Wikipedia:Reference desk/Archives/Mathematics/2010 February 16

= February 16 =

Round 0.45 or 0.49 or 0.50
For 5.55555 would be closer to 6? for 9.45 rounds to 9.5 bt I dunno if 45 cound round the 9 to a 10 or it pulls it back down to 9. In 8th grade math somebody told me .49 or lower I round back down .51 or above I round the number up. Would .50 bring the whole number up or keep it down? --69.229.36.56 (talk) 00:19, 16 February 2010 (UTC)


 * What? --COVIZAPIBETEFOKY (talk) 01:27, 16 February 2010 (UTC)

This is the famous "primary school mathematics". Most primary school "mathematics textbooks" dictate that 9.45 rounds to 9, and that 5.55555 rounds to 6. The number 9.5 will round to 10 because, for some weird reason, "primary school mathematics" thinks that 9.5 is closer to 10 than it is to 9. So .50 brings the number up but technically speaking, that convention makes little sense. When "rounding numbers" be sure to note that a decimal is rounded to the whole number nearest to it; therefore, .49 or lower is "rounded down" because .49 is closer to 0 than it is to 1, and .51 or above is "rounded up" because .51 is closer to 1 than it is to 0. And your "primary school mathematics teachers" will probably tell you that .50 is rounded up, but they are sure to be clueless if you ask them why (I am clueless as well but I am not the one who created or practiced that convention...). PS T  02:18, 16 February 2010 (UTC)
 * I like to think of it this way: If you're piloting a plane, and you get half way with half a tank of fuel left, are you going to go back to fill up or carry on to your destination? This has no bearing on mathematics, but perhaps the assumptions that underlie the convention (see half-full glass) are related. Or, I might be spouting nonsense. — Anonymous Dissident  Talk 11:59, 16 February 2010 (UTC)
 * See Rounding. And always rounding up has no great merits compared to rounding down. Rounding to even will give better overall behaviour, there is an interesting story there about a stock exchange always rounding down. Dmcq (talk) 02:29, 16 February 2010 (UTC)


 * @PST: the obvious answer is that in the absence of any pressing arguments for either direction, rounding .50 up is preferable because it makes a simpler rule ("round it down if the next digit is 0 to 4, and round it up if it is 5 to 9").—Emil J. 11:46, 16 February 2010 (UTC)


 * I think the poster was thinking that perhaps rounding twice in succession with an intermediate precision should lead to the same result as rounding once to the final precision. There's no reason for that to hold and 'double rounding' is an implementation problem with java on ix86 pc's because of them sometimes holding floating point in registers to higher precision than they store them. Dmcq (talk) 12:00, 16 February 2010 (UTC)


 * I always used to explain that the rule (for science) is to round up from half-way because the percentage error will then always be smaller, but that, in statistics, a different rule such as "round to even" would help to avoid bias.   D b f i r s   22:52, 16 February 2010 (UTC)


 * By the same idea one can sometimes defend rounding with the division at the geometric mean of the two possibilities so when rounding to either 9 or 10 the mid point is 9.486 Dmcq (talk) 23:08, 16 February 2010 (UTC)

Cauchy Sequence Ring and Null Sequences - Square root of 2
Hi all,

I've been playing around with the Ring of Cauchy Sequences in the rational numbers C and the subset of Null sequences N (tends to 0 as n tends to infinity). I've shown that N is a maximal ideal in C, which means C/N is a field - 2 members in this field are equal if they differ by a null sequence I believe, a.k.a. they share the same limit as n tends to infinity - so we can find a subfield of all the sequences which have a rational limit and identify that with the rationals themselves: however, I'm told that the solution x2=2 has a solution in this field, and I have no idea why. (Assuming I've got the right subfield - the problem i'm doing says deduce C/N is a field with a subfield which can be identified with Q - that must be the limits of the sequences, right?) - So would anyone be able to explain to me why the solution x^2=2 exists in this field, yet of course doesn't in the rationals themselves?

