How did anyone do math in Roman numerals? (2017)

How did anyone do math in Roman numerals? (2017)
washingtoncitypaper.com

Never actually used any of it so most of it has evaporated from my memory along with calculating square roots by hand, but it's nice to know at least enough to be able to look up the information if I want it.

The base with best radix economy (except for e) is 3. But roman numerals are better still.

For representing 0-999, you would need 19 base-3 digits, but only 15 roman numerals (plus a symbol for zero).

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that ... a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility ... Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle—they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

John Allen Paulos discusses the power of notation in his book Beyond Numeracy:

A German merchant of the fifteenth century asked an eminent professor where he should send his son for a good business education. The professor responded that German universities would be sufficient to teach the boy addition and subtraction but he would have to go to Italy to learn multiplication and division. Before you smile indulgently, try multiplying or even just adding the Roman numerals CCLXIV, MDCCCIX, DCL, ANDMLXXXI without first translating them.

Numbers may be eternal and invariant, but numerals, the symbols used to represent numbers, are not, and the above anecdote illustrates how easy it is to take for granted the Hindu-Arabic numerals we use today. The history of numeration systems is a long one extending from prehistoric times to the adoption in the Renaissance of our present system. The heroes of the story are the nameless scribes, accountants, priests, and astronomers who discovered the principles of representing numbers systematically.