(April 30, 2013) As I entered the building, L, one of my Math Circle lurkers, asked excitedly, “Are you studying The Pythagorean theorem?” I never erase the board work at the end of class so that L (and other building users) can study and think about it throughout the week. L does not participate in our Circle, but follows it vicariously through the cryptic writings on the board. I filled him in on last week’s session, today’s plan, and some recommended self-study topics. “Cool!” he said. He left as our participants entered.
“Pleeeeeeeeease, can we hear about the person who was killed because of the square root of 2,” immediately pleaded G. This wasn’t my plan for today, but I did want to tell this story at some point. The class decided to vote on whether to change the topic order. As the yeas had the majority, I began to tell the tale of Hippasus. I read from three sources: James Tanton’s Thinking Mathematics VOLUME 1: Arithmetic = Gateway to All, Bob and Ellen Kaplan’s Hidden Harmonies: the Lives and Times of the Pythagorean Theorem, and a Wikipedia article.
“Wikipedia?!” C scolded.
We conversed about the accuracy of different types of sources, then returned to the story… er, the alleged story. Despite many famous anecdotes, little is known about this ancient Greek philosopher. And the stories about Hippasus are contradictory. After I told a number of these possibly apocryphal stories, people were ready to return to the “certainty” of math.
“How big is the square root of 2?” I asked. The students were able to ballpark it, but no one knew its exact value. I reminded them of Pythagoras’s belief that all numbers could be represented by ratios of integers. The group began calculating possible ratios that might generate our goal number when squared. They tried 1.5, 1.4, 1.45, and 1.43 before someone had the courage to announce that “this is really boring!”
“I have a calculator,” whispered A with a sly grin. Faces lit up. He and C quickly used it to generate 1.414213562373095. Now the deep mathematical questions began to flow:
- “How do we know whether there’s more to it after the 5?”
- “How do we know whether it has a repeating pattern?”
- “How do we know whether repeating patterns can be expressed as fractions (i.e. rational numbers)?”
- “How do we know if non-repeating patterns (i.e. irrational numbers) exist, and if so, whether they correspond to fractions?”
We stopped to ponder. The students hypothesized that the calculator had truncated the actual number. The students agreed that the pattern did not repeat because they Had Been Told So at various times in their math careers. They wondered, though, how Those Who Know actually knew. We imagined mathematicians of old spending their entire lives trying to compute it. That didn’t sit right, though, when the goal is certainty. There must be a logical proof, the kids reasoned. “How did Hippasus prove it, if he really did, and can we prove it?” they asked. I told them yes, we can prove it, and we will.
The students helped each other to understand how rational numbers can repeat in their decimal forms. P wondered how that works in reverse: can you take a repeating decimal and calculate its fractional equivalent? (Yes, you can.) Then I asked the kids for a decimal that would never repeat. No one could come up with one. I hinted, but still nothing. So I showed them one: 1.123456789101112131415… “If you could do that so easily,” asked someone, “then just how many irrational numbers are there? There must be a lot!”
“Do you think there are more rationals or more irrationals?” I queried. Opinion was divided, so I started writing variations of the above example on the board. Most quickly agreed that there must be more irrationals than rationals, but R wanted proof. She also asked about the difference between real and imaginary numbers, and whether negative numbers can have square roots. The group talked about how interesting it would be to do a Math Circle on Number Theory to further explore some of these questions. “Maybe next year!” said someone hopefully. (Maybe!)
We then returned to Hippasus and the Pythagorean theorem. I told of the two schools of Pythagoreans: the mathematikoi (the knowers, or learners) and the acusmatikoi (the hearers, or listeners). We made the analogy between the Pythagorean schools and the schools/groups in the modern children’s novel The Mysterious Benedict Society, which all but one of our participants had read. The kids asked how the Pythagoreans assigned members to the groups. I promised to research this.
With 15 minutes left and a room full of fried brains, I passed out some papers with hints at various visual proofs of the Pythagorean Theorem.* I gave no instructions other than to try to make sense of them. Some students worked together, some individually. Some progress was made as A solved one without realizing it. The class ended with students crowding around him, explaining to him (and marveling over) what he had done.
Then the students filed out. I closed the door, turned off the lights, and made sure to not erase the board, so that all interested lurkers can have fun with the math again throughout week.
*The photos below show which visual proofs we started during this session.