Wikipedia:Reference desk/Archives/Mathematics/2008 March 1

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March 1

Lebesgue Integration Question

My question deals with the relationship between Lebesgue and Riemann Integration on the real line. If one has a continuous function $f:R\to R$ such that the improper Riemann integral over $R$ is infinite, then is the Lebesgue integral over $R$ also infinite?

A concrete example would be a function such as $f(x)=min(1,1/|x|)$. Is it possible for this function's Lebesgue integral to be finite? —Preceding unsigned comment added by 74.15.5.191 (talk) 01:19, 1 March 2008 (UTC)

No. Bottom line is that the only time the Lebesgue and Riemann integrals differ is when the integrand is so discontinuous that the Riemann integral is not defined (because the upper and lower integrals differ). Now watch, someone will come up with some hypothesis I forgot to include, but this is the usual case anyway. --Trovatore (talk) 01:25, 1 March 2008 (UTC)

Oh, since you actually asked two questions, I should specify that "no" is the answer to the second question. The answer to the first question is "yes", at least if the value of f is always nonnegative. --Trovatore (talk) 01:26, 1 March 2008 (UTC)

Ok I think I see why now - I thought the idea about "so discontinuous" would only work on compact intervals, but it can be applied to the whole real line as well, the proof would be something like this: If the Riemann integral is infinite, then for any a > 0, we can find a compact interval such that the integral over that interval is greater than a. But on compact intervals, for continuous functions, Riemann and Lebesgue are equivalent. So we have a series of compact intervals on which the Lebesgue integral is growing without bound... Hence the Lebesgue integral is infinite. 74.15.5.191 (talk) 01:45, 1 March 2008 (UTC)

So you do need the hypothesis that f is nonnegative. Otherwise consider the function that takes the value 2 on the interval [2n,2n+1) for every natural number n, and the value −1 on the interval [2n+1,2n+2). The improper Riemann integral converges to +∞, but only because you're taking the intervals in a fixed order. If you try to take the Lebesgue integral you wind up with ∞−∞. I don't remember for sure how the definition of Lebesgue integral deals with this sort of case; it isn't really very important so it might differ from author to author. --Trovatore (talk) 02:15, 1 March 2008 (UTC)

In the Lebesgue theory f integrable implies $|f|$ integrable, so this function is not Lebesgue integrable on the reals.Silverfish70 (talk) 12:14, 1 March 2008 (UTC)

Well, in context we're allowing integrals to evaluate to (signed) ∞, and the absolute value of the function I described is integrable in that sense. You're right of course that it's not an L¹ function. --Trovatore (talk) 19:44, 1 March 2008 (UTC)

Large Numbers

This may seem like a simple question to some but, how can a number be larger than all numbers (i.e. $\aleph _{1}$) yet not be infinite? Thanks, Zrs 12 (talk) 01:49, 1 March 2008 (UTC)

In ordinary mathematical usage, $\aleph _{1}$ is infinite. Once in a while you'll come across a context in which "infinite" is used to mean absolutely infinite, and in this case $\aleph _{1}$ may be described as transfinite to disambiguate. --Trovatore (talk) 01:55, 1 March 2008 (UTC)

Cool, thanks. Zrs 12 (talk) 02:13, 1 March 2008 (UTC)

Arc Symbol

Hello. Is there a symbol for circular arcs? If so, what does it look like? I could not find anything on the Internet. I regret the simplicity of this question. Thanks in advance. --Mayfare (talk) 05:44, 1 March 2008 (UTC)

As far as I'm aware no such symbol is in generally understood use in mathematics. If you need a symbol, you might use ◝, unicode U+25DD, ("upper right quadrant circular arc"), but you would have to explain its meaning. Or if no confusion can arise, appropriate ∡, unicode U+2221 ("measured angle"), which is not in common use at least in mathematical texts. Again, you would have to explain the meaning to which you put this symbol.  --Lambiam 12:33, 1 March 2008 (UTC)