Many thanks, 82.6.96.22 (talk) 08:21, 16 February 2010 (UTC)


 * Let $$x\in \mathbb{R}$$ be arbitrary, and let $${(x_n)}_{n\in \mathbb{N}}$$ be a sequence of rational numbers converging to x (any such sequence is necessarily Cauchy). The coset of N (the ideal of null sequences) by $${(x_n)}_{n\in \mathbb{N}}$$ "behaves like" the real number x relative to the other elements in $$C\setminus N$$. Therefore, we would expect that the field $$C\setminus N$$ is in fact isomorphic to the field of real numbers (which contains the field of rational numbers as a subfield). Therefore, you are indeed correct that the field of cosets of N by elements in C that have rational limits is isomorphic to $$\mathbb{Q}$$.
 * If you would like a more formal proof of $$C\setminus N\cong \mathbb{R}$$, consider the function $$f:\mathbb{R}\to C\setminus N$$ defined by $$f(r)=s_r+N$$, where $$s_r$$ is a sequence consisting only of rational terms that converges to (the real number) r. If we consider an arbitrary Cauchy sequence c in C, it must have a real number limit x, whence $$s_x-c\in N$$ (the limit of this difference is zero by standard limit laws); hence the element $$c+N\in C\setminus N$$ is the image of x under f, and the map f is surjective. By a similar "coset type argument" one can see that the map f is also injective (for completeness, note that $$s_x-s_y$$ converges to $$x-y$$ (and is therefore in N if and only if $$x=y$$). It should now be clear that f is an isomorphism (being a bijective homomorphism).
 * Now that $$C\setminus N\cong \mathbb{R}$$ has been established, it is easy to see that a solution to the equation $$x^2-2=0$$ exists in $$C\setminus N$$; in fact, it can be shown that "many more" polynomial equations have solutions in this field, as there is nothing special about the given equation in the previous argument. Hope this helps (and feel free to ask any furthur questions if necessary). PS  T  09:35, 16 February 2010 (UTC)
 * Oh I see - I tried to phrase the question as the original problem was worded, but it wasn't specific which field it was referring to which had a solution to x^2=2: I assumed they were talking about the subfield isomorphic to Q having a solution to x^2=2; do you think they meant the whole field C/N which is isomorphic to R having a solution? I just assumed it was the subfield because x^2=2 having a solution isn't really remarkable in R, but have I misunderstood things? Thanks very much - 82.6.96.22 (talk) 20:00, 16 February 2010 (UTC)
 * ( I spent the day thinking about it and couldn't work out a way to correspond any rational-limited sequence to x^2=2, so it'd make me feel a lot better to know I'd been trying to answer the wrong question! Although then, it seems more appropriate to ask why would we expect it not to have a solution in the first place?) 82.6.96.22 (talk) 20:08, 16 February 2010 (UTC)
 * Well, if a field F is isomorphic to $$\mathbb{Q}$$, it should have "the same algebraic structure as $$\mathbb{Q}$$". Therefore, if $$x\in F$$ has the property that $$x^2=1+1=2$$, the image of x under an isomorphism from F to $$\mathbb{Q}$$ (let us denote an arbitrary such isomorphism by "f") also has the same property (since $$[f(x)]^2=f(x^2)=f(2)=f(1)+f(1)=1+1=2$$). In particular, the problem would not be correct if it required one to prove that a field isomorphic to $$\mathbb{Q}$$ contains a solution to the equation $$x^2-2=0$$. Does this answer your questions?
 * My first thought when you posed the problem was that it had little to do with analytic concepts (such as Cauchy sequences and limits) but more to do with algebra (algebra "disguised" within analysis, if you like). Thus I approached the problem by looking at it algebraically rather than analytically. PS  T  08:09, 17 February 2010 (UTC)
 * Yes, I admit when I asked it, I was surprised that there could be something like an isomorphism preserving structure but somehow failing to retain such a fundamental property as the irrationality of square roots in Q - the question makes much more sense that way, thankyou ever so much for the help :) 82.6.96.22 (talk) 08:26, 17 February 2010 (UTC)


 * If I understand the notation correctly, f does not work; for C is the ring of Cauchy sequences over the rationals, so (for example) $$f(\pi) = (\pi, \pi, \ldots, ) + N$$ is not a member of $$ C / N $$.  As for actually finding an isomorphism between $$ C / N $$ and $$ \mathbb R$$, this would depend on how $$\mathbb R$$ is defined (seeing as I would define it as being equal to C / N that makes the problem very easy for me...).  Eric.  131.215.159.171 (talk) 23:58, 17 February 2010 (UTC)
 * You are right; apologies to the OP for that silly mistake on my part. The assertion, namely that $$C/N\cong \mathbb{R}$$, remains correct; we merely need replace $$f(x)=(x,x,x,\cdots)$$ by a sequence with rational terms converging to x (such a sequence exists by the density of the space of rational numbers in the space of real numbers). The mistake is now corrected above. Thanks Eric, PS  T  00:43, 18 February 2010 (UTC)

