Rival ways of splitting off the fractional part, and why the point won

0:00Almost everything about decimal notation follows from one decision. Split the unit into ten, and the places to the right of it are forced: tenths, then hundredths, then thousandths. You could not have chosen otherwise, once the splitting was settled. But there is one part of the notation that follows from nothing at all. The mark itself. The little dot that shows where the whole part stops. That was not derived. It was agreed, and it took a very long time to agree on.

0:35Here is a claim that catches almost everyone: decimals are much older than the decimal point. Not by a few years. By something like eight hundred of them. What follows is the account as it is usually told, and the dates in it are approximate. It starts with the fractions themselves, and with no separator anywhere. Fractions whose denominators are ten, a hundred, a thousand. Three tenths. Forty seven hundredths. Not written with a point. Written as fractions.

1:12These were in use among Indian astronomers and mathematicians a very long time ago. The eighth century is where this account begins, with Śhrīdharāchārya, who wrote on arithmetic and algebra. So the mathematics was already there and working. Tenths and hundredths were being added, compared and reasoned about. And nobody had yet needed a mark to sit between the two parts of a number. That is the thing to hold on to. The idea came first, by centuries, and the notation caught up afterwards.

1:51Move forward about two hundred years, to somewhere around the year nine fifty. The first detailed account of decimal notation in something close to its modern shape. It is credited to the Arab mathematician al-Uqlīdisī. And the title of the work he set it out in is worth a moment. It translates as a book of chapters on Indian arithmetic. So the account of the notation and the account of where it came from are the same sentence.

2:24The mathematics had travelled, and it was named for where it travelled from. But look at what he actually wrote on the page, because it is not what you are expecting. There is no point in it anywhere. Here is a number as he set it down. Nought, nought, five, nine, three, seven, five. Seven digits in a row, and above the first of them, a small upright stroke. That stroke is doing the entire job.

2:54It sits over the last digit before the fraction begins. So everything from the next digit onwards is the fractional part. Read it that way and the number is nought point nought five nine three seven five. Which is exactly what we would write today, with the mark moved down and turned into a dot. Same digits. Same quantity. A different device for saying where the split falls. And that is the first of several devices, because for a long time nobody could agree.

3:29By the fifteenth century there were at least three of them in use. The first you have already seen: a stroke over the last whole number digit. The second was simply to write the two parts in different colours. Whole part in one ink, fractional part in another, and no mark between them at all. The third is the strange one, and the most interesting. A small raised numeral, sitting above the digits, saying how many fractional places there are.

4:02So nought point three six is written as the two digits three six, with a small two above them. The two does not mark a boundary. It counts. Three rival devices, and all three of them work perfectly well. And the counting one can do something the others cannot, which is worth seeing. A mark that sits between two digits needs a digit on each side of it. So to write nought point three six with a point, you have to write that nought.

4:35It is not measuring anything. It has no value. It is there so the mark has somewhere to stand. The raised count does not need it. Three, six, with a two above. Two digits instead of three, and no borrowed nought anywhere. That is true of every number smaller than one, without exception. So the device that lost was not the worse device. In this one respect it was the better one.

5:03Which should make you curious about how the winner actually won. The sixteenth century is where it comes to a head, and it comes to a head as a split. Two mathematicians reach for a point. John Napier, in Scotland, and Christopher Clavius, in Germany. And a third reaches for something else entirely. François Viète, in France, uses a comma. Not a mistake, and not a lesser choice. Just a different one, made at the same moment.

5:37A point and a comma are equally capable of saying where a number splits. Neither is more correct than the other, and no amount of arithmetic will tell you which to prefer. So the split did not get settled. It got inherited. And it is still not settled today, which is the part people find hardest to believe. Several countries write the two parts of a number with a comma between them.

6:06Take a thousand and a half. Here it is written one way. One, comma, zero zero zero, point, five. The comma is grouping the digits of the whole number. The point is separating the parts. Now here is the same number as it appears elsewhere. One, space, zero zero zero, comma, five. A space groups the digits, and the comma separates the parts. The marks have swapped jobs. Same number. Same quantity. Two entirely reasonable agreements about which mark does what.

6:44And now the reason this matters, which is not tidiness. Take just the whole number part. One, comma, zero zero zero. Where the comma groups digits, that is a thousand. Where the comma separates the parts, that is one, followed by three zeros of a fraction. One point zero zero zero. Which is one. The same four characters, meaning a thousand in one place and one in another. Out by a factor of a thousand, and nothing on the page tells you which world it was written in.

7:21So the lesson is not that notation is arbitrary and therefore does not matter. It is the opposite. Because the mark is agreed rather than forced, everybody using it has to agree. Which leaves one more question. Why does there have to be a mark at all? Here are three digits. Seven, nought, five. Without something to tell you where the units place ends, that is not yet a number.

7:50It could be seven hundred and five. It could be seventy point five. It could be seven point nought five. Three different quantities, the same three digits in the same order. Four two has the same trouble, in a smaller way. Forty two, or four point two. Something has to say where the whole part stops. That much is genuinely forced. So here is the line worth drawing, and it is the point of all of this.

8:22Some of decimal notation could not have been otherwise. Some of it is a decision somebody made. Forced: that the unit is split into ten, because the whole place value system already counts in tens. Forced: that the places after the split go tenths, hundredths, thousandths, in that order. Forced: that something must show where the whole part ends, or the digits do not name a quantity. Chosen: what that something looks like.

8:54A stroke above a digit, a change of colour, a raised count, a point, a comma. Change the mark and every answer stays exactly the same. Change the split and every answer moves. That is how you tell the two apart, and it is worth being able to tell, for every notation you will ever meet.