You could use something like ${\widehat {AB}}$ (preferably a curve over the top, I can't find one in LaTeX) for the arc from A to B and ${\overrightarrow {AB}}$ for the chord from A to B. It's not universal, but it's easy to understand. You should always explain your notation the first time you use it if it isn't 100% standard. --Tango (talk) 13:21, 1 March 2008 (UTC)

I'd recognise the symbol " ${\widehat {AB}}$ " but wouldn't expect it to only apply to circular arcs..87.102.83.246 (talk) 14:54, 1 March 2008 (UTC)

Wolfram http://mathworld.wolfram.com/Arc.html suggests "arc AB" for the the curve on the perimeter connecting A and B —Preceding unsigned comment added by 87.102.83.246 (talk) 14:58, 1 March 2008 (UTC)

Yeah, a hat isn't ideal, I wanted an arc over the top, but LaTeX doesn't seems to have one (that I could find). --Tango (talk) 15:01, 1 March 2008 (UTC)

It's not available here, but the mathabx and yhmath packages (see CTAN) seem to define a \wideparen{AB} command. --Tardis (talk) 18:28, 3 March 2008 (UTC)

Is there any discovery that could destroy mathematics?

Could a discovery be made that would destroy mathematics, in the same way that we don't have Humorism anymore because of modern medicine? What would such a discovery look like? —Preceding unsigned comment added by 79.122.77.105 (talk) 07:52, 1 March 2008 (UTC)

New notation constantly destroys (old) mathematics, replacing it with a new mathematics. The use of the positional system slowly replaced the roman numerals. The use of computers has replaced the use of mathematical tables and slide rules. The use of complex exponentials is slowly making trigonometry to vanish from the litterature. The next step might be that programming languages makes the traditional mathematical notation obsolete, such that the square root sign and the integral sign and the bar used for fractions will be used no more. Bo Jacoby (talk) 09:36, 1 March 2008 (UTC).

Most unlikely - mathematics (can be/) is founded in logic, and it's rock solid in that respect. That which is already known cannot be disproved.. In this respect it differs from science theories.

I suppose if we discover some sort of edenic utopia, or it's found that thinking or using a pen causes cancer people might give up on it..(joke)87.102.79.228 (talk) 09:48, 1 March 2008 (UTC)

Or maybe they'll find a vaccine against mathematics...

Throughout the history of mathematics you find preoccupations or misconceptions that are hard to understand now (Is it meaningful to add squares to cubes/represent quantities by letters/imagine non-Euclidean geometries? Do imaginary numbers/infinitesimals/actual infinities exist?). Presumably, a few centuries from now future mathematicians will look at what is considered normal now and wonder: what were they (i.e. we) thinking? Perhaps mathematics can be founded in logic, but which logic? How can we know that that logic offers the "solid rockbottom" you want as foundation? And even if mathematics can be founded in such logic, as it is practiced it isn't.

I can imagine the standards of proof going up, as attempted in Project FlysPecK for Kepler's conjecture, with the typical present-day proof becoming to be considered unacceptably non-rigorous, just like we do not accept most pre-Cauchy/Weierstrass arguments purporting to establish limits anymore as proofs. A proof transported from a far future back to today might, conversely, be utterly incomprehensible to us. Even if the field is still called "mathematics", present-day mathematicians may be considered then like today's chemists look at alchemists.  --Lambiam 12:00, 1 March 2008 (UTC)

On a side note, I have often been occasionally bothered by the idea of sufficient rigor for a proof, it would seem to me, that if one can understand all of the notation in a give problem (and in it's proof) then the question of sufficient rigor is one that can be answered through simple logic. Take for example a final result like $y={\frac {x+3}{x}}$ I would content that one should not have to write $x\neq 0$ as it is directly implied by the notation, if one understands what that a fraction is another notation division. (I should also say that somewhere in the proof one had to know the preceding fact.) A math-wiki (talk) 12:54, 1 March 2008 (UTC)

If you're given the fraction a priori, then you may not need to specify that x in non-zero, but usually you would start with something like xy=x+3 and manipulate it into the fraction, and that manipulation requires you to *assume* x is non-zero, and you should always specify your assumptions. --Tango (talk) 13:09, 1 March 2008 (UTC)

