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

2. 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

3. 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

4. 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.

5. 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.

6. 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 =

7. 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.

8. 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.

9. 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 =

10. 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!

11. 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.

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

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

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

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

16. 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.

17. And so forth…
ad infinitum.