Today's typecast is brought to you by the Optima Super, made in USSR-occupied Germany in the very early 1960s:

This device is of the so-called "2:5" configuration, meaning two beads above the bar and five below. Each bead above the bar has a value of 5 when moved down toward the bar; each bead below the bar has a value of one when moved up toward the bar. This implies each place value can hold values from zero to 15, more than our modern decimal place-value system requires.

The underside of the abacus, showing its Japanese-style construction:

This is the second "hybrid" Chinese/Japanese abacus in my collection with colored beads on metal rods:

*The new one is on the left, the old one on the right. The old one has a wooden plate on the bottom, steel rods and a sticker indicating it was made in Japan, whereas the new one has brass rods and lacks the plate. They're both 13 rods, enough for multiplication and division.*

Here's an example addition problem to show how the modern 1:4 Japanese soroban requires more abstract thought to operate. First the 1:4 version. The problem is 8 + 7.

*First we enter eight on the soroban, by lowering a five bead and raising three one beads.*

*To add seven, this requires performing a "tens complement" problem. We first subtract three, the tens complement of seven*

*Then we raise a single bead on the rod to the left, which has a value of ten. The result is fifteen*

Now let's do the same problem on the 2:5 abacus, this time we won't be using complementary arithematic, as it wasn't used in ancient times. Instead, when they had too high of a value on each rod, they simply "regrouped" the beads until it made sense.

*We enter eight as usual*

*Next, we enter seven directly on the same rods, no complementary steps required!*

*Next, we have to "rationalize" the result so they make sense in reading the values. First, the two five-beads are removed and replaced by a single ten-bead on the rod to the left*

*Continuing to rationalize the answer, we next take away the five one-beads and replace them with a single five-bead. The answer now easily reads as 15*

This process of rationalizing the result in order to conform to the place value system was used as far back as the ancient Greeks, with their counting board (literally

*abacus*in Greek). In their version, rather than beads on rods they used counters on a board marked with columns for the place values. Multiple numbers were "cast" upon the board, then regrouped and rationalized to form a readable result.

I've always fount the abacus fascinating. I've never seen one in real life anywhere.

ReplyDeleteI just noticed as I did not enter my name in the comment that I think yours is the only Blogger blog that does not post comments as anonymous with a Blogger/Gmail ID.

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