From śhūnyatā to śhūnya: turning nothing into a number you can compute with

0:00Here is a problem left over from writing numbers by position. Two hundred and five needs a two in the hundreds, nothing in the tens, and a five in the ones. On a ruled board that middle column is empty, and you can see that it is empty. Take the ruling away and what reaches the page is a two and a five. Which is exactly what twenty five looks like.

0:27So the writing needs a mark that says this column is empty. Several civilisations arrived at one, independently, and it solved that problem completely. It also did nothing else at all, and that is the story. Babylonian scribes had such a mark. So did the Maya, on the other side of the world. Both wrote numbers by position, both hit the same ambiguity, and both solved it the same way: a symbol meaning nothing goes here.

1:02But watch what that symbol can and cannot do. It can sit in a numeral and keep two hundred and five apart from twenty five. That is the writing job, and it does it perfectly. Now try to add seven to it. There is no answer, because the mark was never a quantity. It is punctuation. Multiply by it and the same thing happens. A placeholder passes one test and fails the other, and those two tests are independent. Passing the first buys you nothing towards the second.

1:40The missing step came from somewhere unexpected, and it helps to see why it was easier to take in one place than another. In the Indian tradition there is a word, shunyata, meaning emptiness. It appears in the Upanishads and in Buddhist writing from earlier than the seventh century before the common era. And it is not a lack. It is a state that people work towards. That is a genuinely different attitude. If emptiness is something you can seek and describe, then emptiness is a subject. It has properties.

2:19A tradition that already talks about nothingness that way has very little difficulty, later, admitting nothing as a number. The clearest picture of it comes from Patanjali, in the Yoga Sutras, around the third century before the common era. The mind, in that account, is full of movement. Fluctuations, called vrittis. Thoughts arriving, thoughts leaving, the surface never still. The practice is to quiet them. Not to empty the mind, but to still the ripples, and what follows is control of mind, body and senses.

3:00Notice the shape of that. Stillness is the goal, and it is described as an accomplishment rather than an absence. The empty surface is the one worth having. Hold that thought against the arithmetic to come, because the definition of zero has exactly this shape. An idea that useful does not stay in one field. Emptiness turns up as a working concept in architecture, where the space you leave is as much of the design as the stone you place. In literature, where a pause carries meaning. In linguistics, where a silence does work.

3:43Three fields, and none of them arithmetic. Then it reaches mathematics, and it arrives there through two people in turn. Aryabhata first, whose own dated work is from four ninety nine of the common era. Brahmagupta next. This is the part that is easy to get backwards. The concept did not come out of the mathematics. The mathematics was the last place it went. Along the way, the symbol itself settles down.

4:16In a manuscript found at Bakhshali, from the early centuries of the common era, there is a bold dot used to mark the empty place. It has a name: bindu. It is a small thing to look at, and a real step. Before it, an empty column is a blank space, and a blank space on a worn page could be anything. A dot is positive evidence. Somebody put it there on purpose.

4:45So now there is a mark, and it is a good mark. But look carefully at what has changed, because the answer is: only the writing. This is the point the whole topic turns on, so it is worth being exact about it. Take the dot and put it in a numeral. It works. Two hundred and five is now distinguishable from twenty five, on the page, for ever. Now put the dot in a sum. Seven plus the dot. There is nothing to compute: a dot is not a quantity, and no rule says what to do with it.

5:24Compare that with the mark you were taught in school. Seven plus nought is seven. Seven times nought is nought. Those are answers, and they are answers because somebody decided what they should be. A mark is notation. A number is notation plus consequences. Zero was not invented on the day somebody drew it. The consequences arrive in six twenty eight of the common era, in a work called the Brahmasphuta Siddhanta, by Brahmagupta.

6:00They do the thing nobody had done. They define it. Zero, they write, is what is left when a quantity is taken away from itself. In symbols: a minus a. Run it. Five take away five. Seventeen take away seventeen. A thousand take away a thousand. Every one of them lands on the same place. And notice what kind of statement that is. Not a description of an absence, but the result of an operation on a definite quantity, giving a definite answer.

6:37There is something hiding in that definition which is easy to walk straight past. The definition mentions a quantity, a. So you might expect one zero for each quantity: a five sized nothing, a seventeen sized nothing. It does not. Take every quantity from one to five hundred, subtract each from itself, and collect the answers. Not five hundred answers. One answer, five hundred times. That is what makes it a number rather than a family of them. The definition is true of every quantity at once, and what it names does not depend on which one you used.

7:20Compare it with a rule that does depend on its input, like taking one away instead. Five hundred quantities, five hundred different answers. That is not defining a number. Now the part people memorise, which they should not have to. The first rule says that adding zero leaves a number as it was. That looks like a convention somebody chose. It is not. It follows from the definition, and here is the whole argument.

7:53Take b plus zero. Zero is a quantity taken from itself, so write it that way: b plus, in brackets, a minus a. Regroup: b plus a, then take a away again. Add a quantity and take the same quantity away, and you are back where you started. Three steps, no extra assumptions, and it works for every b and every a you care to pick. Suppose you doubted it. Suppose you thought adding zero shifted everything by some fixed amount. Try every shift from minus twenty to plus twenty against the definition, over three thousand six hundred pairs. Forty one candidates go in. Exactly one survives, and it is nought.

8:43The second rule goes the same way. Taking zero away leaves a number as it was, for the same reason and by the same argument. The third rule is the interesting one. Any number multiplied by zero gives zero. People treat that as obvious. It is not obvious, it is decided, and it is the most consequential of the three. Watch. Once every number times zero is zero, ask what you would have to multiply zero by to get seven. Search every candidate you like. There is no answer, because everything you try gives zero.

9:25So dividing seven by zero has nothing to return. Not a hard problem, not an unknown value. There is no such number. And it fails a second way, which is stranger. Ask instead what you multiply zero by to get zero. Now everything works. Search four hundred and one candidates and all four hundred and one succeed. So one division by zero has no answer at all, and the other has every answer at once. They fail for opposite reasons.

10:01Which is why, when the rules were written, there were three of them and not four. Adding. Subtracting. Multiplying. There is no fourth rule for division, and its absence is not an oversight. The condition you will meet later, that the bottom of a fraction must never be zero, is not a separate fact to learn. It is this one, arriving again.