Constructing Square Roots of Positive Integers on the number line unraveling the root spiral of Theodorus Larry Francis 3/25/15

Start with a horizontal number line. Extend a vertical number line up from the origin. Plunk in a unit square with its lower left corner at the origin. The sides of the unit square are , and so is the area, since . Since the area is , the sides must have length = since .

Now we have on the number line. Let’s go for . Draw a diagonal of the unit square from the origin to the opposite corner. By the Pythagorean Theorem, the length of the diagonal is .

Now to locate on the number line. Swing an arc with its center at the orgin and its radius = the diagonal = . This quartercircle intersects both the horizontal and vertical number lines at a distance of from the origin.

Now we have . Let’s go for . If we had a right triangle with one leg = and the other = , the hypotenuse would = . We’ll do that now by extending the right side of our unit square vertically until it intersects a horizontal line through on the vertical axis.

One leg = , and the other leg = . So we know the hypotenuse = . Here’s the right triangle we wanted. Here’s our triangle another way. One leg = , the other leg = , So the hypotenuse must = .

Now to locate on the number line. We swing an arc with its center at the origin and its radius = . This quartercircle intersects both number lines at a distance of from the origin.

The number 4 is a perfect square: its root is 2. So finding the square root of 4 should be a good test of the accuracy of our constructions. We extend the right side of our unit square as before and draw a horizontal line through on the vertical axis.

The diagonal of this rectangle will give us the length we are looking for. A right triangle with one leg = and the other leg = , will have its hypotenuse = .

Now we swing an arc with its center at the origin and its radius = and look to see if it intersects our number lines at as it should. Yep! We’re good!

We ought to be able to work a little faster now. We’ll extend the vertical line on the right side of our unit square further and draw the horizontals step- by-step as we go.

For we’ll need the diagonal of a rectangle. Once we get that, we can swing our arc and locate on the number line.

For we’ll need the diagonal of a rectangle. Once we get that, we can swing our arc and locate on the number line.

For we’ll need the diagonal of a rectangle. Once we get that, we can swing our arc and locate on the number line.

For we need the diagonal of a rectangle. Then we can swing our arc and locate on the number line.

gives us another test since 9 is a perfect square gives us another test since 9 is a perfect square. We’ll need the diagonal of a rectangle. Then our arc locates on the number line.

And so forth… ad infinitum.