Every fraction has one place on the number line

0:00Here is a number line. Zero, one, two — evenly spaced, and nothing in between them. Everything you have met so far lives on the marks. Whole numbers, sitting at regular intervals. And the gaps look empty. That impression is what this section destroys. But first, notice what the line already gives you for free. The distance from zero to one. That is one unit, and it is the thing every fraction in this video gets measured against.

0:32Up to now a fraction has been an amount of something. A share of a pancake. A piece of a slab. On the line it becomes a number, with an address, sitting alongside one and two as an equal. Start with the simplest cut. Take the gap from zero to one and split it into two equal parts. One new tick appears, halfway along. Now draw a bar. Start at zero, end at that tick.

1:03How long is that bar? The gap is one unit cut into two equal parts, and the bar covers one of them. One half. And the tick it ends on gets that name too. That is the template for the whole section. Cut the unit into equal parts, draw a bar from zero, count the parts it covers. Everything that follows is that same move with a different number of parts.

1:31One distinction trips people up, and it is worth being careful about. Ask yourself what the fraction actually is here. Is it the tick, or is it the bar? It is the bar. The fraction is the length from zero. The tick is just where that length happens to stop. This matters the moment there is more than one tick. Cut the gap in three and you get two new ticks.

1:58Read ticks instead of lengths and a bar ending on the second tick gets called one third, because the first tick is the one that got labelled. So say it in the right order every time. Start at zero. Count the steps. That count is the fraction. So, three equal parts. Two new ticks in the gap. The first one is one third. One step out of three. The second — take a moment. Two steps out of three. Two thirds.

2:30And there is the trap in plain sight. Two ticks, but the second one is two thirds, not one third. Draw the bar and it becomes obvious. From zero, past the first mark, ending on the second. Two steps. Notice what did not change. Zero, one and two have not moved. All that changed is how finely we chose to chop the gap. So chop it more finely, and watch that stay true.

3:01Cut the unit into five. Four new ticks, and the steps are narrower than the thirds were. A bar ending on the second tick is two fifths. On the fourth tick, four fifths. Now cut the same unit into eight. Seven new ticks, narrower again. One eighth, two eighths, three eighths, and on it goes to eight eighths, which is back at one. Two things changed and one thing did not.

3:29The step got smaller and there are more of them. But the unit, zero to one, is exactly the length it always was. Now it is your turn with a pencil, and the question is sneakier than it looks. Draw a number line and mark three lengths on it. One tenth, three tenths, and four fifths. The obvious move is to cut the gap into ten. One tenth and three tenths land straight away.

3:58But four fifths is written in fifths, not tenths. Do you need a second line? You do not. Four fifths is four steps of one fifth — and one fifth is two of the tenth-steps. So four fifths is eight tenths, and it lands exactly on the eighth tick. No new line needed. That is the hidden lesson. Fractions written differently can share one set of ticks, if you pick the cut that suits them both.

4:30Which raises a question worth stopping on. How many fractions fit into the gap between zero and one? Guess before I answer — most first guesses are a number. Here is how to think about it. Take zero and one. Put a mark halfway — one half. Now take the gap that is left, from one half to one, and halve that. Three quarters. Halve what remains. Seven eighths. Halve again — fifteen sixteenths. Again — thirty-one thirty-seconds.

5:02There is no step at which you are forced to stop. There is always a gap left, and a gap can always be halved. So there is no end to them. That gap is not nearly empty — it is crowded beyond counting out. Here is a sharper version, and it kills a stubborn idea. The idea is that fractions come in a queue, so that after one half there is a next one.

5:29There is not. Name any fraction you think comes right after one half, and I will fit one in between. Say you pick three fifths. I take the average of one half and three fifths, and I get eleven twentieths, which sits between them. Pick eleven twentieths instead. The average is twenty-one fortieths. Also between them. You can do this forever. I am not being clever — averaging two numbers always lands between them.

6:01Whole numbers have next-door neighbours: five is followed by six, with nothing in the gap. Fractions do not. Now draw a bar that overshoots. Halve the unit again, so there is a tick at one half. Draw the short bar to it — one half, as before. Then draw a longer bar in a different colour, and let it run past the one. It ends at the next half-step, one half beyond one. What is its length?

6:31Count the steps, exactly as before. Each step is a half. One, two, three of them. Three halves. Nothing new happened. The rule did not change at the one mark, because it was never about the one mark. It was always: start at zero, count the steps. That works just as well past one as before it. And we can push it further, with four bars at once. Cut into fifths, then draw four bars that all overshoot, each one step longer than the last.

7:06The first ends one fifth past one. Count from zero: five steps to reach one, then one more. Six fifths. The next is seven fifths. Then eight fifths. Then nine fifths. And look where the next one would land. Ten fifths — which is two. Sitting exactly on a whole-number mark. So fractions are not visitors in the space between whole numbers. They are spread along the whole line. Some of them land precisely on the whole numbers, because a whole number is just a fraction that came out even.

7:44So here is what the number line has just proved, and it is more than it looks. Every fraction has exactly one place on this line. One point — not a region, not an approximate area. That is the sentence worth keeping. And it changes what a fraction is. Before this section, a fraction was an amount of something — of a pancake, of a slab, of a strip. You could not set it beside the number two and ask which is bigger — not the same kind of thing.

8:18Now you can. Two thirds and one and a half and two are all just places on one line, and you can see which comes first. That is the point of putting them here. A fraction has stopped being a quantity of something and become a number. Three things to carry out of this. One. The fraction is the length from zero, not the name of the tick it lands on. Count steps from zero, every time.

8:48Two. You choose how finely to cut the unit, and the line does not mind. Halves, thirds, fifths, tenths — same line, different step. Three. There is no end to the fractions in any gap, however small the gap. Between any two of them there is always another. And one caution, because you may see this written down somewhere. Saying there is no end to them is the right thing to say. Do not reach for a fancier word — there is a technical vocabulary here, it does not mean what it sounds like, and none of it is needed yet.

9:26Next time we sort these bars into the ones that fall short of a whole and the ones that go past it — and the two kinds have names.