SKStephenson

I've been busy ... Here's a link to the series of my works on Ancient Computers: http://bit.ly/gLD5MS -Steve

They work for me. Unfortunately, some mathematicians have noticed the rules 'don't add up' for some reason, and although it isn't proven, there appears to be a quality to finite values that we gloss over in our desire for a simple decimal set of rules. There's been an article in the science press on this poin t just lately. Fascinating stuff. But hey, I still end up with the expected number of pennies at the end of the day. Usually, anyhow... The "article in the science press" is To infinity and beyond: The struggle to save arithmetic by Richard Elwes in the 16 August 2010 issue of New Scientist magazine. This magazine appears to be like the U.S. magazine, Discover. Both seem to be science popularizers aimed at the general public with articles often sensationalized to attract attention and sell magazines. Not willing to pay $72/year for a subscription to this drivel to read the article, I found an analysis of it by Mathematics Professor Andr

> they didn't bother with clever stuff like division or multiplication If by "they" you mean the aristocracy, I could agree. But the "educated slave" had better "bother" with multiplication and division ... correctly! And the aristocrat should know enough to check the work of that slave, just as you check your tax return prepared by your accountant. The Emperor, short changed on the taxes you paid, might not respond too well to the excuse, "My slave made an abacus mistake." > Our numeric system makes that easy. Except that many of my precalculus and calculus students still have trouble. They often have to reach for their electronic calculators to do the simplest arithmetic. Just like the Roman student would have reached for his abacus (see Ancient Computers). > the rules of arithmetic in our modern day are not necessarily perfect or even correct The rules of arithmetic ARE CORRECT! We trust them everyday when we drive a car, fly in an airplane, ride in a bus or train, etc., because those devices were designed using mathematical models that rely on the correctness and absolute predictability of arithmetic rules. All mathematical rules have been proven through formal theorems based on very few demonstrably reasonable assumptions. That formal process started around 300 BC with Euclid's Elements. Steve

I've changed the links in my two previous posts to my latest article on abaci: http://www.ieeeghn.org/wiki/index.php/Ancient_Computers The article uses the Roman Hand Abacus, and its modern look-alike the Japanese Soroban, to prove that the Romans used The Salamis Tablet's structure for their line abaci, both decimal and duodecimal. It uses Roman Numeral subtractive notation to prove that The Salamis Tablet treated a number as containing both positive and negative parts. It uses a calculation of Frontinus in The Aquaducts of Rome to determine the promotion factors of the Roman duodecimal abacus. Noted is that the Roman duodecimal abacus could be a simple and direct subset of a more ancient abacus used by the Babylonians to do sexagesimal calculations. The lack of a radix point symbol (decimal point) in their positional sexagesimal numbers recorded on clay tablets, indicate that the Babylonians probably used the second smaller grid on The Salamis Tablet to keep track of radix point shifts, entering every number on the abacus as a fractional part and a radix point shift part (what we call an exponent). Having (re)discovered the structure and methods-of-use of these ancient computers, the conclusion is reached that ancient peoples as far back as 2300 BC had technology to do rapid (for them) arithmetic calculations on any numbers of interest to engineering or business. In decimal mode, The Salamis Tablet can accommodate positive or negative fractions with 10 significant digits and positive or negative exponents with 3 significant digits. In duodecimal or sexagesimal modes, The Salamis Tablet can accommodate positive or negative fractions with 5 significant digits and positive or negative exponents with 2 significant digits. Please read the paper and let me know your comments. Thanks, -Steve

That web page shows an "incorrect" use of subtractive notation, i.e. IIX (or was it XXC?). But the rules were put in place, I believe, after the Romans, in Medieval times. Subtractive notation is actually used to represent counters on a counting board style abacus, e.g.: IIXXooI = 18 5/6 (-).....(+) -----|o---o X .....|..... V ---oo|----o I .....|..... S ---oo|----- o [i would like NOT to use the periods, but don't know how to keep spaces from being deleted.] Preferred representations on an abacus: IIV, IV, IIX, IX, etc. This conserves space by dramatically reducing the total number of counters and by reducing the number of counters on a line or space. More explanations here: http://www.ieeeghn.org/wiki/index.php/Ancient_Computers

Ruthe, Re: Last two beads on Roman Hand Abacus. I'm glad I found your post with a google search. I've been struggling for the last two days trying to reconcile the expert stated value of 1/3 of 1/2 with the logical abacus value of 1/12 of 1/12. Now I'll ignore the experts. OTOH, do you know of any scholarly sources that support our position? And that will probably mean a different interpretation of the symbol next to the 1/3's. Somewhere in the thread there was also a question on whether abaci were used prior to the Romans. The answer to that is YES. Well documented is its use by the Greeks, esp. google The Salamis Tablet. It is thought that the abacus was actually used and invented by the Sumerians or Akkadians. Using some historic clues, and The Salamis Tablet as a base, I've put together some thoughts on how these ancient cultures might have used an abacus. You can see it here: http://www.ieeeghn.org/wiki/index.php/Ancient_Computers. Suggestions are welcome. -Steve