Saturday, 29 March 2008

The History of Casting Out Nines

In my youth, back in the dark ages before calculators, one of the common mathematical tools we were taught was called casting out nines. It seems it is not common anymore, perhaps the errors students make with calculators no longer submit to the simple check of this ancient method. If you happen to be of the generation who have been kissed on the forehead by the Gods of Electronics and actually don't know the method, you can find some notes on my page, and also here is a brief video from YouTube.

I have a note on my MathWords page on the subject from a respected math historian (Albrecht Heefer) that tells me, "Casting out nines is believed to be of Indian origin, but it does not occur before 950. Maximus Planudes called it 'Arithmetic after the Indian method". Along the way I seem to have a note from him telling me that I can find more confirmation on the web site of David Singmaster, the famous historian of mathematical recreation; but while searching there, I seem to have a note that claims the first mention of casting out nines was by the Latin writer Iamblichus in 325 BC... But he was talking about Nichomachus a Pythagorean who lived around 100 AD.

Now the common thought, or at least as I thought I understood it, was that the inventors of the hindu-arabic numerals had developed casting out nines and it sort of made its way into the west with the introduction of the Arabic numbers. Leonardo of Pisa, the famous Fibonacci whose bunny sequence you remember from school (of course you do, 1, 1, 2, 3, 5, 8, 13, 21...... That sequence) was a major influence in bringing both to the west with his famous book, the Liber Abaci, (the book of calculating) around 1202.

But the fact is that the general public held on to their Roman numerals for several centuries, and legal documents had to have them in some areas up into the 15th century.. Now the problem, at least for me, is that it seems much less likely that someone would develop casting out nines using Roman numerals.. see if you are using Arabic numerals, you take a number and add up the digits... 2534 would give 2+5+3+4 = 14 and then adding 1+4 = 5 so we know that if you divide 2534 by nine, you get a remainder of 5. Now in Roman numerals we write 2534 as MMDXXXIIII ... It just looks less likely to jump out.. Ok, maybe if they never used the D for 500 and V for five, I can see it becoming obvious, so maybe that is how it came about. If you write the Roman numbers with only unit (that's how math types say ONE) multipliers, like M for 1000 or X for 10 or C for 100, then all you would have to do is count the number of digits (not add them up). For example MMCCXII has seven digits, so the number 2212 should have a digit root of 7, which it does. And for really long numbers, you could throw away groups of nine in the same way we do with casting out nines..... MAYBE... but I wonder..

Anyway, I'm still looking for that Rogue Scholar out there who happens to have the original of Nicomachus' "Introduction to Arithmetic" laying around on his bookshelf and would like to translate for me to explain where he says it came from (if indeed he did).

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