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Mark C. Chu-Carroll at the Good Math, Bad Math blog is writing a series entries on manual computing devices. He's done with the slide rules and on to the abacus.

As someone who grew up with the abacus, I think I can add some real world feel to Mark's rather dry and algorithmic depiction of abacus arithmetic.

I learned the abacus before I started elementary school, when I was about six or seven. The illiterate nanny of the family in the downstairs apartment taught me the abacus. (You should see the books she kept for the household purchases—it's all pictures and numbers. A milk bottle represents milk, a pig's face is pork, a babies head is a haircut for a child.)

Any way, working the abacus is like playing a music instrument. A normal abacus has thirteen vertical rods, each representing a digit. Each rod is divided into upper and lower decks by a beam. The upper deck houses two beads that can be moved towards or away from the beam. Tow lower deck houses five.

The abacus is a hybrid binary-quintary with per bit carry device that simulates a decimal system. It is normally used as a big-endian device—the left most rod is most significant. For each individual rod, the lower five beads form a quintary digit with carry. It can represent 0 (no beads up to the beam), 1 (one bead to the beam), 2 (two beads), 3 (three beads), and 4 (four beads). The last bead in the lower deck is the carry bit. When it is up (five beads up) all beads should be pushed down (off) and a bead in the upper deck should be pushed down to the beam. The upper two beads form a binary bit with carry. It can represent 0 (no beads down to the beam) and 5 (one bead down to the beam). The other bead in the upper deck is again the carry bit. When it is down (two beads down to the beam), both beads should be pushed up (off) and a bead in the lower deck of the rod to the left of the current rod should be pushed to the beam.

Thus the greatest number that can be represented on each rod without the help of the carry beads is 9, forming a simulated decimal system. However, in a pinch, the carry beads can be used to represent numbers up to 15.

The right hand is used to play the abacus. The thumb and the index finger are used to control the lower beads. The middle finger is used to control the upper beads.

The actual arithmetic operations are carried out by following a set of rhymes that's easy to remember.

Addition

The rhymes for addition is intuitive:

one up one
one down five off four
one off nine carry one

two up two
two down five off three
two off eight carry one

three up three
three down five off two
three off seven carry one

four up four
four down five off one
four off six carry one

five up five
five off five carry one

six up six
six off four carry one
six up one off five carry one

seven up seven
seven off three carry one
seven up two off five carry one

eight up eight
eight off two carry one
eight up three off five carry one

nine up nine
nine off one carry one
nine up one off five carry one

The first number of each line is the addend. The rest of each line is the action to be carried out. Only one line from each group is applied. Which one is selected depends on the addend already present on the rod.

The "n up n" line is used when the addition results in no carry (e.g., 2+2=4, 2+7=9).

The "n down five off (5-n)" line is used when the addition results in the carry bit in the lower deck (e.g, 4+4=8: with four beads up in the lower deck, to add another four, you push down a five in the upper deck and push off one in the lower deck, resulting in five+three, i.e. eight.)

The "n off (10-n) carry one" line is used when the addition results in the carry bit in the lower deck, which when resolved results in the carry bit in the upper deck (e.g, 7+4=11, 9+4=13.)

The "n up (n-5) off five carry one" line is used when the addition results in the carry bit in the upper deck (e.g., 6+7=13: with one bead up in the lower deck and one bead down in the upper deck, to add another seven, you push up two in the lower deck, push off the upper bead and push up a lower deck bead on the rod to the left.)

It takes about fifteen minutes for a six year old to remember the addition rhymes. And with practice, addition of multiple digit numbers is really child's play. Addition can be carried out either from left to right or from right to left.