Roman Mathematics

[Gaius Octavius]

Log to the base 12? Wanted to poison the prof who tried to teach 'sets'.

Here is my shot (in order) .6; .3; .4; .2; .15; .1333~.

My formula: 1/x X 12/1 = 12/x.

 

Abacus - Part II of the Macaroni Wars?

 

Somehow, I don't think that the Romans used the '12' or '24' system for telling time. I believe that noon was when the sun was directly overhead. Or so many hours after sunrise or after sunset using a sand clock or a water clock.

 

Wasn't it the Babylonians or Sumerians who used the '12' system first?

 

If we could only get an Imperial gallon of gas for the price of a U.S. gallon.

 

Please don't keep me waiting; I'm a sick person. (Enjoy yourselves, wiseacres!)

Edited August 16, 2006 by Gaius Octavius

[Ruthe]

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?

 

Nearly there. Its when you need to do a calculation such as the following.

It is now 9:30 am. You boss/wife/whoever says they will meet you in 5 hours from now. What time is your appointment? Well 9:30 plus 5 = 14:30. If you are familiar with the 24 hour system, maybe you were in the forces or navy, this would be enough. But since most of us work on the 12 hour clock, we must convert this from 14:30 to 2:30 pm by subtracting 12 from 14:30. This is the modulus 12 bit. But of course we still use a decimal number system.

 

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).

 

Quite right, the simple fractions can be read straight from a clock face. But why is 1/8 = .16 and 1/9 = .14? Well the first is easiest so 1/8 = 1/2 x 1/4. So 1/2 of .3 = .1 and a half of .1. Now since 1/2 = .6, 1/2 x .1 = .06 and so 1/8 = .1 + .06 or .16. As for 1/9 = .14 in duodecimal form lets see why. Now working in duodecimal throughout until a final conversion to decimal, we have .14 = 1/10 + 4/100 = 10/100 + 4/100 = 14/100. This in decimal form = 16/144 (10 duodecimal = 12 decimal, 100 duodecimal = 144 decimal). So 14/100 duodecimal = 16/144 decimal. Reducing this to its simplest form, this is 1/9.

 

This looks far too difficult since we are all used to working in decimal and have to keep converting to something with which we are familiar. But if we learnt duodecimal numbering from the start, we would find arithmetic to be far easier than the decimal system. If you want to see why, have a look at the The Dozenal Society of Great Britain.

 

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.

 

The Imperial system, its predecessors and concurrent systems in Europe continued to employ a partial duodecimal basis despite the introduction of the Hindu-Arabic number system, and the simple reason is its practicality in daily commerce. It's much easier to divide 12 into halves, quarters and thirds than ten. The half is just as easy in both systems but even the quarter gives us two decimal places. So if you wanted a quarter of ten, you would get 2 1/2, and as for 1/3 of 10, we all know this is 3.33333.... recurring! With goods and services treated in twelves, the commonest fractions of 1/2, 1/4 and 1/3 are all finite and only one numeric digit in a duodecimal number system.

 

Also, could you recommend an article on the development of the Roman system of numbers? It's all so fascinating!

 

There are many references on the web, but one that appears to be very complete is the one at Wikipedia.

 

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).

 

 

Nope! I have no affiliation with that august body.

 

As for the dismal understanding of fractions in adults, I think this extends to current output of our education system. Why? I don't have the answer to that, except to say the current syllabus may not have the correct emphasis. While I strongly advocate teaching of the understanding of mathematics and why the various operations work the way they do, I still believe there is a place for some rote learning. Pupils today seem to be able to solve problems using a 'cookbook' approach, but they do not understand why things work. Just as deplorable is their inability to do mental arithemetic, mainly beacuse they don't learn multiplication tables by rote. Without them I wouldn't be able to do half the calculations needed daily, but I have the advantage of understanding why these operations work. I can still recite the full 12 times tables from 1 to 12 and yes, I was at school when

[Ruthe]

Log to the base 12? Wanted to poison the prof who tried to teach 'sets'.

Here is my shot (in order) .6; .3; .4; .2; .15; .1333~.

My formula: 1/x X 12/1 = 12/x.

 

Abacus - Part II of the Macaroni Wars?

 

Somehow, I don't think that the Romans used the '12' or '24' system for telling time. I believe that noon was when the sun was directly overhead. Or so many hours after sunrise or after sunset using a sand clock or a water clock.

