The Roman numerals

The Roman system uses I for 1, V for 5 and X for 10. To write a number up to 39, take as many 10s as you can, then 5s, then 1s. So 27 = 10 + 10 + 5 + 1 + 1, written XXVII.

Instead of writing 50 as XXXXX, the Romans gave it a new symbol: L. And just as 4 is written as one less than five, IV, 40 is written as ten less than fifty, XL (though people weren't always consistent, and XXXX turns up too).

Call any number that gets its own new symbol a landmark number. The Roman landmarks are:

(After X, think 'LCD, then M'.)

To write a bigger number, take as many 1000s as you can, then 500s, and so on down. 2367 = 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1, written MMCCCLXVII.

The system seems to have grown out of the ancient Greek one, in Rome around the 8th century BCE, and spread across Europe with the Roman empire.

Compare it with tally marks and the gain is obvious. The reason is worth naming: the Romans group a number not by one size but by a whole sequence of sizes, the landmark numbers. That's the next big step in this story.

But it has a real weakness: arithmetic. Adding is manageable. Try CCXXXII + CCCCXIII: collect the Is, Xs and Cs, then regroup upwards, remembering that five Cs make a D. Fiddly, but possible.

Multiplying is far worse. What is V × L, or L × D? Every pair of landmarks needs its own answer, because the landmarks don't grow by one fixed factor (they go ×5, ×2, ×5, ×2, …). People using Roman numerals relied on a calculating board called the abacus, and only specially trained people could use it.

One caution from the book: don't read these systems as a ladder where each one improves on the last. The real story is far more tangled, and often not clearly known.