Solving sin(x^2)
A few days ago a student who I was tutoring had to find all solutions to $$\sin(3t)=\frac{\sqrt{3}}{2}$$. That was no problem, but it had occured to me that the only way I knew to solve something like $$\sin(x^2)=\frac{1}{2}$$ was by treating it as a composition of functions. So I'm curious is there another relatively elementary way to solve this? A math-wiki (talk) 08:37, 16 February 2010 (UTC)
 * Could you please clarify what exactly you mean? If we determine the set of all z such that the sine of z is $$\frac{1}{2}$$, the set of all square roots of elements in this set would be the solution set of $$\sin(x^2)=\frac{1}{2}$$ (which is, I think, what you meant by "solving the equation by treating it as a composition of functions"). Other than this method, I do not think that there exists an elementary method to solve these sorts of equations (and even if such an elementary method existed for $$\sin(x^2)=\frac{1}{2}$$, it is unlikely it would generalize to $$\sin(g(x))=\frac{1}{2}$$ for an arbitrary function g whose inverse exists and is explicitly known). PS  T  09:45, 16 February 2010 (UTC)

I think I partially answered my own question, wouldn't taking the arcsin (as a multivalued function?, i.e. taking ALL solutions not just the ones the proper inverse function would give) then taking the square root of both sides accounting for +/- solutions to the root on the side without the variable work? That is;

$$\arcsin(\sin(x^2))=\arcsin(\frac{1}{2})$$ (arcsin here is NOT the usual inverse function but rather a different operator all together that give the whole solution set for sin(x)=1/2)

$$x^2=\frac{\pi}{6}+2n\pi, \frac{5\pi}{6}+2n\pi$$

$$x=\pm\sqrt{\frac{\pi}{6}+2n\pi}, \pm\sqrt{\frac{5\pi}{6}+2n\pi}$$

The one problem i have with this is its essentially solving it in the follow manner, just written differently.

Let $$f(x)=\sin{x}$$ and $$g(x)=x^2$$, solve $$f(x)=\frac{1}{2}$$ for x.

Then for all solutions C of $$f(x)=\frac{1}{2}$$, solve $$x^2=C$$.

This is what I meant by solving it as a composition of function (e.g. $$f(g(x))=\frac{1}{2}$$ was our original problem and all the solutions to $$x^2=C$$ are precisely the solutions to $$f(g(x))=\frac{1}{2}$$) A math-wiki (talk) 09:12, 17 February 2010 (UTC)

Sum of reciprocals of cubics
I need to find the sum of the series $$\frac{1}{2\times 5\times 8}+\frac{1}{5\times 8\times 11}+\frac{1}{8\times 11\times 14}+...$$

It's pretty obvious that the nth term is $$\frac{1}{(3n-1)(3n+2)(3n+5)}$$ but I'm not sure if that helps me find the sum to n terms, and the sum to infinity.--220.253.101.175 (talk) 08:43, 16 February 2010 (UTC)


 * I think you can write out that general term as a partial fraction decomposition giving you a sum like A/(3n-1) + B/(3n+2) + C/(3n+5) (you have to solve for the coefficients) and then you can handle those series one at a time. (Hmm, wait, that's probably no good, they would all diverge).  There are powerful ways of doing these sums with contour integrals but that's probably not what you want, and I don't remember anything about how to do it any more. 66.127.55.192 (talk) 10:34, 16 February 2010 (UTC)


 * No continue on with that fractional decomposition idea and write down the first few terms decomposed that way. You should spot something about the terms which makes things easy. Dmcq (talk) 11:44, 16 February 2010 (UTC)


