The Egyptian system, and what a landmark number is for

0:00Here is a ladder of sizes that a real writing system used. One, five, ten, fifty, a hundred, five hundred, a thousand. The steps between them go five, two, five, two, five, two. Somebody chose those, and not carelessly: five and ten and fifty are sensible things to bundle by. But chosen is what they are, and a chosen list has no next entry. At the top, somebody has to think of another one.

0:33So here is the question that ladder leaves behind. Could you get your sizes from somewhere other than a choice? Could there be a rule that hands you the next one? Start with a pebble. One pebble is your first size. Now the instruction. Take ten of the size you just made, and call that bundle the next size. Ten pebbles. That is the second size. Now apply it again — to its own answer, not back to the pebble. Ten of those bundles.

1:06A hundred. Then a thousand. Then ten thousand. You have chosen nothing since the first line. The rule is doing the choosing, and it goes on as long as you keep asking. Look at what that guarantees. Every size is ten times the one before, because that is the only thing the rule ever does. So the list can only be one, ten, ten times ten, ten times ten times ten. Powers of ten, and nothing else.

1:40That is not something you check about the list. It is a property of the instruction that made it. And the position of a size tells you how many times the rule has run. The fourth entry is ten applied three times. Nothing here is sacred about ten. Bundle by five and the same rule gives one, five, twenty-five, a hundred and twenty-five. Ten is a choice. Having a rule is not.

2:10There is a picture hiding in that list. One pebble is a point. Ten of them in a row is a line. Ten lines side by side is a square. Ten by ten — a hundred pebbles, and you can count them. Ten squares stacked is a cube. Ten by ten by ten. A thousand. Point, line, square, cube. The first four sizes, one dimension at a time. That is what a constant step looks like drawn. The uneven ladder has no such picture, because it has no single move to repeat.

2:49Now give each size a sign to draw. A stroke for one. An arch for ten. A coil for a hundred. Then five more, on up to ten million. Eight signs, and there the drawing stops. But notice what has stopped and what has not. The rule has not. Ask it for the next size and it answers a hundred million, without anybody inventing anything. What ran out is the pictures, not the sizes.

3:18That is a different kind of ceiling from one whose largest size is just the largest letter anyone thought of. Writing a number works as before. Take as many of the largest size as will fit, then move down. Three hundred and twenty-four. How many hundreds fit? Three. Twenty-four left over. How many tens? Two. Four left. Then four ones. Three coils, two arches, four strokes. Nine signs, and it is written.

3:53Same procedure, different ladder. Now look very carefully at what came out. Three coils. Two arches. Four strokes. Three, two, four. The counts of the signs are the digits of the number. Not similar to them. The same three numbers, in the same order. And that is not a coincidence about this one. Run it on every number up to a hundred thousand and it holds every time. It has to. Grouping by powers of ten, largest first, is what writing a number in digits already is.

4:32So this is not a foreign notation to decode. It is your own numbers, each digit drawn as that many repeated signs instead of written as one shape. The gap is much smaller than it looks, and what is left of it is what the rest of this is about. One thing is genuinely different, and it is worth being exact about. Here is a numeral in rows. Two coils on top. Then three arches. Then three more. Then six strokes, with one last arch after them.

5:09To read it, do not read the rows. Count the kinds. Two hundreds, seven tens, six ones. Two hundred and seventy-six. The tens sat in three different rows and one came after the ones, and none of that mattered. Read the top row and stop, and you are seventy-six short. Shuffle the nine signs of our first numeral into any arrangement you like — there are one thousand two hundred and sixty — and the value never moves.

5:40Do it to the three digits and you get six different numbers. Position carries nothing here, and everything there. Because the counts are the digits, you can answer something about a numeral before writing it. How many signs will a number need? Add up its digits. Seven hundred and eighty-four needs nineteen. One thousand one hundred and eleven needs four. Say that again. The bigger number needs fifteen fewer signs.

6:12Length has nothing to do with size here. It is the digit sum, so a number can be large and cheap or small and expensive. Seventy thousand seven hundred and seven costs twenty-one. Ten thousand four hundred and fifty-eight costs eighteen. Sort a handful by value, then by cost, and the two orders are not the same. Which brings the honest accounting, and it does not go the way you would expect.

6:44The even ladder did not make numerals shorter. It made them longer. Take every number both systems can write, up to three thousand nine hundred and ninety-nine. The uneven ladder is strictly shorter on three thousand seven hundred and forty-four of them, and the rest are ties. It is never longer. Not once. Three hundred and twenty-four is seven signs there and nine here. The extra sizes are what buy that — the five, the fifty, the five hundred. Half-steps are compression, and giving them up costs length.

7:24So what did the even rungs buy, if not brevity? Two things, and they are exactly the two things that were broken. First, the trade. Regrouping on the uneven ladder means a different number at every rung. Five here, two there, five again. Here it is ten. At every rung. One number, and the rule guarantees it, because every step is the same step. Second, multiplication. On the uneven ladder some products of two sizes landed between the rungs, on numbers with no sign.

8:01Here, every product of two sizes is another size. Ten times a hundred is a thousand. A thousand times a thousand is a million. Nothing falls between, ever. And the rule says why. Applying it three times and then four more is applying it seven times, so multiplying sizes just adds up how many times the rule ran. There is still a ceiling, and two things that look alike need separating.

8:34With eight signs, none ever needed more than nine times, the largest number you can write is eight nines in a row. That costs seventy-two signs, which is a problem of its own. Add one more and the top sign appears ten times. The row-of-marks problem, back at the top of the ladder. But the repair is not a cleverer letter. The rule already knows the next size. It always has.

9:02What is missing is not a size. It is a way of writing one down that does not need a new picture each time. Hold on to that, because it is the whole of what comes next. Step back and ask what these sizes have actually been doing. Three hundred and twenty-four as plain marks is that many marks, and nobody can read it. As signs on this ladder it is nine.

9:31And each sign earns its place. Take the hundreds sign away and that number costs twenty-seven signs more, for one missing picture. That is what a landmark number is for. Not a big number — a size the system bundles by and gives a sign to. Two systems have now been compared on it. One chose its sizes and got short numerals with awkward arithmetic. This one generated them and got clean arithmetic with long ones.

10:04Neither has done the last thing. Both still need a new picture per size, so the pictures grow with how far you want to count. The idea that ends this: stop drawing sizes altogether, and let where a digit sits say which size it means.