From the Exchange

I speak of Math Stackexchange frequently for two reasons: because it is fantastically interesting and because I waste inordinate amounts of time on it. But I would like to again share some of the more interesting things from the exchange here.

Firstly, in my last post on factoring, I spoke of Sophie Germain’s identity. I’ve had a case of Mom’s Corollary* with this – a question was recently asked on MathSE to “prove that $x^4 + 4$ is composite for positive integer x.” How is this done? In one step, as $x^4 + 4 = (x^2 + 2x+ 2)(x^2 – 2x + 2)$. There is the minor task to recognize that for positive integers x, $x^2 – 2x + 2 > 1$, and so the factorization is nontrivial.

Some may be thinking, “What is Mom’s Corollary?” Mom’s Corollary is a situation named by my high school English Teacher, Dr. Covel. It is astounding how often that one repeatedly comes across a new concept right after one has learnt it. In other words, when your mother tells you something, it’s surprising how often her advice will come up within the next 3 days. When it does – it’s a Mom’s Corollary case.

Secondly, there was a question based on an old GRE question.

“A total of x feet of fencing is to form 3 sides of a level rectangular yard. What is the maximum area in terms of x?”

This is not a hard question except that it defies our normal idea of associating optimal areas as squares and circles. But as opposed to doing the typical sort of optimization route, the MathSE user Jonas Meyer gives a solution that allows our intuition to soar. The idea is, to ‘place a mirror’ next to the missing side of the rectangular yard. Then the problem becomes to maximize the area in terms of 2x, and to translate it back to the 1x case. I love it when people see such symmetries and shortcuts in problems. (It’s now a square – super handy).

Thirdly, I learned of a certain paper with some really interesting identities. It is largely known that $\displaystyle \int _0 ^{\infty} \frac{\sin{(x)}}{x} \mathrm{d} x = \frac{\pi}{2}$. It is not as well known that $\displaystyle \int _0 ^{\infty} \left( \frac{ \sin{(x)} }{x} \right) ^2 \mathrm{d}x = \frac{\pi}{2}$ as well. But this paper references the following, absolutely nonintuitive to me fact:

$\displaystyle \int _0 ^{\infty} \frac{ \sin{(x)} } {x} \mathrm{d} x =$ $\displaystyle \int _0 ^{\infty} \left( \frac{ \sin{(x)} }{x} \right) ^2 \mathrm{d}x = \frac{\pi}{2} =$ $\displaystyle \sum_{n = 1} ^ {\infty} \frac{\sin{( n)} } {n}+ \frac{1}{2} = \displaystyle \sum_{n = 1} ^ {\infty} \left( \frac{\sin{( n)} } {n} \right) ^2 + \frac{1}{2}$.

And therefore, also that:

$\displaystyle \int _{- \infty} ^{\infty} \frac{ \sin{(x)} } {x} \mathrm{d} x =$ $\displaystyle \int _{- \infty} ^{\infty} \left( \frac{ \sin{(x)} }{x} \right) ^2 \mathrm{d}x =$ $\displaystyle \sum_{n = – \infty } ^ {\infty} \frac{\sin{( n)} } {n} = \displaystyle \sum_{n = – \infty } ^ {\infty} \left( \frac{\sin{( n)} } {n} \right) ^2 = \pi$.

Finally, I happened to read a post on whether $\infty$ was an odd or an even number. Okay, this is silly, and not really relevant to understanding the concept of $\infty$. But instead of the – infinity is not a number – response, one of my favorite responses was along the following lines: an even number is a number that can be paired off into equal subdivisions. So in a sense, infinity is even, as it can certainly be paired off (associate $n$ with $n + 1$. Of course, it is also the case that $\omega = \omega + 1$.

Not-so-secretly, this was all just a ploy to give myself a bit more time before I finish the next post in the factorization series. That will come later.

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