Why did Roman numerals make arithmetic so hard?

There is no zero and no place value. In our system the position of a digit tells you its magnitude, so adding and multiplying can be done column by column with a small set of memorised facts. In Roman numerals the symbols carry fixed values wherever they sit, so there are no columns to line up and no algorithm to follow. Try multiplying XLVII by XIX on paper and the problem becomes obvious immediately.

The Romans did not attempt it that way. Calculation was done on a counting board or abacus with pebbles - calculi, which is where calculate comes from - and the numerals were used to write the result down, not to work it out. The notation was for recording, and it is perfectly good at that.

The system that replaced it came from India via the Arabic-speaking world, and Europe resisted it for centuries. Florence banned the new numerals in 1299, partly on the grounds that they were easy to alter - a 0 could be made into a 6 or a 9 - where Roman numerals are harder to tamper with. It took until roughly the sixteenth century for the argument to be settled, and the winner is the reason ordinary people can do arithmetic at all.