 Is the universe constructed of numbers, and can mathematics uncover the nature of reality?

In this lesson, students learn about the Pythagoreans, who believed they could understand the universe by discovering its perfect, harmonious mathematical rules. Just as the Pythagoreans did thousands of years ago, students explore how to square numbers and find square roots, confront the weirdness of irrational numbers, and discover what happens when people’s most fundamental beliefs are thrown into doubt.

### Students will

• Find areas of squares for whole-number and rational square widths
• Explore perfect squares of whole numbers and the finite differences between successive squares
• Find exact values of rational roots and an estimated value of √ 2 by testing and adjusting successive approximations
• Define irrational numbers, and learn that √ 2 is irrational
• Consider the nature of mathematical truth and how people respond when their mathematical beliefs are disproved

### Before you begin

Students should know that a square’s length is the same as its width, and they should understand how to find the area of a square by counting the unit squares that comprise it. Students will need to order and compare decimals. It’s helpful, though not essential, for students to know how to calculate a square’s area by multiplying its length and width; this relationship between area and width is explored and reinforced throughout the lesson. Students need not have any experience with square roots or the notation for squares or roots before doing this lesson: in fact, square roots and their notation are formally defined only at the end of the lesson, after students have plenty of experience with them in a geometric context.