# How did Romans do multiplications?

The Romans did not have Indian numerals. Worse still, they did not have the decimal system. Yet, they produced amazing works of engineering and architecture. How was that possible? It's troublesome to make simple sums, but how could they make products and complex calculations?

Is there any math textbook that survived to tell us how they made complex calculations?

• Bear in mind Roman numerals aren't as dissimilar to Arabic ones as they first seem. Apart from the choice of letter depending on position, the "format" of a digit isn't place-dependent, e.g. compare IV, XL, CD. So the way you learned to multiply still works; you just have to mentally work out which letters to use when writing down everything.
– J.G.
Nov 9, 2018 at 19:27
• They didn't - they got their slaves to do it for 'em, saves all that tedious business of multiplying out by long hand in a very inefficient representation of numbers! Nov 11, 2018 at 22:10
• The fact that abacus is a Latin word, even though modern abaci are invariably East Asian, shows that the technique is ancient in the West as well as the East. There is also the fact that calculus is the Latin word for 'pebble' (it's a diminutive of the word for 'rock', from the same root as calcium and chalk) and is also the source of the words calculate and calculation, which all dealt with moving pebbles around on the abacus table. Dec 12, 2018 at 23:07
• As far as I know, no record of a Roman multiplication has ever found, but there is always hope. May I suggest a method that they could have used which would be both simple and quick. All that is required for the Roman multiplier is to learn 7 rules, each rules for each of the number representations, I, V, X, L, C, D, and M. They would also need to know the relative position of these representations, ie IVXLDM. The 7 rules are: I. Copy V. Take 5 copies X. Shift 2 places to the right if possible. L. Shift 2 places to the right if possible and take 5 copies C. Shift 4 places to the right if possi Sep 24, 2020 at 17:02
• [continuation of Alex Lax' comment] [...] possible D Shift 4 places to the right if possible and take 5 copies M. Shift 6 places to the right if possible. Those are the only rules required. It would seem reasonable that a few more representations would make life easy if if they really didn’t exist then you would have to simply fill in as best as you can. So that you would have to have multiple values of M and D. Finally you simply add up all the representation and obtain your answer. If anybody is interested I would be happy to provide examples.
– Danu
Sep 26, 2020 at 23:48

They used abacus. The techniques used for operations with abacus were understood and were basically the same used also until quite recent time also in China and Japan, as far as I know. This does not require indoarabic numerals.

In fact you shouldn't think "Latins". Latin numerals were in use in Europe till the XIIIth century at least.

Liber Abaci by Leonardo Fibonacci is credited as being one of the main sources of introduction of indo-arabic numerals in the Western countries and dates back to 1228. One of the reason of its success is exactly that it explained how using such numerals could improve computations. It is, in fact, a Book of Abacus, as the name says, i.e. a book centered around techniques for algebraic computations.

• Or, also, pebbles-in-depressions-on-a-piece-of-wood version of abacus, rather than beads-on-wires, quite often. Same function, certainly, but not what many people would think of as "abacus". Nov 10, 2018 at 23:38
• I think you mixed it up, in 1200, Fibonacci introduced in Italy and Europe the indian decimal system and , consequently, the chinese multiplication table Nov 11, 2018 at 9:49
• @paul garrett Yes. But in Italy you still call it abacus, so I don't know how I'd say differently. I haven't discussed about chinese multiplication table. I just said that Abacus multiplication techniques are basically the same. "Indian decimal system" is the same thing I call "indoarabic numerals". Fibonacci learned them from Arabs and they're called "arabic numerals" usually, for this reason. Nov 11, 2018 at 12:17
• @paulgarrett ... pebble moved in wooden grooves: this was so common that the Latin word for pebble (calculus) has come to mean "calculation" Sep 19, 2021 at 13:25

You should be more specific when you say "Romans". If you mean ancient Romans, almost no mathematical text survived in Latin from the times before 2nd century AD. From the Roman empire we mostly have Greek texts. (See also Roman engineers). Almost all technical literature which we have from the Roman empire is written in Greek. Greek was also the spoken language of large portion of the empire.

Greek system is described in detail in the book of van der Waerden, Science awakening. The digits of the decimal system were denoted by Greek letters. One had to memorize (as we do) the multiplication table for digits. That's all one need to multiply numbers. For example, $$265\times265=200\times200+200\times60+200\times5+60\times200+60\times60$$ $$+60\times5+5\times200+5\times60+5\times5=70225.$$ To simplify the task, a counting board was used with counting stones. Such a counting board in mentioned in Polybius, for example. For computations with simple fractions a more complicated algorithm was used.

For astronomical computations, Babylonian positional system with base 60, including fractions based on 1/60th was used (but the numbers from 1 to 60 were denoted by pairs of Greek letters). Multiplication tables were used (as Babylonians did before).