Roman Mathematics

[Kosmo]

Greatest advances were made by the use of zero and of + - = . * /

Edited February 13, 2006 by Kosmo

[M. Porcius Cato]

Positional writing is easier to use and one can make complex operations only with paper and pen, but to say that we use this sistem because we have computers is weird.

 

Has anyone ever maintained such an absurdity?

[Kosmo]

I believed that when I wrote itt. Now I've checked and believe it no more. Sorry.

[Sextus Roscius]

I'm not sure how fair an arguement it is to say that arabic numerals built sky-scrapers, while Roman Numerals built the coloseum. Thats not comparing thigns from the same time period at all, who could say what the Romans would've done 1500 years later, and besides, the Arabic Numerals ended up doing the things they did in the hands of non-arabs, so in the end, I think we can come to the conclusion that its not so much the number system, but who's hands it is in. And yes, many scholars of the time who spoke greek often learned the number system of the greeks which was profoundly more simple than Roman Numbers.

 

All and all, this is like saying which is more sophisticated, English or an African clicking language. Opinions vairy, but those among us who are practical will say english, while those of us who can't say that anything isn't sophisticated or better than others in its own aspect would argue for the clicking noises. Simple as that.

[Guest fuzzytraction]

Roman mathematics has always been puzzling: Why would they use such a weird, non-positional system when there was already a much better one (the Hindu-Arabic) in widespread use? Further, given that they didn't employ a positional notation system, how could they multiply?

 

The rationale for their numeric system may have its origins in mercantile security. In the arabic system, a number like 1039 talents can be easily forged to read 9039 talents. Thus, by simply adding a little circle to the 1, it would be possible to defraud someone of 8000 talents. In contrast, to accomplish the same feat in Roman numerals, one would have to change MXXXIX to MxXXXIX, a change which could be more easily detected.

 

Further, Romans didn't always use their numeric symbols to complete computations. More often, they used a Roman abacus, which did have a bi-quinary coded decimal system. What to do if you left your abacus with that little she-wolf whom you were visiting by the Temple of Venus? Well, there was a method, but it was complex.

[Guest fuzzytraction]

 

I may have something of interest but some help is in order. Does anyone know the signs that were used by the Romans? I mean like in add, subtract and divide. To keep this simple - if you put XX over X, what sign was used to tell if the answer was 30 or 10?

In commoditiy trade they used "logos" to identify some trade goods, mainly smelted ore.

I have found something very small with large implications. They may have used two languages. Roman Latin as day to day and Ancient Latin as a business language. Any help would be appreciated.

[Gaius Octavius]

If you lend someone money and charge interest, you can work out how much they owe you exactly to several decimal places, and make more money. It may not be much from one person, maybe the for example you lend someone $10 for a year and charge alittle interest. Say you get $10.61 back, not much, but using a counting board you would only get $10.50 back. Using a counting board or abacus you have to round to the nearest whole number, meaning they were losing money. Now say you lend 100,000 or 1,000,000 people $10 for a year. The customer doesn't pay much more so they don't complain, but as a business man, your rolling in money now.

 

But I don't see why a Roman couldn't convert $10 to 1000 pennies to do the calculation. As long as the monetary system is decimal, their counting board should work fine. Am I missing something?

 

___________________________

PERHAPS, interest on a loan was calculated in this fashion: "I'll lend you $10.00 for one year and after a year is up, you owe me $11.00." Or 10%. You may have met this method in the army or at your local loan shark. $5 now for $7 in a week.

Something tells me that hidden somewhere in the Roman system, the CONCEPT of 'nothing' existed. If they subtracted II from II they got 'nothing'.

Did the Romans have a need for the very large numbers used at some points here?

Fifty years ago, when the dollar price of a municipal or corporate bond was figured from a yield, it went to three decimal places. This was good for a hundred or a million bonds or so. Now tens of millions of bonds are a 'normal' trade so six decimal places are used.

