Reciprocals, and Brahmagupta's rule for dividing fractions

0:00We have now done four divisions with a fraction underneath, and every one came out. One divided by two thirds. Three divided by two thirds. One fifth divided by one half. Two thirds divided by three fifths. So, a fair question. Do we now know how to divide by a fraction? Not quite. We have four answers, and working out a fifth would only give us five. A rule is a different kind of object. One line that settles every case at once, including the ones nobody has looked at yet.

0:38And getting from four worked cases to that one line is not bookkeeping. It is the actual mathematical achievement here, and it has a date attached, which we will get to. But first we have to notice which number was doing the work. Take the last of the four, because nothing about it was easy. Two thirds divided by three fifths. Here is what we did. We asked what multiplies three fifths up to one. The answer was five thirds.

1:12Then we multiplied that five thirds by two thirds, and got ten ninths. Now look at where the effort went. The second step was ordinary multiplication. We already knew how to do that. Everything new happened in the first step, inside one number. Five thirds. So that is the number to stare at. What is five thirds, though? Not where it came from. What is it? It is the number that multiplies three fifths to give exactly one.

1:45That is a description of a job. It is not a recipe for finding anything. And it is the same job every single time. In one divided by two thirds, we wanted what takes two thirds up to one, and it was three halves. In one fifth divided by one half, we wanted what takes one half up to one, and it was two. All four divisions asked exactly the same thing about their divisor.

2:15Something asked for four times in a row deserves a name. It is called the reciprocal. The reciprocal of a number is the number that multiplies it to give exactly one. That is the whole definition. Notice what the definition does not say. It says nothing at all about turning anything upside down. It describes what the number does, and leaves open how you would find it. So five thirds is the reciprocal of three fifths, because three fifths times five thirds is one.

2:51Three halves is the reciprocal of two thirds, for the same reason. And I checked this on ninety-one different numbers. Every one had exactly one partner that took it to one. Never two. So how do you find it? For a fraction, look at what has to happen. Three fifths has a three on the top and a five underneath. For the product to be exactly one, both of those have to cancel away.

3:22So the other number needs a five on top, to cancel the five below, and a three underneath, to cancel the three above. Five over three. The two parts have swapped places. And that is worth saying slowly. Swapping top and bottom is not what a reciprocal is. It is how you find one, when the number happens to be written as a fraction. I ran that swap on all ninety-one of them and multiplied each back against its original. Every product was one.

3:58Which raises a question. What about a number not written as a fraction at all? Five, say. There is nothing to turn over. But the definition still works. What multiplies five to give one? One fifth. And if you write five as five over one, the swap gives one fifth too. So the definition covers cases the swap says nothing about, which is the sign of a good definition. Two more things. One is its own reciprocal, and among positive numbers it is the only one that is.

4:33And taking the reciprocal twice brings you straight back where you started. Every time. Also, a number below one always has a reciprocal above one, and the other way round. Worth holding on to when you judge the size of an answer. So the whole method can now be said in two steps. Step one. Take the reciprocal of the divisor. Step two. Multiply it by the dividend. That is it. Step one is the only part that is new.

5:08Now, a warning, because this is where it goes wrong. It is the divisor that gets turned over. Not the dividend, and not both. In two thirds divided by three fifths, the three fifths becomes five thirds, and the two thirds is left alone. I tried it the wrong way round on all eight thousand two hundred and eighty-one pairs. Turning over the wrong one gets eight thousand one hundred and ninety of them wrong.

5:40Now for the compression. Write it in letters instead of numbers. Call the dividend a over b, and the divisor c over d. The reciprocal of the divisor is d over c. So a over b, divided by c over d, is a over b times d over c. One line. And it covers every pair of fractions there is, including the four we ground out by hand. You will sometimes see it written with the reciprocal first. D over c, times a over b.

6:14That is not a second rule. Two numbers multiplied in either order give the same product, and that was true long before this. I ran the letters against exact division on all eight thousand two hundred and eighty-one pairs. Not one disagreement. So who did that? The general form of these rules, for multiplying and dividing, is credited to Brahmagupta, in the year six hundred and twenty-eight. That claim needs stating carefully, because it is easy to overstate.

6:51It does not say he invented fractions. Fractions were in ordinary use in India more than fourteen hundred years before he wrote. It does not say that nobody had ever divided one fraction by another. What it says is that he wrote the rules in general form. As statements about any fraction, rather than a pile of worked examples. That is the step from four cases to one line, and it is the step with his name on it.

7:23Five hundred and twenty-two years later, Bhaskara restated the division rule using exactly the word we have been using. The reciprocal. One year later, another mathematician drew a picture of why the multiplication rule works. It is a square, with each side one unit long. Cut it into five columns going across, and four rows going down. That makes twenty small cells, all the same size. One column is a fifth of the square wide. One row is a quarter of it tall.

8:00So one cell is a fifth times a quarter. And it is one twentieth of the square, because there are twenty of them, all the same. A fifth times a quarter is a twentieth, and the picture is the whole argument. Read it backwards and you get the division for free. One twentieth divided by a quarter gives back a fifth, the width of the column. These rules did not stay in one place, and the route they took is worth a minute.

8:33Put the dates on a line. Four ninety-nine, Aryabhata. Six twenty-eight, Brahmagupta. Six twenty-nine, that square. Then three more Indian mathematicians over the next century. Around seven fifty. Around eight fifty. Around eight sixty. Eleven fifty, Bhaskara, and the reciprocal. Eleven ninety-two, al-Hassar, working in Morocco. From there the rules moved into Europe slowly, and were not widespread there until something like the sixteen hundreds. That is around eleven hundred years from the first date to the last, for a rule that now takes a single line.

9:18So, back to where we started. Two thirds divided by three fifths. Take the reciprocal of the divisor, which is five thirds. Multiply it by the dividend. Ten ninths. Same answer we got the long way, and that is the point. The rule did not replace the reasoning. It packaged it. One last practical thing. If a number turns up as a mixed fraction, make it a single fraction first.

9:49Two and a third is seven thirds, and its reciprocal is three sevenths. Turning the two and the third over separately gives something else entirely, and it is wrong. And that is all of it. Find the number that turns the divisor into one, then multiply. It was never a trick you had to remember. It is a description of what the division was asking for all along.