Why a base alone still runs out of symbols

0:00A system of sizes built by a rule, one sign for each size, and one instruction for putting numbers together. It works, and the thing that makes it work is easy to miss. No sign ever stands ten times. Not because anybody forbade it — because ten of a sign IS one of the next, so a writer who leaves ten standing has simply stopped halfway. Write every number below ten thousand and look at the numerals: not one of them has ten of anything, and nine really does happen.

0:36So a numeral never grows past nine marks at any one size, however big the number gets. That is a real bound, and it is the only one the system has. Because there is a second thing, and it is not bounded at all. Count the signs. There are eight of them, and the largest stands for ten million. Fill every one of them as full as the rule allows — nine of each.

1:04That numeral is seventy-two marks long, and it reads ninety-nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine. One short of a hundred million. Below that, everything is fine. The system is comfortable, the numerals are short, the arithmetic is easy. At that number, the system stops. Not slows down — stops. And here is what stopping actually looks like, which is not what you would guess. Ask for a hundred million and the writer does not fail. It does what it always does: take as many of the largest as will fit.

1:47As many as will fit is ten. So back comes a numeral: ten of the largest sign, and nothing else. It reads back to exactly the right number. It is not wrong. It is unfinished — ten of a sign has to become one of the next, and there is no next. Every other numeral in this system can be finished. This one cannot, and nothing about the writing tells you so.

2:14The system did not run out of numbers. It ran out of pictures. Why there, and not somewhere else? Because the ladder of sizes never stops. Ask the rule for the size after ten million and it answers instantly: a hundred million. The rule is fine. It is the drawing that has run out. Every new size wants a brand-new sign, and there is no end to the sizes, so there is no end to the signs.

2:47You can always invent one more. Then you need one more after that. That is not a solution. It is the same problem, postponed by exactly one sign. So the thing worth looking at is not the eighth sign. It is the cost that made an eighth sign necessary. There are two costs to writing a number down, and it is worth separating them. The first is how many marks go on the page.

3:16The second is how many DIFFERENT signs the system had to have before you could start. Take nine, ninety-nine, nine hundred and ninety-nine, nine thousand nine hundred and ninety-nine. Marks: nine, then eighteen, then twenty-seven, then thirty-six. Each one is nine more. Different signs: one, then two, then three, then four. Both are climbing. That looks like one cost measured twice, and it is not. One of them is about the number. The other is about how far up the ladder the number has taken you.

3:55Watch what happens when you change the number without changing its length. One. Eleven. One hundred and eleven. One thousand one hundred and eleven. Marks: one, two, three, four. That is one ninth of what the nines cost. Different signs: one, two, three, four. Exactly the same as the nines. So between a thousand one hundred and eleven and nine thousand nine hundred and ninety-nine, the marks go up ninefold and the sign set does not move at all.

4:28The marks depend on the digits. The sign set depends only on how far up the ladder you have gone. A longer numeral does not mean a bigger sign set, and a bigger sign set does not mean a longer numeral. Two costs, and they move independently. Now put the bound back beside them. Take every number below twenty thousand and find the largest number of marks standing at any one size. It is nine.

5:00Take a range with nothing at all in the ones — every tenth number — and look again. Still nine, standing further up the ladder. Go further and it is still nine, because that is what the trading rule means. Now do the same for the sign set. Over those twenty thousand numbers it reaches five, and it reaches five because five sizes were needed. Ask for a bigger number and it will ask for a sixth, and a seventh.

5:30One cost is capped forever. The other has nothing holding it down at all. And it is the second one that decides where the system stops, because the signs are what you have to invent. Make that concrete with a system small enough to watch. Bundle four instead of ten. One, four, sixteen. One, two, three — three marks of the first sign, and no need for anything else. Four. The first sign is full, and the second sign arrives.

6:04Five is one four and one. Six is one four and two. Eleven is two fours and three. Fifteen is three fours and three — as full as two signs can be. Sixteen. Two signs are not enough any more, and the third one arrives, exactly there. Fifteen numbers, and the sign set has already grown twice — which in base ten would have taken you to a hundred. Sixteen is not a coincidence. It is four times four.

6:37Base ten does the same thing and takes longer: its second sign arrives at ten, its third at a hundred. Base two asks for a second sign at two, and a third at four. Line them up — base two through base ten — and the third sign is forced at four, nine, sixteen, twenty-five, thirty-six, forty-nine, sixty-four, eighty-one, a hundred. Every one of them is the base multiplied by itself.

7:08So the smaller the base, the sooner you need another sign: base four needs its third one eighty-four numbers earlier than base ten does. And the marks go the other way — base two never puts more than one of a sign, base ten allows nine. That is a genuine trade, and it is worth saying plainly. A small base keeps its numerals thin — few of any one sign — and burns through sizes quickly.

7:39A large base allows long runs of the same sign and climbs the ladder slowly. Neither escapes. Both need a new sign at every new size, forever. Choosing the base moves the cost around. It does not remove it. Whatever you choose, the sign set grows with the number, and no amount of bundling changes that. Bundling was never the thing that was going to fix it. Before the fix, one last thing the rule does hand you for free.

8:12Multiplying by the base. Take forty-three, written in the five-signs. Three ones, three fives, and one twenty-five. Multiply by five, and every sign moves up exactly one rung: three fives, three twenty-fives, one one-hundred-and-twenty-five. Two hundred and fifteen, without a single multiplication being carried out. That is the same fact as writing a zero on the end of a decimal number, in a system where it does not look like a zero at all.

8:47One rule, any base — and it is worth noticing it works because of the sizes, not because of the signs. So here is where all of it lands. Bundling gave us a cap on the marks: never as many as the base, at any size, ever. It gave us nothing at all on the sign set, which needs a fresh sign for every new size and always will. The wall at ninety-nine million and change is not a shortage of imagination. It is the design.

9:22And the repair cannot be another sign, because another sign just moves the wall. What has to go is the idea that a size must be DRAWN. If where a mark sits could say which size it means, the sign set would stop growing — and that is the next thing to build. Ten signs, and no eleventh, ever. That is what the last idea in this story is for.