you’re most likely to be familiar with this second type of abacus. The best-known form of bead-and-wire abacus, called a suanpan, was developed in China around 200 BC and later exported to Japan, where it’s called the soroban. Both abaci (yes, I will switch plurals on a whim) consist of rectangular frames containing beads on vertical rods, with a reckoning bar dividing the lower and upper beads, also called the earth beads and heaven beads. The soroban has four earth beads and one heaven bead, simplified from the suanpan, which has five earth beads and two heaven beads.

Abacus Parts

Beads are counted by moving them toward the reckoning bar (up for earth beads, down for heaven beads). Each earth bead represents one unit of whatever column you’re in, and each heaven bead represents five units. The columns move from right to left just like we write numbers in the West, so the labeled suanpan above has a one in the ones column, a nine in the tens column (one heaven bead and four earth beads), and a three in the hundreds column.

The easiest thing you can do with an abacus (aside from using it as a maraca substitute) is addition and subtraction. This works the same way as carrying and borrowing, or whatever you call the process of moving your tens, hundreds, etc., over to the appropriate column. Here’s an example, using the Chinese suanpan. (Try it out yourself with this virtual suanpan!)

Abacus Addition.png

Another option is multiplication/division, which is slightly more complicated because traditional technique involves writing over or erasing one of your numbers as you go. (It’s also worth noting that the abacus is mostly a way to keep track of what you’re doing, and won’t save you from having to remember your times tables unless you want to count out groups of numbers by hand. Which you could do, I suppose, it’ll just take a while.) Here’s how you multiply two-digit numbers on a suanpan:

Abacus Multiplication.png

You can see this would take a little getting used to, but it’s really not any different from using scratch paper. Use an abacus and save the trees!

There are a few more cool things you can do with an abacus, like finding square and cube roots, but just the process of finding square roots would be a blog post on its own. (I may do that, actually…) Also, you may have noticed that the suanpan has enough beads on each column to go up to fifteen, if you use both heaven beads (two fives) and all earth beads (five ones). That’s because traditional Chinese weight measurements were in base 16, making the suanpan perfect for all your hexadecimal calculation needs!

You can get some more information and practice with this guide to the Japanese soroban—it goes into some things I haven’t mentioned in this post. For example, it recommends you work left to right for everything, which doesn’t make a lot of sense when you’re carrying numbers on paper but works just fine when all you have to do is add more beads.

More practice with the suanpan, and of course the virtual suanpan to try everything yourself if you’ve misplaced your physical abacus.

Read more about the history of counting boards and abaci here or here, or of course on Wikipedia. This article compares Chinese abacus organization to what some scholars believe to be a Mesoamerican abacus, which is pretty interesting. And we didn’t even get into khipu!

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3 thoughts on “How to Use an Abacus

  1. Pingback: Cool Math Tricks: Finding Square Roots | Something of the Marvelous

  2. Pingback: GPS Alternatives: The South-Pointing Chariot | Something of the Marvelous

  3. Pingback: Why Slide Rules Rule | Something of the Marvelous

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