What "base n" means, and why ten is a choice not a law

0:00Last time a system stopped choosing its sizes and built them with a rule: start with one, take ten of whatever you just made. Every size came out a power of ten, and that felt like the answer. But look at the instruction again. Where is the ten? It is a number somebody put in. Nothing about the rule needs it to be ten. So put five in. One. Five of those, which is five. Five of those, twenty-five. Then a hundred and twenty-five, six hundred and twenty-five, three thousand one hundred and twenty-five.

0:39Same instruction, different number, and a usable ladder comes out the other end. Ten is a choice about symbols. It is not a fact about numbers. Give the new ladder signs, the same way. A triangle for one. A square for five. A hexagon for twenty-five. A circle for a hundred and twenty-five. Then a wave, then an arrow. Six signs for six sizes, and every size is a power of five.

1:11Now notice what did not have to be rebuilt. The steps between the sizes are all equal, so there is still exactly one number to trade at. And writing still works the same way: take as many of the biggest as will fit, then move down. Nothing was special-cased. The whole construction survived the change. Write a hundred and forty-three. How many of the biggest size fit? One circle, and eighteen to go.

1:44How many hexagons — twenty-fives? None. Eighteen is not enough for one. Squares, worth five? Three of them. That is fifteen. Three left. Three triangles. One circle, three squares, three triangles. Seven signs, and they read back to a hundred and forty-three. The same procedure you already know, on a ladder nobody grew up with. Now the definition can be written down, and it needs only two clauses. One: the first size is one.

2:20Two: every next size is the one before it, multiplied by some fixed number. A system satisfying both is a base system, and that fixed number is its base. Ours are base ten and base five, and base ten is the one everybody calls decimal. Now a test with teeth. Take the Roman sizes. One, five, ten, fifty, a hundred, five hundred, a thousand. First clause, fine. It starts at one.

2:53Second clause, no. It climbs by five, then two, then five, then two. Two different numbers, so it is not a base system at all. Here is what the second clause buys, and it starts with addition. Eighty-seven and seventy-eight, in the ten-signs, without turning either into a number first. Pour them together and count each kind. Fifteen ones. Fifteen tens. That is not a numeral anybody would write. So: ten of a sign becomes one of the next.

3:28Ten ones make a ten. Five ones left, and sixteen tens now. Ten of those make a hundred. Six tens left. One hundred, six tens, five ones. A hundred and sixty-five, after two single trades. Now watch how an error hides inside that. Suppose the poured row had been written down with one stroke too many. Sixteen ones instead of fifteen. Regroup it exactly as before. Ten ones make a ten. Six ones left.

4:03And the answer comes out one hundred, six tens, six ones. That is a legal numeral. No sign stands more than nine times. Nothing about it looks wrong. It is a hundred and sixty-six, and the right answer is a hundred and sixty-five. Regrouping does not detect a miscount. It tidies it, and hands you back something that looks finished. So take one sum and do it three ways at once.

4:34Forty-seven plus fifty-six. In digits: seven and six make thirteen, write three, carry one. Four and five and the carried one make ten. A hundred and three. In the ten-signs: four tens and seven ones, five tens and six ones. Poured together, thirteen ones and nine tens. The same two trades, and the same answer. In the five-signs, forty-seven is one hexagon, four squares, two triangles. Fifty-six is two hexagons, one square, one triangle.

5:11Poured together: three triangles, five squares, three hexagons. Five squares make one hexagon — there is the trade — and the squares vanish completely. Four hexagons and three triangles. One trade instead of two, a numeral that looks nothing like the others, and the same number underneath. The number of trades differed. The instruction did not. Ten of a sign for one of the next, or five of a sign for one of the next. One number, whatever the place.

5:47And that number is not a matter of taste. Take the eighty-seven sum and trade it at five instead of ten. You get something that still looks like a finished numeral. No sign standing more than nine times. It reads three hundred and thirty. The number changed, and nothing complained. The trade count is forced. It IS the ratio between neighbouring sizes, so a system with two ratios needs two trade numbers.

6:18One question, though. All of that is about adding. Is any of it actually deep? Here is the fact the whole thing rests on, and it is about multiplying. In a base system, multiply any two sizes and the answer is another size. A hundred times a thousand is a hundred thousand. Twenty-five times a hundred and twenty-five is three thousand one hundred and twenty-five. On the ten-ladder, every product of two sizes that fits is itself a size. Thirty-six pairs, thirty-six landings.

6:58On the five-ladder, twenty-one out of twenty-one. On the Roman list, twenty-two out of twenty-five land, and three fall between the rungs onto numbers with no sign at all. That closure is the difference. It is what turns multiplying into bookkeeping instead of a special case each time. Watch closure do visible work. Multiply a whole numeral by its base, and every sign moves up exactly one rung. Why? Break the numeral apart. A hundred and twenty-one is a hundred, plus twenty, plus one.

7:38Multiply each piece by ten. A thousand, plus two hundred, plus ten. Twelve hundred and ten. Every piece landed on the next size up, because a size times the base is a size. Nothing had to be worked out. That is a shift, and it is why multiplying by ten looks like writing a zero on the end. It is not a trick of notation. It is closure, plus being allowed to break a sum apart before multiplying it.

8:11Closure does have a limit, and it is not in the arithmetic. Take four products of sizes, named by how far up the ladder each one sits. One rung times five rungs. Two times three. Three times three. Four times six. Add the positions. Six. Five. Six. And ten. The eight signs of the ten-ladder reach position seven. Ten is three rungs past the end of them. The answer exists. The rule that makes it exists. There is simply no picture for it.

8:48Written with the signs there are, it would be a thousand copies of the largest one. Same complaint as before, further up the ladder. A base fixes the arithmetic; it does not stop the pictures running out. One more thing a base gives you, and this one you can hold. A board with lines. The bottom line is ones, the next is tens, then hundreds, then thousands. A counter on a line counts once. A counter placed above a line counts five of that line.

9:24Three thousand four hundred and twenty-six. Three counters on the thousands line, four on the hundreds, two on the tens. And six ones — which is not six counters. One counter above the line, worth five, and one on it. Eleven counters where fifteen would have been needed. The board works only because every line is ten of the one below. That is clause two again, made out of wood.

9:54Add on it, and the trade is that same move once more. Two thousand nine hundred and seven, and forty-three. Sweep the lines together. Seven ones and three ones make ten ones — so all ten come off, and one counter goes up to the tens line. Two thousand nine hundred and fifty. So what did a base buy? One trade number instead of one per rung. Products of sizes that are themselves sizes. Multiplying by the base as a shift. And a board where every line behaves like every other.

10:32All of it out of clause two — consecutive sizes standing in a constant ratio. What it did not buy is a sign for every size you can name, and that is the thing that has to go next.