The Hindu number system, and why treating 0 as a digit changed everything

0:00Here is a number you can read without thinking about it: three hundred and seventy-five. Three, seven, five. The left one counts hundreds, the middle one counts tens, the right one counts ones. Nothing about that feels like an invention. It feels like how numbers simply are. It is not. It is the end of a long argument, and the last move is the one almost nobody expects. Because the idea that a sign's position decides its worth was reached more than once, independently, on continents that had no contact.

0:40So the last move was not that idea. Something else was added on top, and it was added twice. This is what each of those two steps bought. Read three hundred and seventy-five slowly, one place at a time. The five sits in the ones place, so it is worth five. The seven sits in the tens place, so it is worth seventy. The three sits in the hundreds place, so it is worth three hundred.

1:10Five, seventy, three hundred. Add those and you are back where you started. That is the whole of it — and notice what did not happen. Nothing new was needed anywhere. Which is exactly why this numeral is the wrong place to look for the invention. Letting position carry part of the meaning was arrived at separately, with no contact of any kind. It happened in Mesopotamia. It happened in Central America. It happened in China.

1:44Three systems, three sets of signs, three completely different ways of drawing a digit. And all three landed on the same arrangement: put the sign somewhere, and the somewhere tells you what it counts. When an idea is found that many times over by people who cannot have copied one another, it is telling you something about numbers rather than about people. So position was not the last step, because it had already been taken three separate times.

2:16The question worth asking is what all three were still missing. All three were missing the same thing, and it is easy to see once you look. What do you write when a place has nothing in it? The first answer anyone reaches for is: write nothing. Leave a gap. So try it. Take every number below ten thousand and write it with the empty places simply left out. Three thousand four hundred and thirty-eight of them stop being tellable apart from something else.

2:51One, ten, a hundred and a thousand all come out as the same single mark. Nine thousand nine hundred and ninety-nine numbers collapse into seven thousand three hundred and eighty pictures, and nothing on the board says which you are looking at. So give the empty place a mark of its own. A sign that says: there is a place here, and it is empty. Not a digit. Just a mark that takes up room, so the places can be counted.

3:23Run the same ten thousand numbers again. Nothing collides. Not one number. Nine thousand nine hundred and ninety-nine numbers, nine thousand nine hundred and ninety-nine different pictures. The reading problem is gone. Completely gone. And here is the part that changes the story: two of those three earlier systems already had one. So if a mark already fixes the reading, what is there left for a digit to fix? Ask the mark the one question it cannot answer. What number is it?

4:01It is not a number. It is a note about a place — a sign saying the place is empty. You cannot lift it out and hand it to someone on its own. Which means there is one number a system like that cannot write at all. Zero itself. The first number it fails on is the very thing the mark is about. A system with a mark can write every number from one upwards, and not the one below them.

4:31That is not a small gap in the notation. That is a number missing from the world. The first promotion is to stop treating that mark as a note and start treating it as a digit. Not nine signs and a marker. Ten signs, all the same kind of thing. It stands in a place the way a three does. It has a value, and the value is nothing. Now the smallest number has a numeral of its own, and so does every number below a thousand — all one thousand.

5:06The reading was already fixed, so measured on collisions this changes precisely nothing. That is the point, and it is why the step is so easy to miss. The gain is not in the writing. The gain is that zero has walked into the number system, and now the number system has to deal with it. The second promotion is the larger one, and you can watch it happen inside a single operation.

5:36Take subtraction. Pick any two numbers from one to ninety-nine and take the second from the first. There are nine thousand eight hundred and one ways to do that. In a system with neither zero nor the negatives, four thousand nine hundred and fifty of them have no answer. Ninety-nine of those are where the two numbers are the same. Seven take away seven is not a question you can ask.

6:04Admit zero and exactly those ninety-nine get an answer. Four thousand eight hundred and fifty-one are still stranded — every case where the larger number is on the right. Admit the negatives as well and the stranded count falls to nothing. All nine thousand eight hundred and one have an answer. And that is what a mathematician in the seventh century saw, and wrote down. Zero and the negatives are not two separate conveniences. Together they are what it takes to close the system.

6:43Here is the test. Take a set of numbers and three operations: add, subtract, multiply. Ask whether any two members, under any of the three, can produce something not in the set. The counting numbers fail. Subtraction walks straight out of them. Put zero in and they still fail, in that same one place, because subtraction is still the way out. Put the negatives in as well and nothing escapes. Not adding, not subtracting, not multiplying. The set stays inside itself.

7:21That property has a name now, and the name matters much less than the test does. It is not a reward for being large, and not something a set simply has or lacks. It is decided one operation at a time. The odd numbers are closed under multiplication — odd times odd is odd, always. But add two odd numbers and you land outside every time. Subtract them and you land outside every time.

7:52So closure is checked operation by operation, and the whole numbers with their negatives pass on all three. Now notice which operation is not on the list. Divide one by another and you walk out constantly — and that is with dividing by nothing already ruled out. Three operations, and it holds. Four, and it breaks. The list is exactly as long as it is allowed to be. One more thing, because it shows the system you write in was a choice.

8:28Before position did any work there were landmark numbers: separate signs for one, five, ten, fifty, a hundred, and upward. Look at the steps between those landmarks. Five, two, five, two, five, two. That alternation is exactly what stops it being a base. A base has one step, repeated. One, ten, a hundred, a thousand, ten thousand. Ten, ten, ten, ten. So the base is a decision, and ten was almost certainly settled on because of hands.

9:03Which invites the obvious question. What if there had been eight fingers instead? Take twenty-five and write it in base eight. Three eights and one left over, so: three, one. In base five, a single mark in the twenty-fives place and nothing in the other two. One, zero, zero. In base two: one, one, zero, zero, one. Two digits, three digits, five digits — and every one of them reads back as twenty-five.

9:37The numeral changed three times. The number never moved. A base is a way of writing, not a fact about the quantity. And every one of those bases needs a sign for an empty place. Base eight needs eight digits, base two needs two, and zero is one of them every time. Two promotions, then. A mark became a digit, and a digit became a number — and it is after the second one that you can take any number away from any other. Which is where everything else begins.