# I’m counting on it

Joe Celko looks at four different abacuses used throughout history

These days, abacuses might simply look like toys for teaching children to count, but before the invention of the calculator, there were few better ways to perform difficult calculations. Across the world, there have been many different designs for the abacus: let’s take a look at four of them.

## Roman abacus

Roman numerals were not designed for computation, but to record numbers after the computation. Computations were done with either tokens (often pebbles) on a board with lines on it or with an abacus. The use of counting boards in shops persisted well into the Renaissance: today, we still refer to a flat surfaces in shops as counters. If caught cheating customers in Renaissance Italy, a merchant’s counting board (or bank) was physically broken (ruptured) by the local authorities. This is where the word bankrupt comes from.

Long before the Renaissance, the ancient Romans used a metal sheet with grooves cut in it as a counting board. Small pebbles or beads sitting in the grooves represented numeric quantities. The Latin for pebble is calculi: this is the root of many mathematical words, such as calculus and calculation. Perhaps the most interesting feature of the Roman abacus is the part that handles fractions: this could represent any fraction built from twelfths.

## Chinese abacus

The Chinese abacus, or suan pan, has large beads, slightly rounded on the edges, strung on rods in a wooden frame. The wooden frame is divided into two sections horizontally. On the top, there are two beads on each rod. This section of the abacus is called heaven. The beads in heaven represent five units for that column. Each column represents a position in a decimal number. In the larger section below the centre dividing bar, we are on earth and each bead represents one unit in that decimal position.

When a bead is pushed toward the dividing bar, it counts as a value in that decimal position. The extra beads are used for carrying values: If I bring all of the beads on earth up to the centre bar, there is a 5 in that column. The five beads on earth should be returned to their starting positions, and replaced with a single bead in heaven. Likewise, if you have two beads in heaven, you should return them and replace them with a single bead on earth in the column to the left. This is basically mechanical scratch paper!

A column carries over when it exceeds the value of nine. This is a mechanical version of the same thing you do when you are adding columns of digits and put a mark by the column to the left of the one in which you are working.

You may have noticed that the extra beads for carrying mean that each column on a suan pan can hold up to a value of 15 units. This means that you can represent hexadecimal numbers. This was very useful when working with early digital computers, particularly those from IBM. The suan pan became a tool for computer nerds in the 1960s and 1970s who were looking at core dumps before it was possible to get octal and hexadecimal options on calculators.

Modern programmers seldom have to work that close to the hardware, so debugging core dumps with a suan pan is probably unknown to them.

In our standard decimal numbering system, we carry over into the next column once we exceed 9. In hexadecimal, we carry over once we exceed 15. Typically the digits 0 to 9 and the letters A to F are used to write the hexadecimal digits 0 to 15.

Every four bits (or nibble) of computer memory can store a value from 0 to 15, and so every nibble can be treated as a hexadecimal digit. The core dumps on early IBM computers essentially printed out all these hexadecimal digits.

## Japanese abacus

The Japanese abacus or soroban is generally smaller than the suan pan, but usually has more rods. The standard design is usually around 17 to 20 rods. The first thing to notice is that the beads have a different shape from the suan pan: they look more like two identical cones glued base to base. These sharp edges make moving the smaller beads easier: you can quickly slide the beads just by putting your finger between two of them. The extra rods also let you keep longer numbers and more of them when you’re working.

The next major difference is that heaven and earth do not look the same on the soroban as they do on the Chinese suan pan. At the start of the 20th century, the standard was to have one bead in heaven and five beads on earth: as in the suan pan, once five beads on earth are raised they should be replaced with one bead in heaven.

There are dots on the crossbar to let you know where decimal points occur when you keep more than one number for your calculation, such as a dividend and a divisor.

Later in the 20th century, the soroban configuration was changed to one bead in heaven and four beads on earth, so we could only represent the digits 0 to 9. There’s a totally untrue story about this change, which attributed it to the “great abacus bead shortage” of the second world war. The truth is the extra bead was never used, as students were already taught to do the carrying without the extra bead.

A more modern soroban

## Russian abacus

The Russian abacus or schoty is not so well known as the Japanese and Chinese versions of this tool, although versions of it appear in Armenia and Turkey, where it was known as a choreba and a coulba. Its real problem is that it was invented in the 17th century, and strictly meant for commerce, not as a general calculating tool.

The schoty has a frame with wires that hold beads, but they are arranged horizontally instead of vertically. Most of the wires hold ten beads, with the fifth and sixth beads being coloured. The rods are slightly curved, so the beads will fall to the right or left side of the wire. The beads are moved right to left. The first beads of the thousand and million wires are also highlighted for easier identification.

The frame is made of wood or metal with 11 wires holding wooden beads. There is no division within the frame, like the other versions we’ve seen.

The eighth row has only four beads on it, with two coloured beads in the centre. This allowed calculations to be performed using the quarter rouble coins that were in circulation. Some older models of schoty had two four-beaded wires (the first and eighth).

Joe is best known for his work with SQL and relational databases, but he was a mathematician first and loves weird recreational things.

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