Roman-Type Systems
In the finger-stick method, only the number one has a symbol -- the stick (when written) and the sound "wa" (when spoken). In the simplest Roman-type system, there are two symbols. The second symbol represents a larger number. For example, X might stand for |||||||||| (ten sticks).
The obvious advantage, and probably the only advantage, is being able to express large numbers more easily.
A two-symbol system must occur first. But the jump from two to three is simple and would be made very quickly. There is no historical record of any culture (that I know of) having just two symbols. It would be worthwhile to explore a system with just two symbols, but your class will also probably quickly make the jump to multiple symbols (especially if they get to be the inventers and name the symbols).
Name
I think these are called additive numeral systems, but that term is much broader than what will be considered here. Of course, the Roman Numeral System is exactly such as system. I think the practice of putting a smaller number before a larger number to signal subtraction is an unnecessary complexity. The exercise also then becomes too much like learning facts instead of exploring something new. So avoid that. Without subtraction, the system will be cleaner and easier to work with. And hard enough.
By the way, Ifrah, in his book From One to Zero, argues that X is a common symbol for ten because of the way sticks would be notched.
Incorporting Their Advances
When I ended my session on the finger-stick method, my class asked some some questions. Jake wanted to know why the Rikitkians did not have a symbol for 100. He of course was trying to solve the problem of not being able to represent large numbers.
As it turns out, that is exactly what the Rikitikian named Jakesahn suggested. Then other people in the class quickly suggested smaller numbers. I made the point that having a new symbol was a large conceptual leap. 100 wasn't very practical as a second number, which is one reason other people filled it in. I called the system Michael Numerals, because Michael suggested V for 5 and X for 10.
Order
In an additive system, the order is usually irrelevant. Of course, in the actual Roman numeral system, as taught today, IV and VI are two different numbers. But assuming that you avoid that complexity (and I think you should), then they have the same meaning -- five plus 1, or six.
So, when the symbols for an additive system are first invented, they could be written down in any order, and people probably would write them down in any order.
However, with time, a particular order is likely to become conventionalized -- people start using the same order, then people start expecting that order and understanding that order, and then other orders seem strange and might even be confusing.
Similarly, there are choices on the components of a number. Using X as 10, V as 5, and | as 1, 12 could be written as all of the different orderings of X||. But it could also be written as all of the different orderings of VV|| and V|||||||, and also as ||||||||||||.
Of course, in our modern system, order is critical in writing-- 369 is a different number than 639. But for most of our ways of representing either numbers or measurements, we have essentially an additive system with a standardized representation. To give a simple example, everyone says 5 feet, 7 inches. There is no ambiguity in saying 7 inches 5 feet, except that it is unexpected, because the standardization is so strong. You can say 67 inches, but 4 feet 19 inches, though correct and unambiguous, would be unexpected and inappropriate.
Experiencing Rikitikian Culture
Given this, one way to try experience Rikitikian culture is to destandardize some of our ways of representing numbers and measures. I started with this problem. Something costs 58 cents. You give the clerk a dollar. What does the clerk give back?
The first person suggests a quarter, a dime, a nickle, and two pennies. In the format of a Roman-type System, that is QDNPP. I ask the next person what the change is, going around the room. If that person gives an answer that has already been given, I ask for a different answer. (You can change the order and/or break the coins down into smaller coins. For example, DPDPDD is a correct answer.)
We in fact have a standard order, but we can violate it without problem -- "four and twenty blackbirds, baked in a pie." However, though unamiguous in meaning, that sounds a little odd, and perhaps is not as easy to understand as if it had been in standard order. That should be food for thought for your students to try to actually experience what it was like to live in Rikitiki when there was a common order but not everyone followed it.
There is also an issue of combining. Supppose that V is 5. Eleven could be written as X| or |X, but it could also be written as VV|, V|V, |VV, V||||||, |||V|||, etc.
The reality is, they are not going to make a law that you have to standardize. So a number could be written any way, and it would be unambiguous. But people are likely to develop a standardized writing, because that would be easier to understand. Also, as one student noted, it is not easy to grade homework if the answers aren't standardized.
Addition
If the numeric representation does not need to be standardized (ordered and simplified), then addition in a Roman-type system is simple. For example, LXV||| + LXV|||| = LXV|||LXV||||. Standardization adds interest to the addition, and reflects common principles in our modern system. For example, 1 foot 7 inches + 1 foot 7 inches = 2 feet 14 inches, but we write this as 3 feet 2 inches. So it is probably very good to let them practice addition with standardization (though a problem or two without standardization is fine).
Too Deep
Note that, like the stick system this comes out of, there is no representation for zero.
Vestiges
There are several vestiges of this in our current culture. Most obviously, Roman numerals are still taught and occasionally used. Second, the "tally" method is essentially the stick method, but there is a line through 4 sticks to represent five. You can imagine people starting to write short-hand for that, and then you have a nice transition from the stick method to a two-symbol Roman-type method.)
Finally, suppose the second symbol is 10, so that 11 is written as X|. Then, essentially, eleven is ten with one left over. The word eleven comes from the Old English word endleofan. (Twelve also seems to be "two left over.")
Fingers
If it hasn't come up yet, you can introduce the value of fingers in determining number. Suppose you gave a student 12 cards and ask the student to write down how many cards there are. The student will count the cards and write X|| (or whatever the symbols are). But the student has counted the cards using English words. What would a Rikitikian do?
If you want to go in this direction, this is another time to try to really think like a Rikitikian. The Rikitikian cannot count. When the Rikitikian gets to ||||||||||, how does he or she know that there are 10 lines?
The answer (almost certainly for ancient cultures) is this. We know that 10 is based on the number of fingers, but we forget about fingers and just think 10. To the Rikitikians, 10 is the number of fingers. They could eleven things by putting 10 of them in one-to-one correspondence with each finger and finding one left over.
I would save this until after you have had them add with this number system.
Addition and Subtraction
Addition and subtraction now involve packing and unpacking, which is critical for understand how to add and subtract in base 10, or in any of the mixed bases used in measurement.
An actual math test, with C for 100, X for 10 and | for 1.
XXX||| + XX|| =
CXX|| + CCX =
XX||||||| + XX||||||| =
CXXXXXX|| + CCXXXXXX|||| =
XXX||||| - X||| =
XXX - ||| =
CX|| - XXX||| =
C||VXX XX||MW
+ XMV + V|M|CXM
CXLV||X WMXCV||XL
- XLV| - XCV|
X|| CX||
+ X| + CLXV||
V XXX|||
+ V + XXX|||
XXV||| MMCCLXXXV||
+ XX||| + MMCCLXXXV||
Multiplication
I suspect that division is too difficult, but multiplication is very feasible, if there are just units of ten. It is more difficult, but not impossible, there are units of five.
This would take a lot of time, but it might capture some of the advantages of having just one base. If a class happened to go that way -- they invented symbols for 5 and 50, then you might mention a school-children revolt in Eastern Rikitiki, where the children threw out the 5 and 50 to simplify multiplication.
There is another thing about multiplication. It is not obvious to me why the second base has to be a power of the first. And of course there are many systems where is it not (inches, feet, yards; ounces, pounds, tons). However, if you try to multiply with a two-symbol Roman-type system, the first thing you are going to discover is the convenience of a single symbol to mean the square of the first symbol. In other words, if X means 10, you are going to want a symbol for X times X, which is 100.
The students are likely to pick a symbol for 5, such as V. If they do, they have the problem that there is no symbol for V x V. One solution is to create one. For example, V x V = T. The problem is, then there is no symbol for T x T, or V x T.
Next: Unique Symbols
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