We mentioned that the square root of two is irrational. Prove that this is so. Here are some hints. Start by supposing that the square root of 2 is a fraction. Put that fraction in lowest terms, say, p/q, where p and q have no factors in common. Square both sides and conclude that p must be even. Then conclude that q must be even, violating the assumption that p and q have no factors in common. You can see the proof in detail in this video.

We mentioned that the set of all subsets of an infinite set constitutes a larger infinity than the original set. Prove this. Here are some hints. Start by supposing that there is in fact a match between the given set and the set of all its subsets. Call any element that is matched to a subset that contains that element included; call all other elements omitted. Now consider the subset A of all omitted elements. Show that no element of the original set could possibly be matched to A by considering whether any such element is included or omitted. Learn more about Georg Cantor's Infinity Paradox in this video here.

Do a web search on the book One, Two, Three,... Infinity by George Gamow. You will find some great testimonials by people saying that the book had a big impact on their lives. It’s an easy book to read, it’s inexpensive, and it’s terrific. Also look for Gödel, Escher, Bach by Douglas Hofsteader. This is a deep, intricate, and thought provoking book.

For videos on other math topics, check out Dennis Wildfogel’s website here.

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