stolfi@jumbo.DEC.COM (Jorge Stolfi) (11/01/86)
Dan Tilque wrote in net.sf-lovers: > > L. Sprague de Camp's _Lest_Darkness_Fall_ is slightly less realistic. > (SPOILER) His protagonist is an archeologist living in Rome who is > transported to the 5th or 6th century. ... He achieves some fortune > by "inventing" the still and the Hindu-Arabic numerals along with long > division (ever try doing long division in Roman numerals?). Aren't you being a little unfair here? I think that arithmetic with Roman numerals is hard only if you try to handle each letter as a separate digit. If instead you break the numbers into multi-letter decimal "digits", you will find that arithmetic is as almost as easy as with arabic numerals. For example, if you want to add MMCMLXXIV (2974) and CDIX (409), you parse the numbers into thousands, hundreds, tens, and units, as MM CM LXX IV CD IX then write them one on top of the other, with like "digits" on the same column, MM CM LXX IV CD IX and then add the columns, as in the arabic system: M X MM CM LXX IV CD IX + ----------------- MMM CCC LXXX III As in the Arabic system, you must memorize a 9x9 addition table (IV+IX = III plus a carry of X, etc.). Note that the same table works for all columns, once you identify M=C=X=I and D=L=V. Note also that many entries of the table are trivial (e.g., I+I = II, V+II = VII, etc.) If we replace the "subtractive" digits IV, IX, XL, ... by their "additive" equivalents IIII, VIIII, XXXX, etc, the addition table becomes even simpler. (I have seen this notation used in a couple of places, but don't know if it was ever used for computations. Note however that this is exactly how numbers would be encoded in an abacus). The other operations can be handled in the same manner. I don't know if the Romans actually used these methods, but I am sure that with a little practice the parsing will become automatic: one will "see" MMCMLXXVI as four digits, not nine letters. At that point, one will probably be able to match corresponding digits without having to write them into aligned columns. The real problems with Roman numerals are the lack of a symbol for zero and the varying width of digits (which makes it hard to align corresponding digits); the finite supply of letters (which puts a bound on the representable numbers); and its general verbosity. Of course, the Romans also lacked many mathematical concepts that we have since incorporated into our number system, such as decimal fractions and negative numbers. In spite of those problems, I don't think that Roman numerals were as cumbersome as they are usually said to be. Considering how long it took for the Arabic notation to be accepted in Europe, I seriously doubt that L. Sprague de Camp's hero would have made a fortune by "selling" them... vale, j. ------------------------------------------------------------------------ DISCLAIMER: Digital Equipment Corp. does not sell or manufacture Roman numerals, and I do not get any royalties for their use.
pat@leadsv.UUCP (Pat Wimmer) (11/03/86)
In article <487@jumbo.DEC.COM>, stolfi@jumbo.DEC.COM (Jorge Stolfi) writes: > > The real problems with Roman numerals are the lack of a symbol for > zero and the varying width of digits (which makes it hard to align > corresponding digits); the finite supply of letters (which puts a bound > on the representable numbers); and its general verbosity. Of course, > the Romans also lacked many mathematical concepts that we have since > incorporated into our number system, such as decimal fractions and > negative numbers. > I read this yarn some time back, so please forgive me if I am a bit hazy on the details. The point LS de Camp was trying to make were the very ones you bring up, wasn't it? Wasn't the Roman attempt at accounting absurdly unwieldy, extremely prone to error, and user hostile when trying to retrieve data? The sheer effort involved in using such a system should not be lightly dismissed. As to the lack of zero, negative numbers, and decimal fractions, these preclude the development of modern mathematics. Can you imagine trying to do calculus with Roman Numerals, and the above limitations? What about complex numbers, without which elementry physics, not to mention elementry electricity is impossible.