 * More details. This case is particularly simple because the coefficients A B C above give you a telescoping series (write the partial sum for n from 1 to m as a linear combination of the three partial sums, respectively with coefficients A B C; observe that they are in fact the same partial sum, up to a shift and up to the first and the last terms, and note that A+B+C=0). So this is just a three-term version of the easier telescopic sum of 1/n(n+1) shown in the link, and you can even write an analog closed formula for the sum for n from 1 to m. The analog, more general situation, where you don't have cancellations, may be treated using the logarithmic asymptotics for the finite sums $$\scriptstyle\sum_{n=1}^N 1/(an+b)$$, that produces an exact value for the sum of the series.         (1/60)          --pm a  15:26, 16 February 2010 (UTC)

What's the name for a holomorphic function (e.g. a logarithmic function) not included in any other holomorphic function?
HOOTmag (talk) 09:31, 16 February 2010 (UTC)
 * What do you mean by "included"? Staecker (talk) 15:27, 16 February 2010 (UTC)
 * If "included" refers to inclusion of graphs, that is extension, I'd say "maximally defined" or "maximally extended" or "defined on a maximal domain of analyticity of its". Check also domain of holomorphy for related concepts. --pm a  15:43, 16 February 2010 (UTC)

I mean a holomorphic function which can't be more extended analytically. Is the term "maximally" a common usage for holomorphic functions which can't be more extended analytically? And why shouldn't we call them simply: "maximal holomorphic functions"? HOOTmag (talk) 16:28, 16 February 2010 (UTC)


 * As far as I see, the adjective "maximal" with "function" or so, usually refers to other partial orders than extension, and I suspect that "maximal holomorphic function" may leave some doubts on its meaning (although personally I would vote for it). By the way, "maximal solution" in the context of ODEs also sounds ambiguous (some use it the sense of extension, some in the sense of pointwise order). "Maximal holomorphic extension" (of a function/germ) or "maximally extended holomorphic function" is longer but doesn't seem to need explanation, and in fact reflects a standard general usage (e.g. "maximally extended" gets more than 20,000 google results). So, if you need a short form to be used several times e.g. in a paper I'd suggest to give the explicit definition first. Of course, the best should be finding an expression from an authoritative source. --pm a 17:55, 16 February 2010 (UTC)


 * In Google, the expression "maximally extended" appears in various contexts, including physics (like in "maximally extended universes"), geometry (like in "maximally extended polygons"), and the like. However, if you say that this expression "reflects a standard general usage" (in our context of holomorphic functions) then I accept your testimony.
 * How about: "maximally extended analytically"? Is it a common expression as well? Anyways, it's less ambiguous, isn't it? HOOTmag (talk) 18:31, 16 February 2010 (UTC)


 * Yes.. Also in google "maximal holomorphic extension" gives a hundred of results, among which some books and papers that may give you some hint, e.g. --pm a  21:59, 16 February 2010 (UTC)
 * THANKXS. HOOTmag (talk) 01:42, 17 February 2010 (UTC)

Sobbmub (talk) 15:47, 16 February 2010 (UTC)How can we solve it?Method please
18*4/6+77-5


 * Get a calculator. Turn it on.  Type 18.  Press x.  Type 4.  Press ÷.  Type 6.  Press +.  Type 77.  Press -.  Type 5.  Press =.  I believe the calculator will display 84. --  k a i n a w &trade; 15:49, 16 February 2010 (UTC)
 * Not if it is a reverse Polish notation calculator! Nimur (talk) 15:54, 16 February 2010 (UTC)
 * Depending on the reason the question was asked, order of operations might help. - Jarry1250 [Humorous? Discuss.] 15:58, 16 February 2010 (UTC)

Sobbmub, please do not re-post the same question multiple times. You've already received answers here. Nimur (talk) 16:06, 16 February 2010 (UTC)


 * If you search using google just stick the expression in as your search. The nice people at google will quickly work it out on their calculators and send the result back to you. Much easier and faster than asking the wikipedia reference desk. Dmcq (talk) 17:06, 16 February 2010 (UTC)
 * Aren't they supposed to be imps?—Emil J. 17:18, 16 February 2010 (UTC)

Cancel the 18 and the 6 to get 3, BEFORE multiplying. It's not strictly necessary to do it that way, but in some contexts it's useful, so I make a habit of it. Michael Hardy (talk) 22:05, 16 February 2010 (UTC)

math conversion
is there a website where I can learn how to convert a fraction into a percentage, convert percentage into a fraction, convert percentage into a decimal, convert decimal into percentage, convert decimal into fraction and convert into fraction into decimal? —Preceding unsigned comment added by 74.14.118.34 (talk) 16:20, 16 February 2010 (UTC)


