(May 7, 2013) It is said that Pythagoras promoted the belief that every number can be expressed as a ratio of whole numbers. This idea was still a bit confusing to our Math Circle participants, who tried to brainstorm some number that couldn’t be expressed this way. “How about 1.43?” they asked. “3.5 over 4?” “3.5792 over 4?” That last number, combined with the repeating decimals of last week, finally convinced them that some weird-looking decimals could be made to look pretty.
Then, as a post script to last week’s session, I read aloud a sensationalized account of the Hippasus execution story. The students not only enjoyed hearing the dramatic reading, but also enjoyed pointing out questionable details. One sentence in particular caught their fancy. To paraphrase: The Pythagoreans discovered the irrationality of the square root of 2. Could this be true? What assumptions must you state to make this true? Would the following two assumptions be sufficient?
- “Hippasus discovered the irrationality of the square root of 2.”
- “Hippasus was a Pythagorean.”
Hmmm…. What if those supporting statements are not true? What does it mean to assume something as true? Are you allowed to do that? People were really confused, so I suggested the math strategy of finding a simpler example of the same concept. “Do you know who Gabby Douglas is?” I asked. The boys weren’t sure so the girls filled them in. “Let’s assume to following 2 statements are true:”
- “Gabby Douglas won the gold medal.”
- “Gabby Douglas is an American.”
“If those statements are true, can we safely draw the conclusion that The Americans won the gold medal?” This example helped everyone to understand the Pythagorean example. Assumptions and clarifying questions started flying. The students agreed that one would need more than those two statements to prove the veracity of the proposition.
“Does this tangent we’re off on have anything to do with math?” challenged R.
“It has everything to do with math,” I replied. “Why is it important?” I asked the rest of the group. They supplied an explanation of its relevance. Then we moved into a proof of the irrationality of root 2, for which the students had been asking over the last few weeks.
“You mean you’re just going to show us a proof without us figuring it out for ourselves?” asked G incredulously. She was right to be incredulous. I don’t think I’ve ever shown the students anything explicitly. But, I explained, I wanted them to see what a formal proof looks like. I promised to give them their own proposition to prove next time. I also explained that this would be a proof by contradiction - an attempt to prove the opposite and find a contradiction.
“You mean that our goal is to fail, right? clarified P. I concurred, and began to demonstrate the proof. Instead of using variables immediately, I used words, and promised a recap with variables. The algebra was a bit tricky, but we muddled through. The real sticking point was accepting that “top2 must be even” is a logical consequence of “top2 = 2bottom2.” Can you make that logical leap? We had three camps:
- R and P wanted to do a subsidiary proof.
- N and A wanted to use induction (a term we discussed) to generalize from specific examples.
- G and C were happy accepting it without question because they knew it was a rule that someone else had proven.
So we discussed “stepping stones on the path to proving a theorem,” or lemmas: all of the above methods are sometimes used at times to justify a lemma. Then we were suddenly and unexpectedly out of time. The kids were disappointed, but I promised to finish the proof next time, and to discuss the following questions that came up during class:
- Was irrationality really a secret that the Pythagoreans tried to keep? (Today’s sensational article seemed to suggest this.)
- How did Pythagoras react upon first hearing the proof of the irrationality of root 2, if in fact he did hear of it?
- Could the Pythagoreans have proved the existence of irrationality had they really wanted to?
- Was Hippasus a rogue Pythagorean?
- Within the Pythagoreans, was there a custom for reporting individual accomplishments as group discoveries?
- Was this proof originally intended to be by contradiction?
- Could we hear more about Themistoclea, Pythagoras’s reported teacher? (Her name came up when the students said, “Surely there were no female Pythagoreans, right?”)
See you all next time!