Unfortunately, no. There’s the stuff that gets mailed directly to the people who submit questions.
Then there’s stuff that gets archived and is accessible online.
Unfortunately (or fortunately, as the case may be), we don’t put some of the funnier or more insane stuff on the website. That would detract from the primary educational nature of the site and the snark probably wouldn’t be good for the kids.
It’s a shame, because there’s some good stuff we get that’s not just math teachers who don’t necessarily understand math.
There are quite a few cranks out there who try to get us to sign off on their insane stuff. Mostly stuff like proofs of the Goldbach Conjecture, the infinitude of twin primes, poor attempts at trisecting angles, alternate (and usually poorly done) proofs of Fermat’s Last Theorem, the non-existence of odd perfect numbers, the four-color theorem, the Riemann Hypothesis, and some other well known unsolved mathematical problems.
I’ll grant there are a few, well-intentioned people who are genuinely curious if they’ve gotten a new result and usually take the bad news well. Others (and the type should be familiar to anybody who’s ever seen a Witnessing thread on this site) aren’t looking for a sounding board so much as confirmation in the annals of math history.
And then there are the misguided few who want to know how to get a publisher to take them seriously but won’t allow anybody to actually review the work for fear of having it “stolen”.
Back in school (high school and/or college, in the US), it was ‘discouraged’ for one to give an answer with a radical in the denominator. It was like giving an answer of 25/50 in middle school - you were supposed to simplify it.
While it’s clear that 1/2 is simpler than 25/50, it’s difficult to claim sqrt(2)/2 is simpler than 1/sqrt(2). It’s not simpler. It simply conforms to an arbitrary standard.
Nitpick: Fermat’s Last Theorem and the Four-Color Problem are both solved. What people are looking for there is elegant solutions (by some definition of “elegant”), which may or may not exist (probably not, or they probably would have been found already).
Yes, those particular problems have been solved, which we explain along with some historical details, but it often on deaf ears.
We get a regular assortment of cranks claiming to have “better” solutions or more elegant solutions and even a few people claiming the current proofs are flawed. As can be imagined, most submissions (the ones that aren’t paranoid about having credit stolen from them) are really bad. Actually, they remind me a lot about that one thread on how relativity is wrong. Poor understanding of the fundamentals yet claim to find flaws based on popular explanations of the conjectures/theorems.
Those guys are still a lot more fun than the ones trying to trip us on the 0.999… = 1 thing, or Cantor’s diagonal argument, or trying to give us reasons why they’ve got a good definition for division by 0. Wrong and crazy can be fun. Just plain wrong (and stupid to boot) is not fun.
The angle trisection and parallel postulate proofs are also kind of fun. Turns out people really, really want to live in a Euclidean universe (again, reminds me of the relativity is wrong thread).
Hijack: I never understood the fascination/obsession with deriving the parallel postulate from the other four postulates. If you want the parallel postulate so badly, just take it along with the other four (or use a different way of formalizing the same thing altogether; say, as the axioms of an inner product space). Why single out that one as needing to be shown from the rest? Why not try and show postulate #3 from 1, 2, 4, and 5?
(And, of course, at some point one has eliminated redundancies and none of the postulates left will be discardable. Why was it so shocking to realize that this one particular postulate of this one particular very minimal group of postulates was non-redundant?)
At least in the context of Euclid, the parallel postulate looks a lot more like a theorem—like the other statements that Euclid did derive from the postulates—than like the “self-evident truths” that the other four appear to be.
The really amusing thing there is that some of Euclid’s postulates can, in fact, be derived from the others. I’ve personally stumbled across a method for the “extend a line segment” one, and I’ve heard that one of the others can be dispensed with if you change the definitions appropriately.
What’s the difference? (Apart from that sqrt(x) is noninvertible in some cases, but I still think it’s perfectly reasonable to read f(x) = x/sqrt(x) as meaning the same thing as f(x) = sqrt(x), except in unusually formal contexts)
Yes, that’s what I was acknowledging with “sqrt(x) is noninvertible in some cases”.
I’m perfectly happy to take f(x) = g(x)/h(x) to be acceptable notation for “f is the unique continuous function which when multiplied by h yields g”. It’s no worse an “abuse of notation” than many other common practices.
As others have mentioned, it is like simplifying fractions. Just as it is preferred to write a “1” instead of “197/197”, it is preferred to write “2-sqrt(3)” instead of one of the infinite number of equal expressions such as “(50843527-29354523sqrt(3))/(13623482-7865521sqrt(3))”. The latter will even give some calculators problems.
When doing real world number crunching where all you care about is numerical results radial in denominator can be better or it can be worse. Although they don’t normally teach this, the best way to solve a quadratic equation with quadratic formula is by getting one root by a formula with radical in the numerator and the other by a formula with radical in the denominator. In calculus it is common to have formulas with radical kept in the denominator, such as derivative of arcsin(x).