Erdös: The Man Who Loved Only Numbers
Assignments for Class #3
• Read pages 95-130 in the text.
• Watch these videos
Nice Proof of Pythagorean Theorem
Six (6!) Visual Proofs of Pythagorean Theorem
Pythagorean High Jinks
False Proof
• A "proof" that all triangles are isosceles (what?). Try to find out what is wrong with this proof: https://www.youtube.com/watch?v=Ey7wEVVqZVw
• How did Ramanujan get Hardy's attention?
A closer look at Ramanujan's first letter to G.H. Hardy: https://youtu.be/XFsuRxospbU
Questions to Think About
• State the Pythagorean theorem.
• A mathematical proof demonstrates that a statement is true in all instances that fit the premises of the proof. What are the premises of the Pythagorean theorem?
• Do you find the visual proof of this theorem convincing? Are there any questions you feel compelled to ask, in order to be sure that this statement is true in all appropriate instances? Here is the gist of the visual proof:
• See an algebraic proof of the Pythagorean theorem HERE. Give your probably rusty algebraic muscles some exercise, and work through it with pencil and paper. Do you find it any more or less convincing than a visual proof?
Other Resources
• A brief description of the significance of Ramanujan's work: https://youtu.be/LNhfUSqJdZ8
• If you enjoy visual proofs like those for the Pythagorean theorem, you might enjoy this book:
Proofs Without Words: Exercises in Visual Thinking, Roger B. Nelson
• The algebraic proof mentioned in the Assignments is contained in an extensive Wikipedia entry on Pythagoras's theorem and the wide variety of proofs that have been devised, some by people famous in other areas besides mathematics. You might be interested in reading more of that entry, HERE.
• Soon we will encounter some numbers that are mind-blowingly large, but that actually occur in mathematical reasoning. One instance is this one:
Skewe'sNumber https://images.app.goo.gl/KFkhMjvZ8WkcffkF8
Preview of class # 4
In his autobiography, Bertrand Russell wrote about a very important problem he encountered in his work, an example of which is "giving a person a piece of paper on which is written: 'The statement on the other side of this paper is false.' The person turns the paper over, and finds on the other side: 'The statement on the other side of this paper is false.' It seemed unworthy of a grown man to spend time on such trivialities, but what was I to do?"
Groucho Marx said that he would not be a member of a club that would have him as a member.