dant@tekla.tek.com (Dan Tilque;1893;92-789;LP=A;60aB) (11/04/86)
In article <487@jumbo.DEC.COM> stolfi@jumbo.UUCP (Jorge Stolfi) writes: >Dan Tilque wrote in net.sf-lovers: >> >> L. Sprague de Camp's _Lest_Darkness_Fall_ is slightly less realistic. >> (SPOILER) His protagonist is an archeologist living in Rome who is >> transported to the 5th or 6th century. ... He achieves some fortune >> by "inventing" the still and the Hindu-Arabic numerals along with long >> division (ever try doing long division in Roman numerals?). > >Aren't you being a little unfair here? I think that arithmetic with >Roman numerals is hard only if you try to handle each letter as a >separate digit. If instead you break the numbers into multi-letter >decimal "digits", you will find that arithmetic is as almost as easy as >with arabic numerals. > >[ example of addition in Roman Numerals deleted ] I agree that addition is not difficult in RN's, and subtraction is not much worse. This is especially true when you consider that the Romans had some type of abacus. There are two ways to do multiplication with RN's: multiple additions and a halving and doubling scheme similar to base 2 multiplication (yes the scheme was known in the ancient world). Long division was another story. I have heard that some early universities gave degrees in long division using Roman Numerals. (In some ways this is still going on; you have to take a class which teaches elementary arithmetic to get a teaching degree.) >If we replace the "subtractive" digits IV, IX, XL, ... by their >"additive" equivalents IIII, VIIII, XXXX, etc, the addition table >becomes even simpler. (I have seen this notation used in a couple of >places, but don't know if it was ever used for computations. >Note however that this is exactly how numbers would be encoded in an >abacus). > The early RN's did not have the "subtractive digits"; they were a later development (I'm not sure exactly when they were added). > >The real problems with Roman numerals are the lack of a symbol for >zero Exactly. > and the varying width of digits (which makes it hard to align >corresponding digits); the finite supply of letters (which puts a bound >on the representable numbers); Originally, not all the RN's were letters. The D and M were later changes to the Roman Numerals. The original symbols were somewhat like this: |) 500 /* note the similarity to D */ (|) 1000 (|)) 5000 ((|)) 10000 ((|))) 50000 (((|))) 100000 This was about as far as they went (I am reproducing these from memory. I can't remember where I saw them (some book on the history of numbers) so they could be wrong.) The D and M come along in the Middle Ages; evidently they did not need the larger numbers. These symbols could be extended indefinitely, but would quickly get very cumbersome. > and its general verbosity. Of course, >the Romans also lacked many mathematical concepts that we have since >incorporated into our number system, such as decimal fractions and >negative numbers. > >In spite of those problems, I don't think that Roman numerals were as >cumbersome as they are usually said to be. Considering how long it >took for the Arabic notation to be accepted in Europe, I seriously >doubt that L. Sprague de Camp's hero would have made a fortune by >"selling" them... > De Camp's hero did not really make a fortune teaching long division, but he did make some money from it. Read the book if you want to know how he did. It also has a fairly realistic description of what life was like in the late Roman/early Middle Ages period of Italy. > vale, > > j. > >------------------------------------------------------------------------ >DISCLAIMER: Digital Equipment Corp. does not sell or manufacture Roman >numerals, and I do not get any royalties for their use. Dan Tilque dant@tekla.tek.com DISCLAIMER: Roman Numerals are not supported by Tektronix Logic Analysers. Any illusion to the contrary is strictly prohibited.
dgary@ecsvax.UUCP (D Gary Grady) (11/06/86)
Dr John Tepaske, of Duke University, has done extensive research on the
accounting records of the Spanish Empire and has even created a computer
data base. A curious point is that almost all the records are in a sort
of upside-down lower-case Roman numeral notation. Frequently the
margins are filled with computations in Arabic numerals but the actual
entries are in Roman. Shows you how little accountants have changed
:-)!
By the way, De Camp finished writing Lest Darkness Fall on his
honeymoon!!
--
D Gary Grady
(919) 286-4296
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