Yes, it is; other notation conventions like frames or bars around the sides of a number symbol were also used to indicate large numbers.
The idea that the Romans somehow wouldn’t have been able to express “one million” numerically without laboriously writing out one thousand copies of a symbol for one thousand is silly. Like everybody else, Romans wrote numbers for the quantities they needed, sometimes inventing specialized notation conventions that were seldom used in everyday practice.
Now, you can certainly make the case that specialized notation conventions rapidly become cumbersome and hard to keep track of compared to a very general systematic notation like decimal place-value numerals. But that’s not at all the same problem as actually not being able to write large numbers.
There is still some positional logic that complicates things, like the “1s” value before or after the “5s” or “10s” value, like “IV” and “VI”… where the value of the “I” may be “take away 1” or “add one”. Likewise “IX” or “XI” or “XL” or “LX”.
I’ll agree that Base-12, in theory, is more useful than Base-10, but the same is not true of 60. I’d hate to memorize the time tables in that base, or do any sort of everyday arithmetic by hand.
IMO, Friedo nailed it. Positional notation, combined with the invention of zero as a placeholder for “no units of this size”, enables discussion of numbers of arbitrarily large or small size. Imagine discussing the size of the US Federal deficit or even the GDP of the Netherlands using Roman numerals! And positional notation is not base-limited. “203” has a valid meaning in hexadecimal, a different number than in base-ten… what would be represented in base-10 as “515”: two units of 256 (16[sup]2[/sup]) plus 0 units of 16 plus 3 units of one.
There’s a geek pun, by the way, to the effect that considering octal notation, Hallowe’en equals Christmas: 31 OCT = 25 DEC
I imagine some other system than 1,2,3 might be evolved to represent positional digits – I tried, as an experiment, figuring out how to represent base 60 in a quickly-grasped way. But none ever was before Arabic notation was adopted virtually worldwide.
? What don’t you like about the way the Babylonians themselves represented base-60 positional digits? Cuneiform numerals.
A glyph for 1 and a glyph for 10 are combined additively to make all the “digits” from 1 to 59, and “digits” are ordered in the successive base-60 places to represent a number.
Very quickly grasped, IMO: pretty much at first glance you can tell what any base-60 “digit” is, and the place-value principle is no more complicated than it is in base 10.
And in fact, if it’s the cuneiform numerals you don’t like, Greek and Arabic alphabetic numerals were also routinely combined with the base-60 positional system in astronomical calculations. You’d write, say:
delta, iota, and then gamma with a line over it
to indicate 460 + 101 + 3*1/60 = 250.05. Simple as pie.
Personally, I’m fond of the Mayan system. It’s a base 20 positional place-value system, very similar to what we use, but it only needed 3 symbols. Zero had a special symbol, and then the digits from 1 through 19 were constructed from the other two symbols using a method similar to Roman numerals. A dot is 1, two dots are 2, and so on. A horizontal line is 5. A horizontal line and a dot is 6, a horizontal line and four dots is 9, two horizontal lines is 10, and three horizontal lines and four dots is 19.
If the Roman’s didn’t have this strange shorthand as replacement for 4 repeats, they would have had a system kinda like the Egyptians–a specific symbol for certain quantities repeated as necessary. It’s not base-10 and it’s not really base-5, but it could be used for easy counting much like the Egyptian style.
The Egyptians do also have the symbol-as-specific-quantity characteristic that Little Nemo mentioned. One advantage of that is that the digits can be written in any order and the number is the same. Just like 234=100+100+10+10+10+1+1+1+1 in any order, the Egyptian numbers could have been written in any jumble of the digits, although it would probably look weird to them. HHTTTOOOOO, TTTHHOOOO, and HTOHTOOTO theoretically could all be understood as 234.
Isn’t the positional subtraction a medieval invention and not actually used by the Romans? (Or at least the Western Romans? I’m not sure what notation the Eastern Romans used.)
I think I also once saw a number “base” that could be used to express any rational number, using the same notation as it does for integers. Though I think it safe to say that neither of those is ancient.
There was an older set of symbols that the Romans used for large (well, larger than C) numbers. They don’t teach it today so most people aren’t aware of it. It’s composed of verticle lines with semicircles (or Cs, forward and backward) on either side of them. IIRC, it goes like this:
The verticle line usually extended above and below the semicircles. I’m not sure if they ever used more than 3 semicircles on a side, but I wouldn’t be surprised if someone had.
Note that the use of D for the conventional Roman numeral 500 comes from this set of symbols.
Also note that these symbols are not especially good ones, which is probably why they fell out of use. They’re too similar to each other, requiring one to be counting semicircles possibly multiple times within a single number.
Seems to me like you’re thinking that any base other than the one you’re used to is difficult. I agree - but only once that “you” becomes generic. The 5-10 of Roman numerals is as natural as the both-hands-used of our system.
The OP asked for advantages, not arguments over bases. As used in Europe, roman numerals had one overwhelming advantage: familiarity. When hindu/arabic numerals were first introduced in Europe by Leonardo Fibonacci (yes, that Fibonacci–he learned about them while living in North Africa, helping to manage his business operations there) they were widely rejected as “too hard to learn”. It took a generation to change. Like the introduction of the metric system in Canada. It happened nearly 40 years, but my doctor still records my weight in pounds and we still by wood by the inch. And 4 x 8 plywood sheets are here for the foreseeable future.
I was going to mention this. One possible advantage is that it makes addition and subtraction of small numbers very easy - just add or remove dots and lines.
I used to program a computer that used an oddball numbering system called biquinary.
This was a basically decimal system. Each computer “word” consisted of 10 decimal digits, and each individual digit consisted of 4 bits. But those bits had place values of 1, 2, 4, and 5.
Digits from 5 through 9 were encoded using the 5-bit. Thus, for example, 6 was 5+1 and NOT 4+2. The other 6 bit combinations were called “un-digits”, and some of them were used for special purposes.
The computer was a Univac SS-90. (The “90” refers to the 90-column punch card format that it used. There was also a SS-80 model that used the more familiar IBM 80-column punched cards.)
Two of these computers were given to the University of California (Berkeley), donated by a lumber company in Fort Bragg (Ca.) – presumably as a tax write-off when they upgraded their shop to IBM 360’s. They were stuck into the basement of the engineering building, where a small clutch of nerdy students camped out playing with them.
Some years ago I read an article about a proposed notation that had some logarithmic component to it (maybe something along the lines of “float” computer variables). Does this ring a bell with anyone?
I think the motivation was that many real-world quantities change multiplicatively and are thus more naturally expressed logarithmically.