 * Try one of these, perhaps:  . Or just search for it. —Bkell (talk) 16:56, 16 February 2010 (UTC)

Though I'm sure some of those sites describe it better. Hope this helps! Chris M. (talk) 17:54, 17 February 2010 (UTC)
 * convert fraction into decimal - Divide numerator by denominator. ex: 3/4 = 3 divided by 4 --> .75
 * convert decimal into percentage - Multiply the number by 100. ex: .75 * 100 = 75%
 * convert fraction into a percentage - Convert fraction to decimal, then decmial to percentage. ex: 3/4 = .75. .75 * 100 = 75%
 * convert decimal into fraction - in the simplest terms, just put the decimal as the numerator. ex: .341 = .341/1.  This can then (sometimes) be simplified. ex: .5 = .5/1.  .5/1 * (2/2) = 1/2, but sometimes the decimal over 1 is the best you can do.
 * convert percentage into a fraction - Divide percentage by 100 ex: 75% = 75/100
 * convert percentage into a decimal - convert percentage to fraction, then convert fraction to decimal. ex: 75% = 75/100 = .75
 * convert percentage into a decimal - convert percentage to fraction, then convert fraction to decimal. ex: 75% = 75/100 = .75

Least squares and oscillating solutions
Hi all,

I'm running a least squares algorithm for multilateration compensating for error and noise, meaning I have an overdetermined system. The algorithm is on this paper.

Picture

The target is the middle black dot (0.6,0.8) and there are 4 sensors in a square from (0,0) to (1,1). The circles indicate the samples from the sensors, sampled from a normal distribution with mean the distance between the sensor and target and common variance.

In this scenario the solver oscillates between the two solutions $$S_1$$ and $$S_2$$ (after 10,000 iterations at least). I suspect it may be because the fourth circle doesn't intersect with any of the others so there are multiple solutions.

Is there a way to obtain a solution that converges to a single solution when such a scenario arises? Is it even meaningful?

Thanks in advance. x42bn6 Talk Mess 16:29, 16 February 2010 (UTC)


 * Two possibilities emerge to explain your oscillation: (1) failure to converge for numerical reasons (i.e., incorrectly implemented or inefficiently slow converging LSQR algorithm); and/or (2) failure to converge because of physical problem setup (i.e., actual existence of two global minima).
 * If you have four receivers, you have an overdefined system. This means you have a null space which defines that set of solutions that are least incorrect - and you are solving for the least squares error.  I think the solution should be well-defined - a single, unique point with minimal error.  LSQR should be navigating that nullspace to converge at a specific point.  One way to tell if the issue is your descent algorithm or your problem formulation is to print the value of the error (residual) at every iteration.  Is it decreasing, or is it remaining the same when you iterate up to 10,000 steps?  If it continues to decrease, you have a slow convergence, and you might switch to a better solver (such as a conjugate gradient solver).  If the residual is not decreasing, you have identified the null-space of your physical problem, and must specify some other stop-condition (e.g. maximum iteration number) or redefine the error-criteria for your physical problem.  Sometimes in such source-location-detection problems, your setup is symmetric - meaning that there is an inherent ambiguity between (for example) sources directly in front and sources directly behind your receiver array.  Do your S1 and S2 solutions show some kind of physical symmetry like that?  Nimur (talk) 17:00, 16 February 2010 (UTC)


 * Another thing to check - are you using double-precision or single-precision math? In practice, a lot of oscillation and instability is caused by machine roundoff.  This can occur even if you have the appropriate accuracy in single-precision to describe your data - but your residual may be inaccurately calculated, screwing the algorithm up.  Nimur (talk) 17:02, 16 February 2010 (UTC)
 * I'm using double-precision, Java doubles. The errors $$\|T-c\|$$ where c is the current "guess" oscillate between 0.246303038971375.. and 0.253732400873977.. but don't seem to go down or converge.  I may have simply programmed it wrong.
 * Except for these irritating cases I get something which is sensible. A 2D histogram gives a normal distribution-like 3D surface centred about the true position, so I know the algorithm works in some sense.  x42bn6 Talk Mess  17:14, 16 February 2010 (UTC)
 * Hm. I'll take a look at that paper in a little more detail to decipher the algorithm and see if I can't spot an obvious trouble-spot.  Have you tried other simulation inputs and obtained the same oscillation?  Nimur (talk) 22:50, 16 February 2010 (UTC)

