Roman numerals: grouping a number into tens, fives and ones

0:00You can already read this system, at least a little. One is a mark. Two is two marks. Three is three. Then something changes: four and five and six stop being marks and start being letters. There is a rule underneath that, and it is not memorisation. Take as many of the biggest thing you have as will fit, then move down to the next biggest, and keep going. The letters are what falls out at the end.

0:29And the whole character of this system — everything easy about it and everything awful — comes from one property of the sizes it uses, which looks like a detail. Start small. Write twenty-seven. How many tens fit? Two. That is twenty, with seven to go. How many fives fit into seven? One. Two left. Then two ones. So twenty-seven is ten, ten, five, one, one. Write the letter for each, largest first, and you have it.

1:04Notice what you did: you never had to know the numeral. You found the pieces, and the pieces have names. That is the rule for anything up to thirty-nine, and it never needs more than three of the same letter in a row. So what about fifty? By the rule so far, fifty is five tens. Five X's in a row. You could write that, and people did. But five identical letters is the row-of-marks problem again — you would have to count them.

1:38So fifty gets a letter of its own. L. One sign instead of five, and notice how many it replaced: five. That number is going to matter in a moment. And the system now has a new size to group by, which is what makes bigger numbers writable at all. Now a wrinkle, and it is an honest one rather than a mistake. Forty, by the rule, is four tens. XXXX.

2:08But people also wrote it as one ten LESS than fifty — a small sign in front of a big one means take it away. So forty becomes XL. The same trick gives four as one less than five, IV, instead of four marks. Both forms were used. Not one right and one wrong — both, by the same people, sometimes on the same object. Worth saying plainly, because the shorter form gets taught as the only correct one, and that is a modern tidiness.

2:45Here is the full set of sizes the system groups by. One, five, ten, fifty, one hundred, five hundred, one thousand. Seven of them, and each has a letter. Now do what the flat list hides. Ask how many of each makes the next one up. Five ones make a five. Two fives make a ten. Five tens make a fifty. Two fifties make a hundred. Five hundreds make a five hundred, and two of those make a thousand.

3:17Five, two, five, two, five, two. That is the whole video. Not one step size — two, alternating, all the way up. Every system you use climbs by the same factor each time. This one does not. Before we count the cost, watch the rule do something impressive. Two thousand three hundred and sixty-seven. Biggest first. How many thousands? Two. Three hundred and sixty-seven left. Hundreds? Three. Sixty-seven left. One fifty. Then one ten, one five, and two ones.

3:56Ten pieces, and every one of them has a letter. A four-figure number in ten signs, by a procedure you could teach in a minute. The system is genuinely good at this. Which is why it is worth being precise about where it is not. You can add two of these without turning either into a number first. Take two hundred and thirty-two and four hundred and thirteen, in their own letters. Pour them together and count each kind.

4:28Five ones, four tens, six hundreds. Now tidy from the bottom. Five ones become one five. The tens are fine, four is under the limit. And the hundreds — six of them. Five hundreds make a five hundred, so trade five for one D and keep the one left over. The answer has a D in it. Look at what you started with: the biggest letter in either numeral was C.

4:56The largest sign present is not the largest sign reachable, and that is where people lose this sum. But look at what you had to remember to do that. Five ones for a five. Then TWO fives for a ten. Then five tens for a fifty. Then two fifties for a hundred. Those are not four facts. They are the steps between the landmarks, read off the same ladder. Five, two, five, two — the same alternation, showing up as the thing your hands have to do.

5:31And it has to be remembered separately at each rung, because there is no single number that works. Trade at five everywhere on a different sum and you end up with three fives and three fifties still standing. That is not a numeral. One ladder, two step sizes, and the arithmetic inherits both. Addition survives that. Multiplication does not. Multiply five by fifty. Two hundred and fifty. An ordinary number, and the system has no sign for it — you write two hundreds and a fifty.

6:10Now multiply ten by a hundred. One thousand, landing exactly on a landmark. One letter. So some products land on the ladder and some fall between the rungs. Which ones? Take every pair of landmarks whose product the ladder could hold at all. Twenty-two of the twenty-five land on it. So the ladder is not unruly everywhere. Only three pairs miss. And every one of those three involves a landmark that a five-step landed on.

6:43The two-steps are fine. It is the fives that throw products off the ladder. There is a second problem, and it is blunter. Multiply fifty by five hundred. Twenty-five thousand. The largest sign this system has is one thousand. So twenty-five thousand is twenty-five of them in a row. Nothing but M, twenty-five times. The row-of-marks problem again, at the top of the ladder instead of the bottom. Every fix this system found was a new sign for a new size, and that works until you run out of signs.

7:23Answering bigger numbers by inventing more letters does not solve the problem. It postpones it. And people knew. The hard arithmetic was done on a counting board, by people trained to use one. One thing to be careful about before going on. It is tempting to lay these systems in a line and call each an improvement on the last. That is not what happened. The real history is tangled, much of it unknown, and systems overlapped for centuries.

7:59The Romans were not bad at arithmetic. They had a tool, and the tool worked. We are comparing notations, not people — how much work a way of writing makes you do. And on that question this system has one weakness, which we can now say exactly. It is not the letters, and it is not the subtraction trick. It is that consecutive sizes are not related by a constant factor.

8:29Compare. Build a ladder that climbs by ten every time: one, ten, a hundred, a thousand. Same reach. But now ask the two awkward questions again. What do you trade at each rung? Ten. At every rung. One number to remember, not two. And which products land on the ladder? Every single one that fits. Not most. All of them. That is the same test that gave three misses before, and on this ladder it gives none.

9:01One property, and it fixes both things at once. So take stock of what this system did get. It grouped by a sequence of sizes instead of one, which is what makes a four-figure number fit in ten signs. It found that a new size deserves a new sign, and it made the writing mechanical: biggest first, work down, letters at the end. What it never did was make the sizes climb evenly.

9:32So an adder carries two rules instead of one, a multiplier keeps falling between the rungs, and the whole thing stops dead at its largest letter. The next system in this story climbs by ten every time, and the difference is immediate. Its landmarks multiply into each other cleanly, which turns multiplication into something you can do by doubling. The advantage will not look like better letters. It will look like a ladder with even rungs.