# Did the Romans really use the binomial formula to calculate products?

I'm not quite sure if this is the right place to ask this question (in fact, I was redirected to this SE from the Math Stackexchange), but it's probably more fitting than the original posting place.

I was told the following as a motivating story in an undergraduate intro to algorithms class years back. Unfortunately, I have no means of obtaining a source on this account, and now wonder about its veracity:

In Roman times, when going to the market, it would be a commonplace problem to multiply e.g. a price with a quantity of an item to calculate the total price. Since people weren't able to do grade-school multiplication, let alone in their heads, this posed a serious problem. One solution would have been to produce boards that showed a multiplication table. But the quadratic space (weight, production time, etc.) requirements made this not viable.

However, Romans could do addition, subtraction (using some aids, possibly), and division by two, relatively easily; at least compared to outright multiplication of arbitrary numbers. So the clever solution the Romans came up with was to produce boards (of linear size in the size of the range of numbers to be captured) that contained a list of squares. Then, they could employ $$ab=\frac{(a+b)^2-a^2-b^2}2$$ to perform multiplication, reading off the squares from their table and adding/subtracting as they knew how to do it.

Is this something the instructor made up, or has anyone ever heard of this practice? If so, I would be extremely happy about a source for this.

• I could not find anything on the anecdote, but Romans did use tables of squares (Hilton, Roman Elementary Mathematics), and the formula, in geometric form, is already in Elements II.4, so it is possible. Sep 13, 2023 at 11:08
• The romans used abakus to multiply, it seems it was very common, probably also n market places. You find some still in museums Sep 16, 2023 at 14:28