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Pythagoras was an Ancient Greek mathematician who loved to put all the numbers he encountered into different categories. Archeologists recently uncovered Pythagoras' island of Samos and found several puzzling artifacts that indicated he was working on a major new category of numbers.

Your job is to understand what kind of numbers Pythagoras was trying to categorize. One artifact found on the island was the following tablet with three columns. The second and third columns were badly damaged. The first column shows the areas of different sized squares and the second shows the lengths of the sides of the squares. The third column has some strange symbols in it that may point to the new categories of numbers Pythagoras was making.

**Pythagoras' Mystery Tablet**

**The square areas on the tablet are:**

**
4
5/36
3
1/9
2.25
9
2
25/16**

The tablet shows that Pythagoras was looking at some relationships in a square to discover a category of new numbers. Use the following applet to continue his work.

**Questions:**

- Use the applet to answer these questions:
- If side length = 4, area = ?
- If side length = 3/4, area = ?
- If side length = 2.5, area = ?
- What is the relationship between the side length and area of a square?

- Use the applet to find the corresponding side lengths for each area
on the mysterious tablet. Describe your strategy for finding each
side length (or explain why you weren’t able to find one).
- If area = 9, side length = ?
- If area = 3, side length = ?
- If area = 1/9 (0.111111...), side length = ?
- If area = 5/36 (0.138888...), side length = ?

- Think about the difficulty of computing some of the side lengths
versus others. Explain what Pythagoras was doing in the third column when he put the side lengths into two categories of numbers.

**Bonus Question:**

How would you translate the two category symbols (**~** and *****) in the third column into English?