Uniting rationals and irrationals into an unbroken line

0:00Here is a question that gets asked and then quietly dropped. Mark the whole numbers on a line, and then fill in every fraction you can think of. Halves, thirds, hundredths, millionths. They crowd in so tightly that between any two of them you can always squeeze another one. So have you filled the line? It certainly feels as though you must have. The answer is no, and the distance between those two sentences is the whole of this video.

0:34First, though, let us be fair to the fractions. They really are packed. Take every fraction whose bottom number is under thirteen. Between minus two and two alone, there are a hundred and eighty five of them. And between any two you pick, there is always another: their midpoint. Sixty five thousand seven hundred and three pairs were tested, and every single pair had its midpoint sitting strictly between the two.

1:04So squeeze harder. Look only at the hundredth between one point four one and one point four two. Even holding the bottom number down to a hundred, thirty fractions are crammed into that sliver. There is no room anywhere. That is what it means to say they are dense. Now a procedure. Start with the stretch between one and two, cut it into ten pieces, and keep the piece whose square straddles two. That gives you one point four to one point five.

1:39Cut that into ten and keep the right piece: one point four one to one point four two. Do it again: one point four one four to one point four one five. Every step multiplies your accuracy by ten, and every end you write down along the way is a fraction. So run it fifty times. The interval is now one part in ten to the fiftieth wide. And at every single one of those fifty steps, two is still strictly inside it. Never on an end. Always in the middle, untouched.

2:17The procedure closes in for ever, and never arrives. Before you blame the procedure, aim it somewhere else. Point it at four instead of two. First step, and the right hand end of the interval is exactly two, and two squared is exactly four. Not nearly four. Exactly. And every step after that, the same. So the procedure can land. It lands the moment there is something there to land on.

2:48And there is a second check with nothing in common with the first. Take every fraction whose bottom number is up to ten thousand and square it. Not one of them gives two. The nearest miss is eight thousand one hundred and nineteen over five thousand seven hundred and forty one, and its square still is not two. Two searches, no shared machinery, and the same answer twice. So here are two properties that sound alike and are not.

3:20Dense says: hand me two different fractions, and I will find one between them. That is a statement about pairs. Gapless says: hand me any point on the line, and I will find a fraction sitting exactly on it. That is a statement about points. The fractions have the first property. They do not have the second. And that is not a technicality. Take a right angle with both short sides of length one. The long side is exactly the length that the subdividing never reached. Draw it, swing it down onto the line, and your compass marks a point that no fraction occupies. The line has a hole in it, and you have just put your finger on one.

4:08Which tells you what the line actually needs. The fractions, and the numbers that fill the holes the fractions leave. Two collections, neither of them enough on its own, and together an unbroken line with nothing missing from it. That union has a name and a symbol. The real numbers, written with a doubled up letter R. Real, by the way, is a name and not a claim. You will see why in a few minutes.

4:39It helps to see how these collections sit inside one another. The counting numbers, one, two, three, sit inside the whole numbers, which add nought and the negatives. And those sit inside the fractions. Three rings, each one inside the next. And here is where very nearly everybody draws the picture wrong. The irrationals are not a fourth ring inside the third. They are not fractions of any kind, so they cannot possibly sit inside the collection of fractions. They sit beside it.

5:15Two regions, side by side, and the two of them together are the reals. Four collections nest. The fifth one does not. Try placing some numbers, then, each in the smallest collection that will hold it. Seventeen. A counting number, so it goes in the innermost ring. Minus twenty six. Not a counting number, but a whole one. Minus seven over three. Neither of those, but a fraction. Seven point three two. That is a hundred and eighty three over twenty five, so it is a fraction too. And if you would rather have a test that never asks you to spot the fraction, divide it out. Seven point three two stops. Nought point one three five stops. One seventh gives a block of one four two eight five seven, over and over. Stopping or repeating means a fraction.

6:14And the root of a hundred and one. A hundred and one is not a perfect square, so that one belongs in the region next door. Put both kinds on a single axis, and go looking for the join. Here are twenty three marks, and four of them are irrational: minus the root of ten, the root of two, the root of five, and pi. Now read them left to right, in order of position rather than by which side of the axis somebody wrote them on.

6:48Rational, rational, rational, irrational. Then nine rationals in a row. Then irrational, rational, rational, irrational, rational, irrational, and four more rationals. There is no seam. No stretch of the line is reserved for one kind. They are threaded through each other everywhere you care to look. There is a pattern in the way these collections got built, and it is worth saying out loud. Every stage added exactly what the stage before it had no way to say.

7:25Counting numbers cannot say nothing. So, nought. Counting numbers and nought cannot say less than nothing. So, the negatives. Whole numbers cannot say a piece of one. So, fractions. And fractions, as you have now watched happen, cannot say the length of that diagonal. So, the irrationals. Every time, the same shape of move. Something you needed to write down could not be written, so the numbers grew until it could.

7:59Which raises a fairly obvious question. Is it finished? Try one more. What number, multiplied by itself, gives minus one? Take a positive number. Positive times positive is positive. Take a negative one. Negative times negative is also positive. And nought times nought is nought. Those three cases cover the entire line between them, and not one of them can produce a negative. That was run on fourteen thousand two hundred and nineteen numbers, and none of them squared to a negative, and on three thousand seven hundred and twenty one pairs of the sign rule underneath it, with no exceptions at all.

8:47So the root of minus one is not sitting somewhere on the line waiting to be found. There is no room for it anywhere on the line. And that is the honest ending. The reals are complete in one exact sense. There are no gaps left between them. The procedure that failed on that diagonal succeeds on every real number, because every real number is already there. But complete always means complete for the questions you have actually asked.

9:21Ask for the root of a negative and the line has nothing to offer, and nobody stretched it further to make room. They stepped off it instead, into a second dimension of number, marked with the letter i. The line is finished. Number is not.