Why India needed names for powers of ten

0:00A port city, a dock, and a warehouse behind it. Ships come in loaded and go out loaded, and somebody has to keep the account. Now try keeping that account the way a herder keeps theirs: one notch for each thing. A stroke for every jar, every bale, every ingot that crosses the quay in a season. It works, and it is useless. Not because it is wrong, but because nobody can read it. A row of strokes running the length of a wall tells you nothing you can act on, and comparing two such rows is worse.

0:39The pairing that settled a herd cannot settle a warehouse. What had to change was not the counting. It was the words. It helps to know what was actually on those ships, because the cargo is what made the numbers big. Terracotta pottery, fired and stacked. Lapis lazuli, the deep blue stone, carried in from the mountains. And cotton, which the Indus cities were growing and spinning while most of the world had never seen it.

1:12Two of the cities that handled it were Lothal, on the coast, and Harappa, far inland. The trading partner across the sea was Mesopotamia. Three kinds of cargo, two cities, one long sea route. Enough traffic to generate quantities nobody wants to write as strokes. But before a quantity can even be recorded, the two people trading have to agree what it is. Put one heap of grain on the table. I weigh it against my stone, and I make it sixty. You weigh the very same heap against yours, and you make it eighty.

1:53Neither of us is lying. We are using different units, and a number that depends on whose stone you used is not a number anybody can trade on. Check every heap from one upwards against those two stones and the readings never once agree. Not for a single heap in five hundred. Fix the unit, though, and something quietly enormous happens. Quantities become numbers that add. Two loads together weigh what the two of them weigh apart, every time, and an account becomes possible.

2:30Here is the kind of sum the dock produced. A merchant trades bags of spices for copper ingots. The rate is fifteen ingots for every two bags. They arrive with twelve bags. How many ingots do they leave with? Work it through the rate first. Fifteen ingots for two bags means seven and a half ingots for one bag. Twelve bags at seven and a half each comes to ninety.

2:58Or never mention a half at all. Twelve bags is six lots of two bags, and each lot fetches fifteen, so six fifteens. Ninety again. Notice what happened in the first route. The rate per bag was not a whole number, and the answer was. Halves turned up in the middle of a problem whose question and whose answer are both whole. Back to the warehouse, and the real problem. These accounts have to be said out loud.

3:31The answer that took hold in India was to give every power of ten its own name. Not a name for each quantity you happen to need, but a name for ten, for a hundred, for a thousand, and onwards up a ladder. The ladder has one rule and it never varies. Each rung is exactly ten times the rung below it. Divide any rung by the one under it and you get ten, with nothing left over, every single time.

4:03That sounds like housekeeping. It is the whole invention. The Vedic texts carry that naming up to the twelfth rung, and the name it gets there is parardha. Ten to the twelve. Write it out and it is a one followed by twelve noughts, which is thirteen digits altogether, not twelve. The count of digits is always one more than the power, and that off by one catches people every time.

4:33If thirteen digits is hard to feel, there is a second name for exactly the same number. A lakh is ten to the five. A crore is ten to the seven. A lakh crore, then, is ten to the twelve. Same number, two naming systems, no disagreement anywhere. And that is already a very large account. It does not stop there. In a Buddhist text called the Lalitavistara, the naming is carried out to the fifty third power of ten, and that rung is called tallakshana.

5:11That text is traditionally placed in the fourth century before the common era, though the dating is genuinely argued over. Ten to the fifty three has fifty four digits. Divide it by parardha, the twelfth rung, and you are still left with ten to the forty one. So why name it? Not because anyone was counting that high. Because a scheme that can reach the fifty third rung at all is a scheme with a rule, and that is a completely different kind of thing from a long list of words.

5:47Here is what that rule buys you, and it is the point of the whole topic. Suppose I hand you the values of the names and tell you nothing else. You can recover the base from the words alone. Take each name, divide it by the one before, and look at what you get. Ten. Ten. Ten. Ten, all the way up. One answer, repeated. The scheme is telling you, without being asked, that it is built on ten.

6:17The base is not extra information stored somewhere beside the names. It is carried inside them. Which means anybody who has learned to say these numbers has already learned place value, before ever seeing it written down. To see that this is a real property and not just a description, watch a scheme that does not have it. Roman numerals. One, five, ten, fifty, a hundred, five hundred, a thousand. Run the same test: divide each by the one before.

6:53Five. Two. Five. Two. Five. Two. Two different answers, alternating. There is no single step, so there is no base carried in the symbols. Not a criticism of the Romans, who calculated perfectly well. It is a statement about what the symbols can do on their own. Ask a scheme like that to line up in columns and it has nothing to line up on. There is a second difference, and it shows up when you ask each scheme for something it has not been given.

7:28The Roman set has a largest thing it can say. A thousand. Past that you are inventing, because there is no rule telling you what the next symbol would be worth. Ask the ladder of powers for the fifty fourth rung, though, and it answers immediately: ten times the fifty third. Ask for the hundredth, the thousandth. Same answer every time, ten times the one below. A list of names runs out where the list ends. A rule does not run out at all, which is why the naming could be carried past whatever any text happened to record.

8:11Now write that speech down, and watch what the writing is forced to become. If every rung is ten times the last, you do not need a fresh symbol for each rung. You need somewhere to put a digit, and an agreement that moving one step to the left multiplies it by ten. Two hundred and five is a two in the hundreds place, nothing in the tens, and a five in the ones. Read it by the places: for each digit, multiply what you have by ten and add the new one. Two, then twenty, then two hundred and five.

8:48The base has moved out of the words and into the positions. Which is only possible because the words carried it in the first place. And that leaves exactly one thing broken. On a counting board, that middle place is a column with no pebbles in it, and you can see it. Rub out the ruling, and what is left on the page? A two and a five. Which is what twenty five looks like as well. Two hundred and five, twenty five, and two hundred and fifty all collapse onto the same mark. Three different numbers, one written form.

9:28And it is not a rare accident. Of the first thousand numbers, two hundred and seventy one stop being written uniquely, and only a hundred and eighty one of those contain a nought at all. The other ninety are numerals like twenty five, with nothing missing themselves, ruined by the ones that do. So the positional system arrives with a hole in it. Something has to stand in a place that holds nothing.