Wikipedia:Reference desk/Archives/Mathematics/2009 October 26

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October 26

Logs

I suck at anything having to do with mathematic logs, so you might have to dumb down your answers to my question... :-)

I want to convert some decibels to w/m². So, how do I do that? I have 110db = 10log(I/10e-12) So, how do I get that log out of there and solve for I? Dismas|^(talk) 04:53, 26 October 2009 (UTC)

Set both sides as an exponent of the log base (which is 10 in this case), so 10¹¹ = 10^{log(I/10e-12)}. Rckrone (talk) 05:32, 26 October 2009 (UTC)

And of course, exponentiation and logs cancel so you end up with $10^{11}={\frac {l}{10e-12}}$. More generally, sound intensity of x decibels is equivalent to $10^{x/10}I_{\textrm {ref}}\,\!$. -- Meni Rosenfeld (talk) 05:42, 26 October 2009 (UTC)

Did you really mean $10e-12=10^{-11}$, or rather $1e-12=10^{-12}$?

PS. I assume you are a logrolling master? -- Meni Rosenfeld (talk) 05:42, 26 October 2009 (UTC)

Now that I see how you do the math markup...To clarify, I meant this: $110=10\log \left({\frac {I}{10^{-12}}}\right)$ I know that you guys think that you're explaining this well but I just don't understand what you're saying.

And yes, I'm pretty good at logrolling, actually.  :-) Dismas|^(talk) 05:55, 26 October 2009 (UTC)

Let's use the letter t to refer to ${\frac {I}{I_{0}}}={\frac {I}{10^{-12}W/m^{2}}}$. You have $110=10\log t$, which means $11=\log t$. Now, "log" here means the common logarithm which is defined as: "$\log x$ is that number y such that $10^{y}=x$". Since $11=\log t$, it means that 11 is the number y such that $10^{y}=t$. In other words, $10^{11}=t$. Since $t={\frac {I}{I_{0}}}$, we have $I=tI_{0}=10^{11}I_{0}=10^{11}10^{-12}W/m^{2}=10^{-1}W/m^{2}=0.1W/m^{2}\,\!$. -- Meni Rosenfeld (talk) 06:10, 26 October 2009 (UTC)

The substitution of t helped a lot! Thanks! Dismas|^(talk) 06:16, 26 October 2009 (UTC)

Diagonalise these 2 QFs at once - must be going wrong somewhere

Hi there all:

I'm looking at the following question -

Find a linear transformation which reduces the pair of real quadratic forms $2x^{2}+3y^{2}+3z^{2}-2yz$ and $x^{2}+3y^{2}+3z^{2}+6xy+2yz-6zx$ to the forms $X^{2}+Y^{2}+Z^{2}$, and $\alpha X^{2}+\beta Y^{2}+\gamma Z^{2}$, where $\alpha$,$\beta$ and $\gamma$ ($\in \mathbb {R}$) should turn out to be integers.

Now that essentially means find a basis for which they're both diagonal, right?

Assuming so, I've found the eigenvalues/vectors of the first form okay, but for the second case I get something like $4,\,{\frac {3\pm {\sqrt {73}}}{2}}$ - is that right? I'm having trouble then calculating the eigenvectors in a non-incredibly-messy way assuming all's okay so far, but I think I might be approaching it wrongly to start with.

Thanks a lot for any help, the more I can get the better!

Spalton232 (talk) 19:08, 26 October 2009 (UTC)

The eigenvalues you specify are correct. The eigenvectors are $(0,1,1),\left({\frac {2}{3}}-{\frac {1}{6}}\left(3\pm {\sqrt {73}}\right),-1,1\right)$, but I have no input on how to find them effectively or other parts of your solution. -- Meni Rosenfeld (talk) 20:54, 26 October 2009 (UTC)

0.9...

There's been some discussion lately at the talk page of 0.999... disputing the accuracy of this article. I am not involved nor do I really plan to be, but I think it's probably better to discuss this here. Is there any way we can make people actually believe the article and its associating proofs? Is it worth investing time into this?

A transcript of the argument:

• First post
• A reply
• Another response

I think it's time we resolved this, personally. ceranthor 19:13, 26 October 2009 (UTC)

We created a page, Talk:0.999.../Arguments, so such arguments don't clog up the article talk page. --Tango (talk) 19:24, 26 October 2009 (UTC)

I don't think that particular discussion is really relevant until we get a response from the OP, explaining his/her stance on the issue, in light of the responses 76.103.47.66 and I have provided.