Be careful: from $xy=x+3$ we can deduce, not assume, that $x\neq 0$. --Tardis (talk) 17:50, 3 March 2008 (UTC)

Historically the discoveries that have "destroyed mathematics" have not been discoveries that proved some part of math wrong, but rather discoveries that destroyed prevailing views about what math is. For example, when the Pythagoreans discovered irrational numbers, it shook a core belief of Greek mathematicians, which was that all numbers could be represented as fractions and that integers were the foundation of the universe. Imaginary numbers and even negative numbers similarly met stiff resistance from people who could not accept their existence. Non-Euclidean geometry was very controversial when first introduced, because it violated people's intuitive ideas about how "geometry" should act. The proofs that it is impossible to square the circle, double the cube, and trisect the angle with compass and straightedge would have been very difficult for the ancient Greeks to accept, who seem to have assumed that these problems were possible somehow, if only one could figure out the necessary steps. (Even today there are some people who refuse to accept these impossibility proofs.) More recently, Gödel's incompleteness theorems show that mathematics has certain inherent limitations, which dealt a death blow to Hilbert's program, the idea that all of mathematics could be reduced to a single consistent set of axioms. It is also known that almost all real numbers are not computable, which essentially means that nearly every real number is impossible for a computer to calculate. —Bkell (talk) 16:10, 1 March 2008 (UTC)

Yes, Gödel was a right annoyance. If I remember correctly, one corollary of his incompleteness theorem is that within some set of axioms, you cannot prove that those axioms are consistent (that using them correctly, they can never lead to a contradiction). For instance, we have to use algebra to prove the consistency of arithmetic, and set theory to prove the consistency of algebra (someone correct me if I'm wrong). But we can never prove that the "top-most" system is consistent without referring to a higher system. -mattbuck (Talk) 18:38, 1 March 2008 (UTC)

What do you mean by "algebra" in that context? Set theory is right at the bottom, you seem to have it at the top... --Tango (talk) 19:16, 1 March 2008 (UTC)

New concepts, especially new concepts that are abstract by their very nature, are difficult for many people to accept. It's human nature (for many humans, anyway) to resist change. When a new concept shatters the very basis that someone has believed in for decades, it is sure to meet with skepticism at best and with militant resistance at worst.

I'm not sure whether a new theory can destroy mathematics, as "destroy" is a very strong word, but it can certainly make irreparable changes thereto. If those changes make mathematics easier to assimilate for the common man or the average student, then those changes can be beneficial and should be embraced by academia in particular and perhaps by society in general. But change for change's sake might or might not be the best thing for society. Jonneroo (talk) 19:24, 1 March 2008 (UTC)

Change for change's sake is not likely to happen. Nobody is going to use whatever new concepts are invented if they don't serve a purpose. If they do serve a purpose, academics are likely to embrace them pretty quickly - society in general can be a little more resistant. I think modern academics are a little different to those of Pythagoras' day - modern mathematicians just want to know the truth, they don't mind what it is. Pythagoras and co. were more interested in proving themselves right, from what I can tell. --Tango (talk) 19:34, 1 March 2008 (UTC)

(ec)I should add, though, that there are likely to be some dramatic discoveries waiting to be made. The capacity of the human mind has hardly been tapped, even in the most brilliant of thinkers. If scientific research someday uncovers some of the mysteries about how the brain works, e.g., how savants with astonishing mathematical talents are able to do some of the amazing things they do, change in mathematical processes and education could be right around the corner. It would be fascinating to see the results of such research. What they find, and how the human mind can process mathematical thought, could lead to dramatic developments in mathematics. Jonneroo (talk) 19:32, 1 March 2008 (UTC)

Science and math are not much more than methodologies. You can prove a finding or theory wrong, but they are subjects that welcome disproofs. I wonder if the OP understands that. Imagine Reason (talk) 21:46, 1 March 2008 (UTC)

Pretty much all of mathematics rests on the Zermelo–Fraenkel axioms of set theory. If those axioms ever turn out to be inconsistent, it would mean that most known theorems are worthless. This would count as mathematics being destroyed in my book. -- Meni Rosenfeld (talk) 23:15, 1 March 2008 (UTC)

