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I wonder how the Romans were able to control so vast territories, peoples, armies, ammunition, taxes, etc., without the help of a number system that allowed easy computing (e.g. multiplication and division) of (relatively) large quantities.

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    $\begingroup$ This is what accountants are for. Even now, with a more efficient decimal system, calculators, etc., firms and individuals employ accountants to deal with such computations. $\endgroup$ – Moishe Kohan Apr 9 at 14:52
  • $\begingroup$ @Conifold Many thanks for the links. They provide some insights. Still I find it hard to understand how such an empire could be "run" with some tricks and the abacus. $\endgroup$ – Leandro Caniglia Apr 9 at 23:56
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    $\begingroup$ People did more with less, the Persians and the Egyptians did not even have the abacus. Having an army of professional scribes helps, and comparing to our modern positional system distorts the impression of the difficulty for those who mastered alternative tricks. $\endgroup$ – Conifold Apr 10 at 0:06
  • $\begingroup$ @LeandroCaniglia, what experience have you in running such an empire? Compared to today, it must be understood that there was generally a lot more local autonomy, and a lot less bookkeeping. $\endgroup$ – Steve Apr 10 at 12:51
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Romans used tricks for arithmetic that worked well enough, see Roman Numerals and Arithmetic on ScienceBlogs. Addition and subtraction was quite simple, as for multiplication:

"Multiplication using Roman numerals is not particularly easy or obvious. You can do the trivial thing, which is repeated addition. But it should be pretty obvious that that's not practical for large numbers. The trick that they used was actually pretty nifty. It's basically a strange version of binary multiplication. You need to be able to add and divide by two, but those are both pretty easy things to do. So here goes:
Given A×B, you create two columns, and write A in the left column, and B in the right. Then:
1. Divide the number in the left column by two, discarding the remainder. Write it down in the next row of the left column.
2. Multiply the number in the right column by two. Write it down in the right column next the the result from step 1.
3. Repeat from step 1 until the value in the left column is 1.
4. Go down the table, and cross out every row where the number in the left column is even.
5. Add up the remaining values in the right column.

Division is the biggest problem in roman numerals. There is no good trick that works in general. It really comes down to repeated subtraction. The only thing you can do to simplify is variations on finding a common factor of both numbers that's easy to factor out. For example, if both numbers are even, you can divide each of them by two before starting the repeated subtraction. It's also fairly easy to recognize when both numbers are multiples of 5 or 10, and to do the division by 5 or 10 on both numbers. But beyond that, you take a guess, do the multiplication, subtract, repeat."

But for the more serious calculations they did have the Roman abacus and reckoning boards. They effectively implemented a base 10 system. Here is a reconstruction of a Roman hand abacus, made by the RGZ Museum in Mainz from the bronze original held by the Bibliothèque nationale de France in Paris. Many counter beads are missing.

enter image description here

From Menninger's Number Words and Number Symbols, p.315:

"For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. The Etruscan cameo and the Greek predecessors, such as the Salamis Tablet and the Darius Vase, gives us a good idea of what it must have been like, although no actual specimens of the true Roman counting board are known to be extant. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more. Above all, it has preserved the fact of the unattached counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called psephoi, the Romans called calculi. The Latin word calx means 'pebble' or 'gravel stone'; calculi are thus little stones (used as counters)."

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