Why didn't the Roman numeral system have a zero digit of its own?

On Wikipedia:

The number zero did not originally have its own Roman numeral, but the word nulla (the Latin word meaning "none") was used by medieval scholars to represent 0. Dionysius Exiguus was known to use nulla alongside Roman numerals in 525. About 725, Bede or one of his colleagues used the letter N, the initial of nulla or of nihil (the Latin word for "nothing") for 0, in a table of epacts, all written in Roman numerals.

Although Romans used the word nulla (nothing) to convey the concept of zero, the Roman numerals lack a zero digit in their system.

This is interesting. If the Romans were aware in the mathematical concept of zero, or at least, some concept of "nothingness", then they would have been likely to have a digit for it. However, they didn't, as the two quotes say above.

Well, then what's the reason? Why did they only use unofficial symbols such as nulla or the letter N to represent the concept of zero, and not an actual zero digit?

• Romans did calculations with stones moved along grooves. ("Stone" is "calulus" in Latin.) After the calculation, you copy down the answer: 3 in the hundreds column and 6 in the units column ... CCCVI No need to show zero in the tens column. Jan 4 '20 at 13:34
• Aug 22 '20 at 14:48
• It's commonly also said that they didn't use a zero, and that's why the distance posts start at "One mile" but at the same time you can't walk "Zero miles": i.e., there's no innate value in "showing" zero in transactions of that kind. Aug 26 '20 at 6:04

The digit 0 was not needed in the Roman numeral system because this is not a positional system. The only case when it was used was when the number was actually zero, which they called nulla.
Roman digits have a fixed value independent of where they are in a number. For example, the letter X means 10 per se. Its value is added to the other digits in a number, with the exception that if it comes before a higher value digit it is subtracted instead of added, e.g. XL means 40 (50-10) and LX means 60 (50+10).
In the Indian or Arabic system however, each digit has a value which depends on its position, e.g. the digit 2 can have the value 2 if it is in the last position, 20 if it is in the next position, or 200, etc... In this positional system you need to fill every position in the number to preserve its meaning. That's why number zero becomes necessary to indicate an empty position.

Because, very simply, their number system was evolved to fit the abacus device, in any of its forms, as they used it.

They had an upper and lower register, the upper being half the register value, whatever that might be, and the lower the "ones" for the register. This allowed them to use fewer single item markers in each column of the abacus making its use both easier and less prone to error (pushing a stone for 5 toward the line between the upper and lower halves of the register and two stones for 1 upward to it was less prone to error than moving seven stones for 1. Also faster.

Accordingly, they had a symbol for one in each power of ten register (yes, like us, they used positions (yes, their system was indeed positional: while one COULD write IIMX for 1012, no one ever would for two reasons that will momentarily be obvious) in the form of columns for each power ten) and a symbol for half the value of the register.

The symbols for half the value of the register were LITERALLY the symbol for the next register, halved in some obvious way. So the value for half a register in a "ones" column was "V", the upper half of "X". This is actually hugely more obvious if one looks at the symbols used before they stopped differentiating them from letters and just used the letters they most looked like.

So why write them in particular order (remember IIMX, above)? So that one 1) Wrote them right off the abacus, left (highest) to right (lowest), not in some mixed way that would only confuse and would be prone to the error of forgetting a column. And in the logical manner of usually the half a register symbol, then the ones for the column, unless one had 9 in the column in which case, they seem to have felt it easier to write, say, XC as in "one less" than a full column (as a full column there (the tens) would equal 10 tens, or 100) and this is the second reason why to not mix as in my example above, would he "II" mean two pebbles in the ones column or combine with the thousand to mean "two less than 1000 (998) instead, and 2) Could therefore "write" them right back onto the abacus in the same exact way.

(We are used to working right to left, "carrying" as we call it, to a final answer. They could work either direction more easily than we as one would usually be snapping the new number in (something being added, say) and if overrunning the 9 they could represent, snapping them all outward and snapping one more in the ones half of the next higher register. That would sometimes run into lots of that going left thing and they did seem to love efficiency, so my bet is on loading right to left usually.)

