Why dividing can make a number bigger

0:00Six divided by three is two. Nothing surprising there. Six things shared into three equal piles, and two in each pile. But look at where the answer landed. Two is smaller than six. And that is not a coincidence about these particular numbers. I checked eight hundred divisions where both numbers are whole. Not one of them had an answer bigger than the number being divided. Not a single one.

0:30So after years of dividing whole numbers you end up with a very reasonable belief. Dividing makes things smaller. That belief is built on eight hundred honest examples, and it is about to be wrong. Here is the division that breaks it. Six, divided by one quarter. Twenty-four. Read that again, because it is genuinely startling the first time you meet it. We divided six by something, and got twenty-four.

1:03The answer is four times bigger than the number we started with. Nothing has gone wrong. Twenty-four is exactly right. And notice that four. Hold on to it, because in a few minutes it is going to explain the entire topic. But first: where does twenty-four actually come from? Dividing by a quarter is asking a question you already know how to answer. How many quarters fit into six? So lay out six units, and start cutting quarter-sized pieces off them.

1:39One whole unit gives you four quarters. So six whole units give you six lots of four. Twenty-four. The pieces are small, so there are a lot of them. That is the whole of it. Make the pieces smaller still, eighths instead of quarters, and you get forty-eight. Make them bigger, divide by two, and you only get three. Now, you might reasonably suspect that six was doing the work here. Six is a decent-sized number to start from.

2:14So let us make everything small. One eighth, divided by one quarter. Every number in that sentence is less than one. The answer is one half. And one half is bigger than one eighth. The answer has overtaken the number being divided again. So it was never about six being big. It is about the quarter. At this point there are two questions worth asking, and they are not equally easy.

2:45First. When does the answer come out below the number being divided, and when does it come out above? Second. How does the answer compare with the number you divided by? The first one has a clean answer, and we are going to get it without doing any new work at all. The second one does not have one, and I want to show you why rather than just tell you.

3:08So take the first. The temptation is to start collecting examples and hunting for a pattern. Do not. Everything we need is already sitting inside what dividing actually is. Dividing by a number is multiplying by the number that takes it up to one. That is not a rearrangement of dividing. That is what dividing is. So: divide by one quarter. What takes one quarter up to one? Four. Which means dividing by one quarter and multiplying by four are the same instruction.

3:48Six divided by a quarter. Six times four. Both twenty-four. I checked that on eight thousand two hundred and eighty-one pairs of numbers. The two never disagreed once. So every division on this board is secretly a multiplication. Which means we do not need a size rule for dividing at all. We already have one for multiplying. Here is the piece that makes the whole thing work. Take a number below one. One quarter, say.

4:22What takes it up to one? Four. Which is above one. Now take a number above one. Two. What takes two up to one? One half, which is below one. It always crosses over. I checked all forty-five numbers below one in my list. Every single reciprocal came out above one. And all forty-five above one. Every reciprocal came out below. One is the only number that stays where it is, because it is its own reciprocal.

4:56So now go back and run our three divisions through that. Six divided by three. Three is above one, so it turns into a third, which is below one. Six times a third is two. Multiplying by something below one shrinks it, and two is below six. Six divided by a quarter. A quarter is below one, so it turns into four, which is above one. Six times four is twenty-four. Multiplying by something above one grows it, and twenty-four is above six.

5:32One eighth divided by one quarter. Same divisor, so it turns into four again. One eighth times four is one half, which is above one eighth. Three divisions, no new rule anywhere, and the prediction came out right every time. So here is the whole thing, in one sentence. Divide by a number between zero and one, and the answer lands above the number you started with. Divide by a number bigger than one, and it lands below.

6:06Divide by one, and nothing moves at all. That is the entire rule. And notice what it is about. It is about which side of one the divisor sits on. I ran it across eight thousand two hundred and eighty-one divisions. Four thousand and ninety-five landed above. Every one of those had a divisor between zero and one, without a single exception. Four thousand and ninety-five landed below, every one with a divisor above one. And the ninety-one that did not move are exactly the ones divided by one.

6:48Now, a warning, because this is where people over-correct. It is very tempting to shorten that rule to: dividing by a fraction makes the answer bigger. That is wrong. Six divided by three halves. Three halves is a fraction. It has a bar in it and everything. But three halves sits above one, so the answer comes out smaller than six. It is four. There are thirty-four fractions above one in my list, and dividing by any of them makes the answer smaller. All thirty-four.

7:25The hinge is one. It was never the fraction bar. Which leaves the second question, and this is the more interesting one. How does the answer compare with the number you divided by? You might expect a rule in the same shape. There is not one, and here is why. Six divided by three is two, and two is below the three. Now keep that three exactly where it is, and change only the six. Twelve divided by three is four, and four is above the three.

8:01Same divisor. Opposite answer. So no rule that only looks at the divisor can possibly settle this. And it happens below one as well. One eighth divided by a quarter is a half, which is above the quarter. One hundredth divided by that same quarter is one twenty-fifth, which is below it. I counted. Sixty-five of the ninety-one divisors in my list do exactly this. The comparison depends on the number being divided, and a rule about the divisor alone simply cannot see that.

8:42One last thing, which is what all of this is actually for. Eight metres of lace, cut into quarter-metre lengths. How many pieces? The work here is not the arithmetic. The work is knowing that this is eight divided by a quarter, and not eight times a quarter. Eight times a quarter is two, and two is obviously not how many quarter-metre pieces you get out of eight metres. Eight divided by a quarter is thirty-two, and the answer being bigger than eight is exactly what you should expect from small pieces.

9:19Compare that with half a metre of ribbon shared between eight badges, which is a sharing, and comes out at one sixteenth of a metre each. Same operation, opposite direction, because the two divisors sit on opposite sides of one. So the question was never whether dividing makes things bigger or smaller. It is what you are dividing by.