A Problem For You May 4, 2005
Posted by Sina in : Daily , trackbackMy exams are finally done. Now we just sit and see the marks roll in. I’m trying to decide whether I should work this summer (if I find a job) or take summer school. Anyone have any advice? Today I solved a very difficult question that I’ve been thinking about for a while now. Here’s the question to keep your brain busy: (-1)^pi = ? Let me know if you find an answer. [For the experts: Try to find (-1)^sqrt2, or (-1)^r for any irrational r] It’s not that easy. Ask me for a hint if you want. It took me a VERY long time to find the answer. "How long?" you ask. Well, I’ve been thinking about that question for over a year now. I mean, we know: (-1)^2 = 1, (-1)^3 = -1, (-1)^(1/2) = sqrt(-1) = i (where i is imaginary). But since we can’t write irrationals as fractions, I began to wonder: what is (-1)^r (where r is irrational)? Is it 1, -1, or imaginary? In particular, I was interested in finding (-1)^sqrt2, and then maybe expanding what I found for all irrationals. So while riding the subway this morning with Jian, I gave the question to him. He had no idea either. But then it hit me! And it turns out it’s not that hard or complicated. So have fun with it.
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