But since they used different symbols for each column (tens position), reading them out left to right (highest to lowest going rightward) did not inhibit them loading right to left as seeing LXX meant activity in the tens column, no question for anyone, and not activity in the ones or ten thousands column. No ambiguity at all.

So, with that background, the need for a "zero" in their number writing DID NOT EXIST and would have served no purpose. An absence of symbols for a column's value meant nothing goes in that column. No need at all for a special symbol to note that: one just simply skips over it/them moving on to the column for the next set of symbols.

Did this mean they never had a need for zero? No, as noted in all the other answers and even the question. Just not in the simple use of the numbers in calculations. Their number system was not positional in its written form, though in practice, they did keep things in order like we would. But the abacus WAS utterly positional and that was where they calculated, not on paper or with a calculator that has to have a way to know a column is empty: their calculator had that in that they just skipped a column as needed.

To make sure references to the abacus are not misunderstood, they mgiht normally have a tray of sand which they'd smooth out when needed, then draw the column lines with a finger, and the upper and lower separator line as well, then place back their sets of pebbles. Nicer "models" might have a paddle to do the smoothing rather than a palm and fingers, a stylus for the lines, and sets of pebbles that were color-coded. Think chess set and expensive chess set. How simple is a tray with sand and pebbles? A more involved setup might be a large sand area where several to many abacus sets might be drawn and pebbled. But they also could have beads or stones on strings in the setup we think of for "abacus." Indeed any arrangement they liked: imagine Alice playing "abacus" rather than "chess"...

One pictures mathemeticians either choosing to work on problems tools existed for, as nowadays, or inventing their own methods, as needed, as nowadays, but no matter their needs and inventiveness, the vast majority of users of numbers would have had no need for or knowledge of, them, as nowadays.

The reason the abacus, and therefore Roman numerals, passed away for most uses is because actual paper came along and gradually became cheap enough to do things like bookkeeping. Mankind is very much driven to choose practical things (outside the field of women's shoes). The Roman numeral system worked for 2,000 years before paper made something different more practical (and cheap enough to be worth doing). It did not get supplanted because it wasn't awfully good at what it did but rather because something better became possible. And the something better (paper) offered a much easier set of methods for doing arithmetic, methods that made the abacus more of a specialty tool. Roman numerals that showed their positional value in their very symbols were no longer needed then either. (Never really noted is that we enlarged the symbol set one needed from seven symbols to 10.)

• That's a very interesting answer. Do you have any sources for the statement: "The Roman numeral system worked for 2,000 years before paper made something different more practical (and cheap enough to be worth doing). It did not get supplanted because it wasn't awfully good at what it did but rather because something better became possible." I find this very interesting.
– Mayo
Sep 2 '20 at 19:16

Simply because the Romans understood nothing as a concept does not mean that they need recognise it as a number per se. After all, every society understands the distinction between having something and having nothing.

Moreover adding nothing to anything leaves that unchanged. This is unlike every other addition, for example, adding two or seven to something. It's perhaps then not suprising that the notion of nothing as a number was slow to arise. It's a question here, not of the mathematical properties of nothing, but of the different ontological status of nothing and something.

Thus, whilst it's commonly said that the Romans didn't have the concept of zero as a number due to their numeral system not being positional, I'd rather say it's simply because of the special ontological status of zero as a number which signifies no thing as opposed to some thing.

We can, for example, easily imagine a Roman numeral system with a zero. But unlike the decimal system, where zero is used positionally, to signify for example, twenty or three hundred and six; a Roman zero would only have a single use, that is to signify nothing and only nothing.

It's worth stressing that this ontological difference between nothing and something disappears once the notion of geometrising number appears. Then zero is a position, as is the number five or twelve. All numbers are then alike, as all positions are alike. This then makes the notion of zero (as well as negative numbers) much more intuitive than that of counting - which measures what is, and what is not. However, Wikipedia suggests that the geometric number line was a relatively recent discovery - which I find odd, given that geometry has been quantified from even before Babylonian times. The very term - geometry - is named after measuring the earth, that is the ground beneath us, after all. It's possibly suggestive of where new research might be worthwhile.