Edited March 29, 2006 by Gaius Octavius

[Ruthe]

Having read all of the posts in this thread, it seems that the myth of Roman inability to perform complex mathematics is being perpetuated. A great deal of the fault for this lies with the very sparse evidence left to show detail of such mathematical skill, However, there are clues that suggest a far greater facility for calculation was a common aspect of Rome's success, although this would be likely to have remained a very closed group of 'Calculators'.

 

As evidence I present a couple of very compelling exhibits.

 

First of these is the Roman pocket calculator (here I am talking about an object rather than the person). None of these have survived to the present day, but there is one depicted in great detail as a relief on one of the triumphal columns in Rome and from which a faithful reconstruction has been made. This can be seen at the site of Prof. Dr. J

Edited July 13, 2007 by Ruthe

[Gaius Octavius]

Ruthe:

 

Thank you. Although I don't really understand how the 'abacus' worked, what you say goes a long way toward understanding how the Romans could build tunnels from both ends, walls from different points and domes.

[Ruthe]

Ruthe:

 

Thank you. Although I don't really understand how the 'abacus' worked, what you say goes a long way toward understanding how the Romans could build tunnels from both ends, walls from different points and domes.

 

Gaius,

 

It is quite straightforward.

 

For the 'decimal' columns (7 to 1 from the left) there are four beads in the lower slot and one in the upper slot. By moving a bead in the lower slot to the top you are adding 1. Two beads = 2, 3 beads = 3 and of course 4 beads =4. At this point to add one more you move the four beads in the lower slot down, and move the bead in the upper slot up. This bead represents 5. Thus with the 4 beads in the lower slot and the one bead in the upper slot, you can represent any value from zero (all beads down) to 9 with all beads up. Of course those adept at using an abacus will be able to add any value in a single movement. For example,if the lower slot has 3 beads up and you wish to add 4, the user would move the bead in the top slot up and move one bead in the lower slot down, leaving the one bead in the upper slot up = 5, and two beads in the lower slot up = 2 giving a total of 5 + 2 = 7.

 

In just the same way that we carry tens from the units column, the Roman calculator would do exactly the same.

 

The second column from the right has two slots with one bead in the top slot and 5 beads in the lower slot. In this way, the Roman could start at zero and move successive beads in the lower slot up until all 5 are up. Adding one more, he would move all the lower beads down and move the bead in the upper slot up. In the case of this column, the upper slot bead represents 6. This column counts unciae. An uncia is the Roman name for 1/12. For fractions, the Romans used a duodecimal system. Why on earth would they do this? Well they inherited it from the inhabitants of Southern Italy or Etruscans. Why would they keep it when they used a basically decimal system. Their method of writing numbers had symbols for each power of 10 and the halfway values i.e I = 1, X = 10, C = 100, M = 1000 and V = 5, L = 50, D = 500. So why the duodecimal fractions?

 

Well, you tell me, in our decimal system, what is 1/2? Simple, 0.5. What about 1/4? Well not as simple because it takes two decimal digits i.e 0.25. OK, what about 1/3? Yeah, that's right 0.333333........... and so on. The Romans didn't like the thought of not being able to represent finitely 1/3, 1/6 and their multiples. So by using a duodecimal system of fractions, they became very adept at dividing land between sons after the death of their father, or working out interest rates, particularly when they adopted a 12 month year.

 

Thus the first column on the Roman abacus was split into three slots, to show 1/2, 1/4 and 1/12 or 2/12 in the bottom slot. In this case we are talking about showing twelfths of an uncia, which is itself 1/12 of 1, whether 1 'As' for weight or money or 1/12 of 1 foot. So with this versatile little pocket abacus, they could calculate down to the level of 1/144 directly, and probably could even reassign the value of each column to represent further duodecimal subdivisions. As shown by the reference in my previous post, the document by Frontinus shows that the diameter of pipes were measured, and therefore by inference were made to a precison of 1/288 of the unit in question.

 

It is due to this system of Roman duodecimal fractions that we retained 12 inches to the foot, 12 ounces to the pound (originally before the avoirdupois pound was changed to 16 ounces). Furthermore, our name for inch and ounce come from the Roman word for 1/12, the uncia. Too bad the French Acadamy didn't adopt a numeric base of 12 for the metric system!!!!

