Where our way of writing fractions comes from

0:00Look at this symbol. Three over four. One number above another, with a line between them. It is so ordinary that it feels like it was always there. It was not. This shape has a history, and inside the history there is an argument: the way you write something decides what you are able to say about it. So start with what a fraction was called, before it looked like this.

0:27In Sanskrit, the word is bhinna, which means broken. Two more words are used for it as well. Bhaga, and ansha. Both carry the sense of a part, or a piece. Notice what those words describe. Not a symbol. An action. Something whole has been broken, and you are holding a part of it. Now here is one of the oldest surviving Indian writings of a fraction. It comes from a mathematical manuscript found near a village called Bakshali, dated to about three hundred of the common era.

1:05One half was written like this. A one, and underneath it, a two. Put today's version beside it. Same two numbers. Same positions. Same idea. One difference, and only one. There is no line. That is the whole of it. The notation you use was already in place seventeen centuries ago, minus a single stroke. And the stroke is the part that arrived last, which is not the order anybody expects.

1:37That form did not sit still in a manuscript. It got used, by mathematician after mathematician. Aryabhata, four hundred and ninety-nine. Brahmagupta, six hundred and twenty-eight. Sridharacharya, around seven hundred and fifty. Mahaviracharya, around eight hundred and fifty. Put them on one axis, with the manuscript at the far left. From the manuscript to Mahaviracharya is five and a half centuries, and across all of it fractions are written in essentially the shape you use now.

2:16Hold on to that, because it kills a comfortable assumption. Older notation does not mean weaker mathematicians. So when did the line arrive? The twelfth century. Around eleven fifty. And not from India. From Morocco, from a mathematician named Al-Hassar. Look at that gap on the axis. Eight hundred and fifty years between the stacked pair of numbers and the little bar drawn between them. The bar is the newest part of the symbol, and it is the part that feels most permanent.

2:53Which is a fair warning about symbols in general. From the inside, they always look like they were always there. Now for the reason any of this matters, and for that we need a rival design. Ancient Egypt and Babylon worked with fractions too. This is not a story about one civilisation having the idea and nobody else. But those systems mainly wrote fractions with a one on top. One half. One third. One quarter. One twenty-fourth.

3:29A fraction with a one on top has a name of its own. It is called a fractional unit. And if fractional units are mainly what your notation hands you directly, then every other amount has to be built out of them, as a sum of several such pieces. That practice has a name today. Egyptian fractions. Watch what that costs. Take nineteen twenty-fourths. A completely ordinary amount, just over three quarters.

4:01In a system where the top number is free, you write two numbers and you are finished. In a system of fractional units, you have to take it apart. One half, plus one quarter, plus one twenty-fourth. Check that, rather than believe it. Over twenty-fourths those three pieces are twelve, six and one. Twelve and six and one is nineteen. Nineteen twenty-fourths. It works. But look what happened to a perfectly ordinary quantity. It became a small puzzle.

4:35And be careful here: this is the same number. The same single point on the number line. Only the writing changed. Here is the deeper cost, and it is the real point of this whole story. When the top number is free to be anything, the symbol shows you two things at the same moment. Underneath, the size of the step. On top, how many steps. And once the count is visible separately from the step size, you can state a rule about counts.

5:08That is what makes a sentence like, make the step sizes match and then add the counts, possible at all. With the top number pinned to one, there is no count to talk about. Every amount is its own little assembly problem. You end up with techniques that work for some numbers, instead of a rule that works for all. Notation decides what you can say. Which brings us back to the man sitting in the middle of that axis.

5:39In six hundred and twenty-eight, Brahmagupta wrote a treatise called the Brahmasphutasiddhanta, and in it, at verse twelve point two, he set down how to add and subtract fractions. In our words: multiply each fraction's two numbers by the other one's bottom number, so both sit over a shared bottom number, then add or subtract the tops. That is the method from last time, written down in the seventh century.

6:11And notice the shape of it. It is not stated for a chosen pair. It is stated for any pair. Rules for multiplying and dividing came from India too, and the ancient Sulba treatises show arithmetic with fractions already in use far earlier. The claim is not that fractions were invented in India. It is that fractions with an unrestricted top number, and the rules for calculating with them, were.

6:44How did it reach everybody else? Through Arab scholarship, over several centuries, and from there into Europe, where it was in general use by around the seventeenth century, and then outward across the world. From that manuscript to general use in Europe is thirteen and a half centuries. So the symbol on your page has travelled a long way. Its shape is from India. Its line is from Morocco. Its route ran through the Arab world.

7:17No committee anywhere in that story ever sat down and designed it. Now a puzzle, because the fractional unit idea turns out to have real teeth. Fractional units that are all the same size adding to one whole is easy. Three thirds. Five fifths. Anything you like. So make it hard. Can you find fractional units that are all different from each other, and total exactly one whole? Start with two of them. And here the answer is no, and you can see why in two moves.

7:54First, one half is the largest fractional unit there is. Every other one has a bigger bottom number, which means smaller pieces. Second, two halves already make exactly one whole, with nothing to spare. So if the two pieces must differ, at least one is smaller than a half, and the total falls short. The closest you can get is a half and a third. Five sixths. One sixth short.

8:23Two different pieces can never do it. Three pieces, then. And now something surprising happens. There is exactly one answer. Here is the reasoning, and every step of it is forced. Start from three thirds. One whole, but all three the same size, so it does not count. To make them different, one of them has to grow. And between a third and a whole there is exactly one fractional unit: a half.

8:54So the largest piece is one half. There was no choice about it. That leaves one half still to cover, using two different pieces, and neither of them can be a half. The largest fractional unit still available is a third. Take it. A half and a third together are five sixths, so what remains is one sixth. One half, one third, one sixth. Watch it on a disc cut into six equal sectors: three, then two, then one.

9:26Three and two and one is six. Six sixths. One whole. Four different pieces, and the answers multiply. There are exactly six of them. Here is one, and the other five are yours to find. One half, one third, one ninth, one eighteenth. Check it over eighteenths. Nine, six, two and one. Nine and six and two and one is eighteen. Eighteen eighteenths, which is one whole. Every one of the six begins with a half, for exactly the reason the three piece answer did.

10:02Go and find the rest. Draw each one as a divided circle, the way we just did. And while you draw, notice what you are actually doing. You are solving a problem that exists only because of how somebody chose to write things down. That is the lesson worth keeping from all of this. Notation is not decoration.