Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!watmath!clyde!rutgers!lll-crg!seismo!nbires!hao!hplabs!tektronix!teklds!zeus!tekla!dant From: dant@tekla.tek.com (Dan Tilque;1893;92-789;LP=A;60aB) Newsgroups: sci.math Subject: Re: Arithmetic with Roman numerals Message-ID: <820@zeus.UUCP> Date: Mon, 3-Nov-86 21:44:57 EST Article-I.D.: zeus.820 Posted: Mon Nov 3 21:44:57 1986 Date-Received: Wed, 5-Nov-86 05:38:39 EST References: <782@zeus.UUCP> <487@jumbo.DEC.COM> Sender: rogers@zeus.UUCP Reply-To: dant@tekla.tek.com (Dan Tilque) Distribution: net Organization: Tektronix, Inc., Beaverton, OR. Lines: 100 Keywords: Roman numerals, arithmetic, Arabic numerals 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.