 

PS Did you know that nearly everybody in the developed world can do modulus 12 arithmetic while using a decimal number system? I'll let you try to work that one out yourself before I give you the answer.

 

PPS What are the decimal represntations of 1/2, 1/4, 1/3, 1/6, 1/8 and 1/9. Then, what are these fractions in a duodecimal system. Clue, the answer to the previous problem should give you some help.

[Gaius Octavius]

Ruthe, thanks for your effort and I promise that I will work on it. I must warn you that I have problems walking and breathing at the same time.

[Favonius Cornelius]

I don't know much about ancient mathematics, but wouldn't the Romans have gained much of their knowledge of it from the Greeks? Can concepts of Greek mathematics play a role in the use of Roman numerals?

[Gaius Octavius]

I don't know much about ancient mathematics, but wouldn't the Romans have gained much of their knowledge of it from the Greeks? Can concepts of Greek mathematics play a role in the use of Roman numerals?

The Greeks also used 'letters' in their counting system, but I believe much more so than the Romans did. Perhaps they also had an abacus. The Romans might very easily have adopted the Greek system. I'll bet that Ruthe can explain.

[Ruthe]

I don't know much about ancient mathematics, but wouldn't the Romans have gained much of their knowledge of it from the Greeks? Can concepts of Greek mathematics play a role in the use of Roman numerals?

The Greeks also used 'letters' in their counting system, but I believe much more so than the Romans did. Perhaps they also had an abacus. The Romans might very easily have adopted the Greek system. I'll bet that Ruthe can explain.

 

A very complete description of Greek number systems can be found at School of Mathematical and Computational Sciences University of St Andrews .

 

As for the abacus, the Roman version appears to have predated even those of China and to my knowledge there were no predecessors. I assume without investigationg further, that any such calculations were performed by other means or at best by methods similar to an abacus but as marks in sand.

 

Egyptian arithmetic is very well presented and explained again at School of Mathematical and Computational Sciences University of St Andrews.

 

So, until some later archeological find contradicts this situation, we are led to believe the Romans themselves and not the Greeks or any earlier civilization were the inventors of the abacus unless they copied from some subjugated area of the world and they didn't bother to mention it. I tend to think it was their own work, not for any afinity for the Romans, but just from their acheivements in building tunnels, buildings, aqueducts and even machines of war. Some form of planning and calculation must have preceeded their construction for them to have been so successful, and so they must have had the mathematical skills to develop such a device.

[M. Porcius Cato]

PS Did you know that nearly everybody in the developed world can do modulus 12 arithmetic while using a decimal number system? I'll let you try to work that one out yourself before I give you the answer.

 

PPS What are the decimal represntations of 1/2, 1/4, 1/3, 1/6, 1/8 and 1/9. Then, what are these fractions in a duodecimal system. Clue, the answer to the previous problem should give you some help.

 

 

Ooooo, fun! A real brain-teaser! Ok, I'll take a shot. Are we using modulus 12 when we read time off an analog clock?

 

That makes the next problem easy, in the duodecimal system, 1/2 = .6; 1/3 = .4, 1/4 = .3, 1/6 = .2, 1/8 (harder) = .16, and 1/9 (also harder)= .14. (Full disclosure--I cheated on the last two: my original answers were .15 and .13, not really sure why they're .16 and .14)

 

BTW, I'm a big fan of the British Imperial system of measurement, even though I use metric in my scientific work, so I greatly appreciate your argument for the Roman system.

 

Also, could you recommend an article on the development of the Roman system of numbers? It's all so fascinating! Are you at St. Andrews? I gave a lecture there a few months ago, and I talked about the dismal understanding that adults have of fractions--their attempts to estimate fractional magnitude are actually worse than if they simply guessed randomly and even worse than children's estimates (for a reason that takes us even further afield).

Edited August 16, 2006 by M. Porcius Cato