Friday, February 4, 2011
Proving the Impossible
In Calculus II today, we were talking about improper integrals and other things that under 10% of humans fully understand, and one integrand came up that our professor said was impossible to antiderive. (+5 if you understood that entire sentence.) As in, it was proven to be impossible to solve. My question is, how can this be so? How can you prove something is impossible? I suppose the convention on this would be trying every single possible method, and if all resulted in failures, well, you've got yourself an impossibility. Take the sum of 2 and 2. It's impossible for 2+2 to be anything besides 4, because 1) 2+2 does not equal 5, or 6, or 7, or -3, or 1.4, or anything that isn't 4, and 2) 2+2 does in fact equal 4. So how does that relate back to proving the impossible? Just because something hasn't been observed, doesn't mean it's not impossible. I've never been able to dunk on a ten foot rim, but that doesn't mean it's impossible for me to. Improbable, yes. Very unlikely, yes. Ever going to happen, probably not. But I mean it's not outside of the realm of possibility. So just because we haven't seen an antiderivative of e^(-x^2), how does that make it so no antiderivative exists?
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To be precise, these functions cannot be expressed as some finite combination of common functions, like sin, cos, e, +, -, etc. The word finite is a key word. If you allow infinite sum, then as you learned in Calculus class, you can express the antiderivative of e^(-x^2) into a infinite series. Thus, the antiderivative does exist, just not finite combination of the common functions we use.
ReplyDeleteTo prove you need more advanced math. But, the basic idea could be by assuming if the antiderivative of e^(-x^2) is a finite sum of those common functions and then find some contradiction.
Also, 2+2 could be 0 or 1. You will learn this in Abstract Algebra. Why 2+2 is 4? It is based on Peano axioms.
Anyway, welcome to math world.
Su
Su, I love you
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