Wikipedia:Reference desk/Archives/Mathematics/2008 August 30

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August 30

Operation of adding a fraction and its reciprocal

For nonzero $a$ and $b$, define $a\odot b={a \over b}+{b \over a}$. It seems to be that for $k\geq 2$ and positive integers $n_{1},n_{2},\ldots ,n_{k}$, if the expression

$n_{1}\odot n_{2}\odot \cdots \odot n_{k}$,

parenthesized in any way, is an integer, then it is 2. Is this true? —Bkell (talk) 05:10, 30 August 2008 (UTC)

Hmm, after thinking about this as adding a fraction and its reciprocal (which didn't occur to me until I had to come up with a heading for this section), I see why it's true. Suppose $x+{1 \over x}=k$ for some integer $k$. Then $x={k\pm {\sqrt {k^{2}-4}} \over 2}$, which isn't going to be rational unless $k=2$. Since the $\odot$ operation can produce only rational numbers if we start with integers, we'll never get any integer answer except 2. —Bkell (talk) 05:39, 30 August 2008 (UTC)

Verhoeff algorithm

I'm trying to use this for some hobbyist thing so I've been looking for an online or offline for Windows easy to use implementation. First came across this [1] which I presume is precomputed check digits for up to 5 digits which is fine since I only need 4. But I wanted to check it and eventually found this [2] (which I later realised is link to on the wiki article). Trouble is, the results don't tend to agree. Could someone who can actually understand the article (or already knows the algorithm) tell me which one is producing correct results, if any? Alternatively links to any offline or online implementation that works is fine Nil Einne (talk) 09:39, 30 August 2008 (UTC)

Affine curvature

Is Affine curvature defined when the speed of the curve is zero? My instinct says it isn't, but I'd like a second opinion. Tompw (talk) (review) 11:52, 30 August 2008 (UTC)

The speed of a curve is a property of the parametrisation, the curvature shouldn't depend on the parametrisation so it will still be defined, you can just pick a different parametrisation in order to calculated it (parametrise by arc length, perhaps, then you know the speed will always be positive). --Tango (talk) 18:40, 30 August 2008 (UTC)

Actually, having a good notion of Eucldiean curvature generally does depend on being able to select a parameterization for which, at least, the speed is nonzero. The affine curvature requires slightly more than this. In order for the affine curvature to be well-defined, a plane curve must support a parameterization in which the velocity and acceleration are linearly independent. Ultimately the definition can be made independent of the parameterization, or at least dependent only on the preferred parameterization with respect to special affine arclength, but the basic non-degeneracy requirement remains. siℓℓy rabbit (talk) 23:55, 30 August 2008 (UTC)

Range and Domain

please could someone answer how to find the range domain and inverse of y=x^3. i tried finding the answer, i got the range as negative infinity and infinity. according to me i suppose that is correct, but somewhere i read the answer as 'R' which is real numbers i guess. which i couldn't understand, someone please tell me how does it become real number? its not affinely extended real number which is R with a dash on top, but it is just R. so please explain. —Preceding unsigned comment added by 128.211.240.72 (talk) 20:07, 30 August 2008 (UTC)

Strictly speaking, the domain could be any number of things, but in context, the expected answer is probably $\mathbb {R}$, the set of real numbers. With this domain, the range is also $\mathbb {R}$, since the cube of a real number is always a real number, and every real is the cube of some real. I'll leave working out the inverse function as an exercise for the reader. Feel free to ask here again if you have more questions not answered in the articles I've linked. Algebraist 20:13, 30 August 2008 (UTC)

$(-\infty ,\infty )$ (or $-\infty <x<\infty$) is just the whole of the real numbers, so you're absolutely right. --Tango (talk) 22:12, 30 August 2008 (UTC)

For the record, there's no problem with having the extended real number system, $\mathbb {R} \cup \left\{-\infty ,\infty \right\}$ as the domain either, so your answer is definitely correct. But as Algebraist pointed out, a person asking "what is the domain of $y=x^{3}$?" most likely means "what is the largest subset of $\mathbb {R}$ that can be the domain of a function $x\rightarrow y$ given by the rule $y=x^{3}$?". -- Jao (talk) 12:26, 31 August 2008 (UTC)

I think the poster saw the answer $\scriptstyle (-\infty ,\infty )$ somewhere and interpreted it to mean $\scriptstyle \{-\infty ,\infty \}$. He/she then saw the answer $\scriptstyle \mathbb {R}$ elsewhere, and did not realize that that is what $\scriptstyle (-\infty ,\infty )$ means. So the short answer is that $\scriptstyle (-\infty ,\infty )$ is the set of real numbers. The more important lesson is that it is no good to know the answer is you don't know what it means. Oded (talk) 17:02, 31 August 2008 (UTC)