Mathmatical formula for infinity symbol
I have a 3D application that allows me to draw different waveforms with the variables X(t),Y(t), and Z(t). I can also plug numbers into Tmix and Tmax. Based on this, how could I draw a figure-8 infinity symbol? (Googling 'infinity symbol math formula' doesn't give me anything relevant. --70.167.58.6 (talk) 17:51, 16 February 2010 (UTC)


 * It's called a Lemniscate. Black Carrot (talk) 18:21, 16 February 2010 (UTC)


 * Mathworld has the parametric equations for one version of the lemniscate. Black Carrot (talk) 18:22, 16 February 2010 (UTC)

Alternate Law of Cosines
1.) c2 = a2 + b2 -2ab cosC,

2.) a2 = b2 + c2 -2bc cosA,

3.) b2 = a2 + c2 -2ac cosB,

Add 2.) and 3.):

a2 + b2 = a2 + b2 +2c2 -2bc cosA -2ac cosB.

Terms in a2 and b2 cancel, leaving

2c2 -2bc cosA -2ac cosB = 0.

Divide out the common factor 2c and move terms:

4.) c = a cosB + b cosA, similarly:

5.) b = a cosC + c cosA,

6.) a = b cosC + c cosB.

Are not equations 4, 5 and 6 alternative forms of the law of cosines? Should they not be mentioned in the article on that law? —Preceding unsigned comment added by 71.105.162.193 (talk) 17:54, 16 February 2010 (UTC)
 * I thought the law of cosines was most useful when you knew the lengths of two sides and the angle between them. If you know two sides (WLOG a and b) and two angles (WLOG A and B), then the sine rule is the fastest, using 180&#176;=A+B+C.  x42bn6 Talk Mess  18:01, 16 February 2010 (UTC)

Glancing at it for a few seconds, it looks correct. Whether the law of cosines can be deduced from these identities is another question. What else besides that can be deduced from them is yet another; in particular, might one use them in solving triangles or for other purposes? Michael Hardy (talk) 18:14, 16 February 2010 (UTC)


 * It's that they contain redundant information: If you know e.g. two of the angles A and B you can deduce the third and only need to know one side to work out all the others. Not only is this wasteful (you have to do extra work measuring four things) but is also more complex as how do you deal with e.g. the values not all agreeing because you measure angles better than lengths? You can do extra calculations to be sure they agree, but then you've done at least as much work as the original formula.-- JohnBlackburne wordsdeeds 18:18, 16 February 2010 (UTC)


 * Equation 4 can be deduced immediately by drawing the perpendicular from C to c, and similarly for the other equations. But there is a proof of the law of cosines here if you take your derivation in reverse.--RDBury (talk) 00:08, 18 February 2010 (UTC)
 * In fact this is one of the proofs of the law of cosines given in the article.--RDBury (talk) 00:10, 18 February 2010 (UTC)

Here's another way to derive it. Recall that the law of sines says that
 * $$ {\sin A \over a} = {\sin B \over b} = {\sin C \over c}, $$

so that for some constant d we must have
 * $$ d\sin A = a,\quad d\sin B = b,\quad d\sin C = c $$

(the constant of proportionality, d, is actually the diameter of the circumscribed circle, but we won't need that fact here). So then
 * $$ a\cos B + b \cos A = d\left( \sin A \cos B + \sin B\cos A \right) = d\sin(A+B), \, $$

and since A + B + C = half-circle, it follows that sin(A + B) = sin(C). Consequently
 * $$ a\cos B + b \cos A = d\sin(A+B) = d\sin C = c, \, $$

and hence
 * $$ a\cos B + b \cos A = c.\, $$

Michael Hardy (talk) 23:53, 16 February 2010 (UTC)

One last question on principal ideals
Hi there everyone,

Another one from me, this is my last question on Ring theory for the fortnight (hooray) and it's a toughie unfortunately.

Let F be a field, and let R=F[X,Y] be the polynomial ring in 2 variables. i) Let I be the principal ideal generated by the element X-Y in R. Show $$R/I \cong F[x]$$ ii) What can you say about R/I when I is the principal ideal generated by $$X^2+Y$$? iii) [Harder] What can you say about R/I when I is the principal ideal generated by $$X^2-Y^2$$?

