The Roman system of number symbols had similarities to both the Greek and Babylonian systems. Like the Greeks, the Romans used letters of the alphabet. However, they did not use them in order, but use just a few letters which they repeated as often as necessary-as in the Baby lonian system. Unlike the Babylonians, the Romans did not invent a new symbol for every tenfold increase of number, but (more primitively) used new symbols for fivefold increases as well.
'Thus, to begin with, the symbol for "one" is 1, and "two," "three," and "four," can be written II, III, and IIII.
The symbol for five, then, is not 11111, but V. People have amused themselves no end trying to work out the reasons for the particular letters chosen as symbols, but there are no explanations that are universally accepted.
However, it is pleasant to think that I represents the up held fin-er and that V might symbolize the hand itself with all five fingers-one branch of the V would be the out held thumb, the other, the remaining fingers. For "six," "seven," "eight," and "nine," we would then have VI, VII, 'VIII, and VIIII.
For "ten" we would then have X, which (some peo ple think) represents both hands held wrist to wrist.
"Twenty-three" would be XXIII, "forty-eight" would be XXXXVIII, and so on.
The symbol for "fifty" is L, for "one hundred" is C, for "five hundred" is D, and for "one thousand" is M. The C and M are easy to understand, for C is the first letter of centum (meaning "one hundred") and M is the first letter of rnille (one thousand).
For that very reason, however, those symbols are sus picious. As initials they may have come to oust the original less-meaningful symbols for those numbers. For instance, an alternative symbol for "thousand" looks something like this (1). Half of a thousand or "five hundred" is the right half of the symbol, or (1), and this may have been con verted into D. As for the L which stands for "fifty," I don't know why it is used.
Now, then, we can write nineteen sixty-four, in Roman numerals, as follows: MDCCCCLXIIII.
One advantage of writing numbers according to this sys tem is that it doesn't matter in which order the numbers are written. If I decided to write nineteen sixty-four as CDCLIIMXCICT, it would still represent nineteen sixty four if I add up the number values of each letter. However, it is not likely that anyone would ever scramble the letters in this fashion. If the letters were written in strict order of decreasing value, as I.did the first time, it would then be much simpler to add the values of the letters. And, in fact, this order of decreasing value is (except for special cases) always used.
Once the order of writing the letters in Roman numerals is made an established convention, one can make use of deviations from that set order if it will help simplify mat ters. For instance, suppose we decide that when a symbol of smaller value follows one of larger value, the two are added; while if the symbol of smaller value precedes one of larger value, the first is subtracted from the second. Thus VI is "five" plus "one" or "six,"' while IV is "five" minus "one" or "four." (One might even say that IIV is "three," but it is conventional to subtract no more than one sym bol.) In the same way LX is "sixty" while XL is "forty"; CX is "one hundred ten," while XC is "ninety"; MC is 44 one thousand one hundred," while CM is "nine hundred."
The value of this "subtractive principle" is that two sym bols can do the work of five. Why write VIIII il you can write IX; or DCCCC if you can write CM? The year nine teen sixty-four, instead of being written MDCCCCLXIIII (twelve symbols), can be written MC@XIV (seven sym bols). On the other hand, once you make the order of writing letters significant, you can no longer scramble them even if you wanted to. For instance, if MCMLXIV is scrambled to MMCLXVI it becomes "two thousand one hundred sixty-six."
The subtractive principle was used on and off in ancient times but was not regularly adopted until the Middle Ages.
One interesting theory for the delay involves the simplest use of the principle-that of IV ("four"). These are the first letters of IVPITER, the chief of the Roman gods, and the Romans may have had a delicacy about writing even the beginning of the name. Even today, on clockfaces bear ing Roman numerals, "four" is represented as 1111 and never as IV. This is not because the clockf ace does not ac cept the subtractive principle, for "nine" is represented as IX and never as VIIII.
With the symbols already,given, we can go up to the number "four thousand nine hundred ninety-nine" in Ro man numerals. This would be MMMMDCCCCLXXXX VIIII or, if the subtractive principle is used ' MMMM CMXCIX. You might suppose that "five thousand" (the next number) could be written MMMMM, but this is not quite right. Strictly speaking, the Roman system never re quires a symbol to be repeated more than four times. A new symbol is always invented to prevent that: 11111 = V; XXXXX = L; and CCCCC = D. Well, then, what is MMMMM?
No letter was decided upon for "five thousand." In an cient times there was little need in ordinary life for num bers that high. And if scholars and tax collectors had oc casion for larger numbers, their systems did not percolate down to the common man.
