30 ms·
Both doubling and halving on a base ten counting board (e.g. of the style commonly used in Europe up until at least the 16th century; Roman counting boards were
by jacobolus 10y ago
Both doubling and halving on a base ten counting board (e.g. of the style commonly used in Europe up until at least the 16th century; Roman counting boards were presumably recognizably similar, though not much physical evidence remains) are very fast and easy.
To halve any even group of pebbles, just remove half of them. To halve a lone pebble resting on a line (representing some power of ten), just move it downward into the adjacent space. To halve a number in a space (representing 5 times a power of ten), replace it by 2 pebbles on the line below, and one pebble in the space below that. These operations can be done very quickly and fluidly with a bit of practice. Maybe not quite as fast as operating a soroban, but not inordinately much slower either.
Historically nobody performed arithmetic calculations by writing down a bunch of intermediate steps on scratch paper using Roman numerals (for one thing, paper wasn’t available in the time of the Romans and both papyrus and parchment were expensive). Roman numerals are just a way of recording the finished answers to calculations done on an abacus (counting board).
- chopin 10y ago>Historically nobody performed arithmetic calculations by writing down a bunch of intermediate steps on scratch paper When I was in elementary school we used a slate and chalk. I think this was quite common when paper wasn't abundant.
- jacobolus 10y agoFrom what I understand ancient Roman schoolchildren did most of their writing on wax tablets. It’s not clear when writing slates were first used; one of the earliest known examples is apparently 11th century India, though I’m not sure if they used chalk or some other writing material. See https://en.wikipedia.org/wiki/Slate_(writing) https://en.wikipedia.org/wiki/Slate_(writing) https://en.wikipedia.org/wiki/Blackboard https://en.wikipedia.org/wiki/Blackboard So to correct your statement: this was quite common in the 18th–20th century, well after the introduction of “arithmetic” and Hindu–Arabic numerals, which were popularized in Europe by Leonardo of Pisa’s 1202 book Liber Abaci (mostly material translated from Arabic sources). Before that (and for a few hundred years after in most parts of Europe), most calculation was done with counting boards. Here’s a famous 1503 woodcut showing a comparison between arithmetic (“Boethius”) vs. abacus (“Pythagoras”): http://www.maa.org/sites/default/files/images/upload_library/46/Plimpton-Smith/8056Reischarithmetic.tif http://www.maa.org/sites/default/files/images/upload_library...