Constructing an irrational length and marking it on the number line

0:00Start with a length that resists being written down. The square root of two. Take the unit and cut it into equal parts. Cut it finer, and finer again. Every length you can reach that way is some whole number of parts out of some whole number of parts, which is to say a fraction, and this number is not one. Here is that failure with a count attached. Write it with one digit after the point, then two, then three, out to twelve. Thirteen writings in all.

0:34Every one of them falls short, never over, and each is closer than the one before. Not one of them lands. The cutting does not get slow here. It never arrives. So stop cutting, and start building, with two instruments. A ruler, which carries marks and can read a scale. And a compass, which carries no marks at all. A compass cannot tell you how long anything is. What it can do is hold an opening, and carry that opening somewhere else without ever reading it.

1:10That is the whole trick of what follows, and it is worth saying plainly. You are about to produce a length nobody can write down, and then move it exactly where you want it, using an instrument that never measures anything. Drawable and writable are going to turn out to be different properties. Step one. On the line, mark the origin, and mark the point one unit along from it. Call them O and A.

1:41Now raise a perpendicular at A. Not at O. At A. That distinction is the single place this construction is most often got wrong, and later on we will run the wrong version and see exactly what it costs. For now there is nothing clever here. A unit laid off, and a right angle raised at its far end. Both are things a ruler and compass do without any argument at all.

2:11Step two. On that perpendicular, mark the point one unit up, and call it B. Join O to B. Now the theorem about right angled triangles does the work. The two legs are one and one. One squared plus one squared is two. So the square on OB is two, which means OB itself is the square root of two. Notice what just happened. We did not find that length, and we did not approximate it. We drew a segment whose square is exactly two, out of nothing but a unit and a right angle.

2:48It is on the page. It is simply not on the line yet. Before moving it, look at what is on the page. Complete the picture with a fourth point, directly above the origin. Now those four points make a square, one unit on every side, and OB is its diagonal. That is worth seeing, because it tells you this is not some special triangle picked to make the arithmetic come out. It is the diagonal of the simplest square there is.

3:22The two diagonals of that square cross at right angles, and each one cuts the other exactly in half. Nothing in the figure is arbitrary. Step three, and this is the one that does the moving. Put the point of the compass at the origin. Open it until the pencil sits on B. Do not read the opening. There is nothing there to read. Now swing the pencil down until it meets the number line, and call the meeting point P.

3:55The distance from O to P is the same as the distance from O to B, because the compass never changed. And the distance from O to B is the square root of two. So P is where that number sits on the line, and it got there without anyone writing a single digit. Why is that transfer exact, rather than a careful guess? Because of what a circle is. Hold a compass at a fixed opening and every point the pencil visits is the same distance from the point of the compass. Not roughly the same. The same.

4:35So the arc is not an approximation being dragged across the page. It is every place that length can reach from the origin, and the number line is one of the things it crosses. The compass copies. It does not read. Which is why an instrument with no scale on it can be more exact than one that has a scale. Now the mistake, because it is a specific one.

5:04There are two corners you might rest the compass on, and two places you might have raised the perpendicular, so there are four ways to run these steps. Two of them carry the square root of two onto the line. Two of them carry one. And the rule separating them is simple once you have seen it. The arc has to start at the corner diagonally opposite the top of the perpendicular.

5:30Rest it on the corner directly below instead, and you are swinging a side of the square rather than the diagonal, and you land on one. Same two instruments, same two steps, wrong length. So where is P, roughly? Between one and two, because one squared is under two and two squared is over it. Short of halfway, because three halves squared is nine quarters, which is more than two.

6:00Squeeze harder and you get it past one writing and short of the next. And here is the thing to be careful about. Both of those writings are fractions. P is not at either of them, and P is not at any writing of that kind, which is the whole reason we had to build it instead of finding it. When a decimal turns up beside this point, it is a description of the neighbourhood. It is never the address.

6:30Here is what makes the construction worth the trouble. Take OB, the length whose square is two, and use it as a leg. Raise one unit perpendicular to it, and apply the theorem again. Two plus one is three. The new hypotenuse squares to three. You did not need a new idea. You needed one more triangle. The same move again gives four, then five, so the square root of five arrives at the fourth triangle, and the square root of ten at the ninth.

7:04To reach the square root of any whole number at all, you need that number minus one triangles. That is the entire recipe. Repeat that move, keeping the shared corner fixed, and the triangles fan out into a spiral. Here is a fan of ten. Every triangle after the first takes the previous hypotenuse as one leg and adds a single fresh unit edge. Which is why ten triangles carry eleven edges marked one. The very first triangle has two of them, because both of its legs are units.

7:44The hypotenuses read root two, root three, root four, and onwards to root eleven. And notice two of those. Root four is two. Root nine is three. The fan runs straight through whole numbers on its way out. Being built this way tells you nothing about whether a length can be written down. Finally, a line with everything on it at once. Twenty three marks. Nineteen of them are ratios of whole numbers, and every single one of those could have been found by cutting.

8:20Four of them could not. Three of those four are built on square roots of whole numbers, one of them negative, and all three are reachable by the construction you have just watched, the negative one by reflecting across the origin. The fourth is pi. Pi is not the square root of any whole number, so this construction never reaches it. It is on the line anyway. And that last part is a promise rather than a proof. It is worth knowing which is which. Two tools reach the whole number roots. Everything else is taken on trust, for now.