 

From The Private Life of the Romans

 

427. Hours of the Day. The daylight itself was divided into twelve hours (hōrae); each was one-twelfth of the time between sunrise and sunset and varied therefore in length with the season of the year. The length of the day and hour at Rome at different times of the year is shown in the following table:

 

 

Month and	 Length of	  Length of	   Month and		  Length of	Length of

Day Day Hour Day Day Hour

Dec. 23 8

[Pertinax]

1. excellent posts Ruthe, I now understand my great emotional attachment to Imperial measures, (and my total inability to mentally assimilate decimal systems). What a splendidly civilised way to divide up a dark winter day, by reducing the hours to manageable sizes.

 

2. I first bought petrol at 77pence per gallon , and that made me mad!

 

Does this instrumentation ring any bells?

http://www.unrv.com/forum/index.php?act=mo...=si&img=907

[SKStephenson]

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

Edited August 11, 2010 by SKStephenson

[SKStephenson]

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

[Ruthe]

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

 

Steve,

I've just found your latest update and had a quick scan of your document of the ancient's uses of the abacus. It will take a little while for me to give it the attention it deserves, but it appears from my rapid scan that you have done a fantastic job in investigating the methods used by the ancients. I will get back to you when I have had fully digested the details. I must admit my long absence from mathematics will require a little remedial effort first.

 

I also saw your previous post and wondered why I never saw it when you first posted it. You asked which source I was referring to in questioning the 'expert' view that the symbol beside the lower slot in the first column which is similar to our present digit "2" is believed to be ¹/₃ of ¹/₁₂. Friedlein¹ states it to be the symbol for a 'sextula' or ¹/₇₂ of an 'As' = ¹/₆ x ¹/₁₂ As or ¹/₆ of an uncia when used on a pocket/hand abacus. He also gives some examples of the use of the Salamis Tablet type or table top version of an abacus with the three separate subdivisions of an uncia and in this case shows the bottom position to have the symbol like a truncated union operator which he again indicates in his table to be the symbol for sextula.

 

What he does not explain is if the value of ¹/₇₂ refers to the value of each counter in that position, or the total value of the two counters. It is only if each counter has a value of ¹/₁₄₄ that a continuous range of values from ¹/₁₄₄ to ¹¹/₁₄₄ or ¹/₁₂ x ¹/₁₂ to ¹¹/₁₂ x ¹/₁₂ can be represented. Thus the two counters would together have a value of ²/₁₄₄ or ¹/₇₂ of an As.

 

The entire volume of Friedlein's book can be viewed online at the University of Strasbourg and the table containing the symbols I noted are on webpage 171 and 172.

 

 

¹Friedlein, Gottfried, Die Zahlzeichen und das elementare rechnen der Griechen und R

Edited September 24, 2010 by Ruthe

[caldrail]

I think the problem with Roman mathematics is that we're accustomed to a different system. They lived with theirs, so it was more intuitive to them, abacus or not. Also I suspect they didn't bother with clever stuff like division or multiplication (you might want to buy an educated slave to handle that onerous problem for you) in the way we do. Our numeric system makes that easy. We apply that system to everything so it guides our manner of doing business. They would have done the same with Roman numerals, and not suprisingly, I scratch my head along with everyone else.

 

As an interesting aside, it turns out that the rules of arithmetic in our modern day are not necessarily perfect or even correct. Apparently some mathematicians have discovered there may be flaws in the system. Like what? It all seems to work as far as I can see. Perhaps that's how the Romans viewed their own system too. What's wrong with it?

[SKStephenson]

... I suspect they didn't bother with clever stuff like division or multiplication (you might want to buy an educated slave to handle that onerous problem for you) in the way we do. Our numeric system makes that easy. ...

 

... the rules of arithmetic in our modern day are not necessarily perfect or even correct. ...

 

> 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

[SKStephenson]

The entire volume of Friedlein's book can be viewed online at the University of Strasbourg and the table containing the symbols I noted are on webpage 171 and 172.

 

¹Friedlein, Gottfried, Die Zahlzeichen und das elementare rechnen der Griechen und R

[SKStephenson]

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.

 

 

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

[SKStephenson]

I've been busy ...

 

Here's a link to the series of my works on Ancient Computers: http://bit.ly/gLD5MS

 

-Steve