That's as close as we get to maths being destroyed, but I don't think it would actually destroy maths - our theorems work so well that even if the ZF axioms are inconsistent there must be something usable about them, I find it hard to believe that our successes to date are pure luck. Perhaps we can make them consistent by just removing one of them, and then only the theorems that rely on that axiom are destroyed (I'm not sure if there are any axioms that aren't used for anything particularly fundamental, so that might not work). Proving the ZF axioms inconsistent would force some major changes to mathematics, but it wouldn't destroy it, we would just have to correct things and carry on. --Tango (talk) 23:33, 1 March 2008 (UTC)

AFAICT, some scientists regard maths as a weaker subject because it is based on axioms. I still maintain, however, that the methodologies are based on disproof as much as they do on any mathematical proofs, and so they can never be totally discarded. To me it's like the existential problems of how we can tell if anything is real. We can't, but so far as we can tell, science will find the best answers, and in the realm of numbers and all their esoteric relatives, so will mathematics. Imagine Reason (talk) 05:58, 2 March 2008 (UTC)

All that being said, it's hard to theorize convincingly about our theories being dead wrong. We won't really know until it happens, if it ever does, and it would almost certainly be something out of left field. Also, regarding the four humors/modern medicine comparison (which was not a single discovery, rather a complete reassessment of how things work), it was not really a setback. It was a huge jump forward. A setback like you're describing would have been all their patients dying for no reason they could guess at. Black Carrot (talk) 21:47, 2 March 2008 (UTC)

(Outdent) I want to comment here on the notion that mathematics is "based" on the ZF axioms. That's a popular misconception, and actually pretty easy to refute rather definitively, just by pointing out that there was plenty of mathematics before Zermelo and Fraenkel. Don't want to take anything away from them -- Zermelo especially was an important figure -- but they didn't invent mathematics. (In fact they didn't even invent set theory, and not even set theory is "based" on the ZF axioms.)

The demonstration that all previously existing mathematical reasoning could be formalized in a single axiomatic framework was a signal development of the twentieth century, but unfortunately a lot of people overgeneralized from that to the (fairly nonsensical) idea that that axiomatic framework defines what mathematics is. That appealed to people who want to have a cut-and-dried answer to the question "what is mathematics?". Unfortunately for those people, the answer they came up with is just wrong. And in fact there isn't any such answer. Mathematics is much more precise and much more certain than almost any other area of human endeavor -- but it is not completely precise nor completely certain. --Trovatore (talk) 08:20, 3 March 2008 (UTC)

I guess it depends what you mean by "based". While they aren't significant in the actual creation of mathematics, they are, formally speaking, the starting point of every mathematical theorem. Few people actually go back that far when proving something, but if you were to trace back through the proofs of everything you would end up at the ZF-axioms. --Tango (talk) 13:38, 3 March 2008 (UTC)

And it doesn't have to be ZF per se. It can be another formulation which is essentially equivalent to it, or just those parts of ZF which are required for a given theory. But the point remains that the ideas embodied in ZF are utilized everywhere, and it's hard to fathom how they might be invalid. A discovery that they are inconsistent would require some serious rethinking, possibly greater than ever known. -- Meni Rosenfeld (talk) 14:22, 3 March 2008 (UTC)

I think Trovatore's point is that there is plenty of math which, if you actually trace back through the proofs, doesn't rely on ZF at all, but rather a different foundation. In practice you don't even need to found your work on classical logic, let alone ZF axioms. There's plenty that can be done, say, in Intuitionistic type theory which allows for non-classical logics, and can fall outside ZF quite easily. That in turn has close ties with Topos theory which provides potential for alternative foundations again (only a relatively naive set theory -- enough to define collections and escape basic paradoxes is sufficient -- is needed, not explicit ZF), from whence things like Synthetic differential geometry arise. None of that makes ZF inconsistent, but it does make it far from unique. Mathematics requires a foundation, but it doesn't have to be ZF, and alternative approaches can give rise to interesting mathematics. -- Leland McInnes (talk) 15:45, 3 March 2008 (UTC)

