Math 409/409G History of Mathematics

1. Math 409/409GHistory of Mathematics Babylonian Approximation of Square Roots

2. Before looking at how the Babylonians approximated square roots, let’s first look at the type of problems that led them to a need to approximate square roots.

3. Standard Babylonian Problem You want to construct a rectangle having a specified area and a specified diagonal. What are the sides of this rectangle? Given: xy= a x2 + y2 = d2 Find: x and y

4. Here’s how the Babylonians solved the system of equations: They recognized that So

5. In a similar fashion, expanding yields So the given system of equations is equivalent to the system

6. Adding these equations and solving for x gives and subtracting them yields

7. Conclusion A rectangle having area a and diagonal d has sides

8. Example What are the measures of the sides of a rectangle having area and whose diagonal measures ?

9. Another example What are the measures of the sides of a rectangle having area and whose diagonal measures ?

10. To approximate the Babylonian way: • First find the largest perfect square a2 that is less than n. • Then expressed n in the form • And then used the “formula”

11. Examples of using this formula 49 = 72is the largest perfect square less than 60, so Similarly,

12. Conclusion What are the measures of the sides of a rectangle having area and whose diagonal measures ?

13. Babylonian proof that Start with a square having side a. Attach a rectangle measuring b/a by a. The area of the resulting figure is a2+ b.

14. Divide the smaller attached rectangle into two rectangles each measuring b/2a by a. Remove one of the b/2a by a rectangles, rotate it, and attach it to the bottom of the square having side a.

15. The area of the resulting figure is still a2+ b. Enclose this figure in a square. The sides of this square measure a + b/2a.

16. The areas of these two figures are approximately equal, so

17. This ends the lesson on Babylonian Approximation of Square Roots