Realizing that there are several other threads with similar inquiries, I need help choosing an appropriate topic for my I.B. mathematics extended essay. The extended essay is written on a topic of the (high school) student's choice and must be between 2000 and 4000 words. For some background information on the assignment, I'll pull information directly from the I.B.O. "An extended essay in mathematics provides students with an opportunity to demonstrate an appreciation of any aspect of the subject, whether it is: ~the applicability of mathematics to solve both real and abstract problems ~the beauty of mathematics as in, for instance, geometry or fractal theory ~the elegance of mathematics in the proving of theorems as in, for example, number theory ~the origin and subsequent development of a branch of mathematics over a period of time, measured in tens, hundreds or thousands of years ~the links between different branches of mathematics and the powerful structures that enable may seemingly different problems to be solved by a single theory ~the way that a branch of mathematics has been born, or has flourished as a result of technology. "The extended essay may be written on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself . . ." My thoughts: Since the essay doesn't need to be exceedingly rigorous, I figured should be able to write a reasonable essay detailing how non-standard analysis can be used to simplify things like the definition of continuity. Recently, I've also been reading about the cardinality and countability of certain sets and I think it would be interesting to try and write an essay on that as well (the elementary stuff doesn't seem too difficult). However, since I don't if either of these would ultimately lend themselves to writing the essay, I'm in need of advice. Any recommendations are greatly appreciated. Thanks!
I recommend "Where mathematics Comes From" by Lakoff and Nunez if you decide to pursue the non-standard analysis path. But, be warned, the original monograph by Robinson is way beyond high school level. The first few chapters of H. Jerome Keisler: Elementary calculus: An Approach Using Infinitesimals is about the right level. You could compare what you learn about calculus at school with what Keisler says - his book was used as a first textbook in University calculus and was well respected. It's now available free online, see Wikipedia article for links and more: http://en.wikipedia.org/wiki/Non-standard_analysis
Thanks, I realize the Robinson's text would be inaccessible for me. I've looked at Keislerâ€™s text before and I was actually going to do a comparison between standard calculus and non-standard calculus. Again, thanks! Any more suggestions are welcome.