The first was fine, I used the isomorphism theorem for rings with the homomorphism taking f(x,y) to f(x,x), which has kernel (X-Y), and for the second I took the same approach, setting f(x,y)=f(x,-x^2) - I got the same solution as (i), is that right? It's the third part of the question I'm having the problems with, since obviously we can't just set $$x^2=y^2$$, because that has ambiguities, and x=y or x=-y aren't valid either. Could anyone suggest what I should do to evaluate R/I in this case?

Thanks very much all :) 82.6.96.22 (talk) 21:43, 16 February 2010 (UTC)
 * You'll have to decide what you mean by "evaluating" R/I. It's certainly not isomorphic to k[x].  Try writing down a basis for R/I as an F-vector space, that might give you a feel for what the ring is like Tinfoilcat (talk) 23:27, 16 February 2010 (UTC)
 * Well, if I could find an obvious analogue to F[x] to which R/I would be isomorphic, then that would be great, but I'm not sure whether one actually exists - I'll try and fathom out a basis and see if I can get anywhere with it. 82.6.96.22 (talk) 08:28, 17 February 2010 (UTC)

(Consider the following in the context of (iii); it may provide a different way of analyzing the problem) Let $$p(X,Y)=X+Y$$ and let $$q(X)=X-Y$$, and consider the cosets $$r=p(X,Y)+I$$ and $$s=q(X,Y)+I$$ in $$F[X,Y]/(X^2-Y^2)$$. Note that $$rs=0$$, $$\frac{r+s}{2}=X$$, and $$\frac{r-s}{2}=Y$$. Consider the ring $$F[X,Y]/(XY)$$. PS T  11:08, 17 February 2010 (UTC)
 * In other words, use the ring version of the Chinese remainder theorem.—Emil J. 12:26, 17 February 2010 (UTC)
 * It's not really the CRT, which only applies if the two ideals are coprime. There's something slightly different going on though: what you get, as PST observes, is a fibre product (pullback).  This example should be an instance of a more general result about fibre products, something like: let R be a ring with max ideal m such that R/m=F.  Let I,J be ideals such that I+J=m.  Then R/IJ is iso to the fibre product of R/I and R/J 129.67.37.143 (talk) 21:16, 17 February 2010 (UTC)

LaTeX question, columns
I want to type up a list of derivatives and integrals, just like the table that is in the front or back cover of a lot of calculus textbooks. That is, a one page sheet, one-sided sheet with just the standard derivatives and integrals, up through arcsine and such, but not hyperbolic functions and their inverses. Any way, what I have in mind is Derivatives on top with 2 columns and then Integrals on the bottom half with 2 columns. Does any one know a good way to do this? I used something called "multicols" but it doesn't line up well. That is, if some formula is taller than others, the two columns don't line up well. I want 1. to be in the exact same vertical position as 10 (or whatever number), then 2 and 11, 3 and 12, and so on. Any help would be appreciated. Thanks. NumberTheorist (talk) 22:29, 16 February 2010 (UTC)
 * Hmm, I guess I could just do a table. That might work nicely and it's simple.  Well, if you have any great ideas, I'd love to hear them but otherwise I'll just do a table. NumberTheorist (talk) 22:50, 16 February 2010 (UTC)
 * I am not sure why you'd want to align rows exactly, especially if the formulas had very different heights, but that is your call. A table would do that.  You can always permute the formulas to approximately match heights within each row.  Baccyak4H (Yak!) 19:25, 17 February 2010 (UTC)
 * It's not that they have very different heights. It's that they have slightly different heights and most are a standard height.  But, with multicols, a few with slightly different heights adds up and then the rows aren't lined up well at all.  Also, in this case, I want the formulas in a certain order that makes sense, like all the trig derivatives will be in a group at the bottom and sin and cos go first as far as those go, and other things like this.  I think a table is a perfect idea and I don't know why I didn't think of it sooner. NumberTheorist (talk) 20:52, 17 February 2010 (UTC)


 * Depending on what exactly you are going for the AMS-math "align" environment might work also. Although it's generally used for lining up a single column of equations, it supports multiple columns as well.  See Reference desk/Archives/Mathematics/2009 September 7 for an example.  Eric.  131.215.159.171 (talk) 23:39, 17 February 2010 (UTC)