One method of penetrating to "five thousand" and be yond is to use a bar to represent thousands. Thus, V would represent not "five" but "five thousand." And sixty-seven thousand four hundred eighty-two would be LX-VIICD LXXXII.
But another method of writing large numbers harks back to the primitive symbol (1) for "thousand." By adding to the curved lines we can increase the number by ratios of ten. Thus "ten thousand" would be (1)), and "one hundred thousand" would be (1) Then just as "five hundred" was 1) or D, "five thousand" would be 1)) and "fifty thousand" would be I))).
Just as the Romans made special marks to indicate tbou sands, so did the Greeks. What's more, the Greeks made special marks for ten thousands and for millions (or at least some of the Greek writers did). That the Romans didn't carry this to the logical extreme is no surprise. The Romans prided themselves on being non-intellectual. That the Greeks missed it also, however, will never cease to astonish me.
Suppose that instead of making special marks for large numbers only, one were to make special marks for every type of group from the units on. If we stick to the system I introduced at the start of the chapter-that is, the one in which ' stands for units, - for tens, + for hundreds, and = for thousands-then we could get by with but one set of nine syrrbols. We could write every number with a little heading, marking off the type of groups -+-'. Then for "two thousand five hundred eighty-one" we could get by with only the letters from A to I and write it GEHA. What's more, for "five thousand five hundred fifty-five" we could write EEEE. There would be no confusion with all the E's, since the symbol above each E would indicate that one was a "five," another a "fifty," another a "five hundred," and another a "five thousand." By using additional symbols for ten thousands, hundred thousands, millions, and so on, any number, however large, could be written in this same fashion.
Yet it is not surprising that this would not be popular.
Even if a Greek had thought of it he would have been re peucd by the necessity of writing those tiny symbols. In an age of band-copying, additional symbols meant additional labor and scribes would resent that furiously.
Of course, one might easily decide that the symbols weren't necessary. The Groups, one could agree, could al ways be written right to left in increasing values. The units would be at the right end, the tens next on the left, the hun dreds next, and so on. In that case, BEHA would be "two thousand five hundred eighty-one" and EEEE would be "five thousand five hundred fifty-five" even without the little symbols on top.
Here, though, a difficulty would creep in. What if there were no groups of ten, or perhaps no units, in a particular number? Consider the number "ten" or the number "one hundred and one." The former is made up of one group of ten and no units, while the latter is made up of one group of hundreds, no groups of tens, and ont unit. Using sym bols over the columns, the numbers could be written A and A A, but now you would not dare leave out the little sym bols. If you did, how could you differentiate A meaning "ten" from A meaning "one" or AA meaning "one hun dred and one" from AA meaning "eleven" or AA meaning "one hundred and ten"?
You might try to leave a gap so as to indicate "one hun dred and one" by A A. But then, in an age of hand-copy ing, how quickly would that become AA, or, for that mat ter, how quickly might AA become A A? Then, too, how would you indicate a gap at the end of a symbol? No, even if the Greeks thought of this system, they must obviously have come to the conclusion that the existence of gaps in numbers made this attempted simplification impractical.
They decided it was safer to let J stand for "ten" and SA for "one hundred and one" and to Hades with little sym bols.
What no Greek ever thought of-not even Archimedes himself-was that it wasn't absolutely necessary to work with gaps. One could fill the gap with a symbol by letting one stand for nothing-for "no groups." Suppose we use $ as such a symbol. Then, if "one hundred and one",is made up of one group of hundreds, no groups of tens, an one + - I unit, it can be written A$A. If we do that sort of thing, all gaps are eliminated and we don't need the little symbols on top. "One" becomes A, "ten" becomes A$, "one hun dred" becomes A$$, "one hundred and one" becomes A$A, "one hundred and ten" becomes AA$, and so on.
Any number, however large, can be written with the use of exactly nine letters plus a symbol for nothinc, Surely this is the simplest thing in the world-after you think of it.
Yet it took men about five thousand years, counting from the beginning of number symbols, to think of a sym bol for nothing. The man who succeeded (one of the most creative and original thinkers in history) is unknown. We know only that he was some Hindu who lived no later than the ninth century.
The Hindus called the symbol sunyo, meaning "empty."
This symbol for nothing was picked up by the Arabs, who termed it sifr, which in their language meant "empty." This has been distorted into our own words "cipher" and, by way of zefirum, into "zero."