In general however, there is a clear trend of disbelief in the equality, and I'm honestly not sure how it can be resolved. The trouble is that any proof of the equality has one of two problems: either it is oversimplified and sweeps some important issues under the rug, and the reader may spot these issues, and come to believe that they are inherent flaws in accepting the equality, or it is too complicated for the non-mathematician to follow, and therefore useless for convincing them of the equality.

I will say that I like 76.103.47.66's argument; it's a nice visual proof, not too complicated, and it makes it very clear exactly how we go about deciding that the equality is true (we have the same amount of pie). Does anybody else have any interest in working on getting that argument added to the list of proofs? Obviously, we would have to make it a bit more concise, and maybe add a picture illustrating it.

By the way, shouldn't this discussion be on the 0.999... talk page? --COVIZAPIBETEFOKY (talk) 21:24, 26 October 2009 (UTC)

I don't know where this should or should not be discussed. That much said, I think that proofs are going to be beside the point for many people on this issue simply because they want to accept their wrong intuition that every decimal number representation is for a number distinct from all others. To reach such people, it is important to make it clear that all mathematical objects, even the simple ones with which they themselves are familiar, are abstractions. Then one can say that two points on a line--a kind of abstraction--that have no distance between them (a distance of zero) are really not distinct points and that this number issue is the same exact thing by the correspondence we all (?) learn in grade school. If one wants to go on and pique the curiosity of the person, then (after he or she is convinced and has been shown how a real mathematical proof works in addition) one can talk about how there is a way to make something containing our ordinary numbers but in which infinitely small numbers not equalling zero exist (in a subject called non-standard analysis), etc.Julzes (talk) 07:54, 31 October 2009 (UTC)

Domains wot change

Hi.

$A=\lbrace {y\in \mathbb {R} |y\neq -1}\rbrace$

$f:A\to \mathbb {R}$, with $f(x)={\frac {2x+2}{x+1}}$.

Firstly, can I do this?

$f(x)={\frac {2(x+1)}{x+1}}$,

$f(x)=2\,$

And if so, what's the deal regarding the domain? I assume the original domain is "preserved" so even though it looks like a harmless constant function, we are still denied $x=-1$. In which case, what happens when kinda do the reverse, and turn $f:\mathbb {R} \to \mathbb {R} ,f(x)=2$ into $f(x)={\frac {2x+2}{x+1}}$ by reasonable algebraic manipulations? Does it "force" a domain change, or is it an illegal move altogether? What about if $f:\mathbb {R} \to \mathbb {Z}$? —Preceding unsigned comment added by 94.171.225.236 (talk) 19:47, 26 October 2009 (UTC)

Those are good questions. I'll begin by saying that defining a function by a formula is not enough - you must specify what its domain is. However, usually we assume that if a domain was not specified explicitly, then an implicit specification is taken from the formula. Thus, the formula $f(x)={\frac {2x+2}{x+1}}$ is valid exactly when $x\neq -1$, so we use it to mean that function whose domain is A (at least, if it's clear we're talking about reals and not complexes or something).

By this convention, the formula $f(x)=2$ refers to a function defined on all of $\mathbb {R}$, which is of course different from the previous function which was defined only on A. By itself, the formula is not a valid representation of the original function. However, you still have the option to specify the domain explicitly - so the function defined as $f:A\to \mathbb {R} ,\ f(x)=2$ is the same function you started with.

For the reverse, the manipulation is illegal. From $f(x)=2$ you cannot deduce that $f(x)={\frac {2x+2}{x+1}}$ since the latter is not defined for $x=-1$ which is in f's domain. Of course, you can choose to move on from your original function $f(x)=2$ and discuss the function $f(x)={\frac {2x+2}{x+1}}$ instead - the functions are very similar, having a domain that differs by one point and being equal in their common domain.

Regarding $\mathbb {Z}$ - depends on whom you ask. Some define functions to include a specification of their codomain, while others don't. By the latter convention, it's the same function whether you say $f:A\to \mathbb {Z}$ or $f:A\to \mathbb {R}$ - the domain is the same and the values in this domain are the same. If you use the former convention, they are different as they have different codomains. -- Meni Rosenfeld (talk) 20:23, 26 October 2009 (UTC)