Absolutely. And even if mathematics as a whole is proved to be fatally globally inconsistent, it will remain "hanging in the air" as a logical system to be studied from the outside as part of a new and exciting field of mathematics, as yet to be constructed by the next generation of mathematicians who will no doubt use its global logical inconsistency as a kicking-off point for creating new structures, rather than regarding it as a death-knell for the field. (Consider, for example, things like ω-inconsistent systems.) -- The Anome (talk) 15:58, 3 March 2008 (UTC)

Actually, my point was not exactly what Leland was saying. My point is really that the whole notion of a "foundation" is misunderstood somewhat, or overstated. Foundationalism in mathematics is itself an error (though the mathematical disciplines that are called "foundations of mathematics" are very valuable and indeed the most interesting part of math in my opinion).

Axiomatic systems have an important role to play in mathematics, but they are not its "foundation" in the sense of something to be specified once and for all, after which everything that follows from them is to be considered apodeictically certain. Nor are they arbitrary -- the "correct" axioms are something to be discovered rather than invented, and this is an ongoing process (large cardinal axioms are in some sense the most recent ones to be discovered and attain wide acceptance). In this regard mathematics is much like an experimental science -- large cardinal axioms in particular are falsifiable in something very much like Popper's sense. --Trovatore (talk) 16:37, 3 March 2008 (UTC)

Just to clarify, I suspect we are not that fr apart in our thinking; I was pitching for an example that mathematics is ultimately pluralistic in foundation (that is, there is no foundation, only choices). -- Leland McInnes (talk) 15:39, 4 March 2008 (UTC)

Well, kind of sort of. Foundations are developed after the fact (which means that "foundations" is not a terribly appropriate word for them) and you have choices in how to do that, but these choices are largely formal and/or expositional. You don't have a choice about what the truth is, but you do have a choice about how to formalize it or express it in language.

So what I'm saying is that set theory in particular is trying to get at one unique underlying reality -- that part of it is not pluralistic. We'd like to know, for example, whether the continuum hypothesis is really true or really false, notwithstanding the inability of our currently accepted axiomatizations to answer the question. But what is pluralistic is that there are lots of ways to describe or formalize the underlying reality or pieces of it. It's kind of like Rashomon. --Trovatore (talk) 20:14, 4 March 2008 (UTC)

I stand corrected. We differ rather radically -- I don't believe there is a True or False answer for the continuum hypothesis, and claims that there is a real answer out there and we just don't know implies a sort of platonist mathematical realism that I have trouble accepting. Mathematics is the model, not the reality, and I can't see any "capital T" Truth to it, just efficacy (as in, some models work better than others). Of course there's a "small t" truth local to any given model. I just don't see that there needs be "One True Model"; merely a plethora of different models that vary to fit the circumstances. On some level, for instance, I find the model of the continuum offered by smooth worlds in smooth infinitesimal analysis to be far more natural and realistic model than the punctiform continuum of classical analysis, and if I had to declare one "True" then it would likely be SIA. That doesn't make classical analysis wrong, just different and still efficacious where is is suitably applied. In that sense I see the continuum hypothesis as a matter of deciding what assumption is going to be most beneficial for the problem you are putting your model toward (i.e. do you want a very rich world of sets, or a relatively tame and constructible world of sets). -- Leland McInnes (talk) 20:24, 5 March 2008 (UTC)

Returning to the original question, what about if someone disproved the Riemann hypothesis, as I understand it quite a bit of mathematics has been built on the assumption that this is true, so a disproof would cause quite an upset. --Salix alba (talk) 17:46, 3 March 2008 (UTC)

Now that's a really interesting question. It wouldn't destroy mathematics itself, but could certainly wreck all kinds of nice theories. Many of those theories are either clearly very close to true, though, or are true and can be proved without the Riemann Hypothesis, as has even been done in some cases, so it might not cause much of a collapse. It would certainly be a disappointment, though, and would be famous as one of the big reasons we try to prove things in the first place. Black Carrot (talk) 18:59, 3 March 2008 (UTC)

Difference between n to the power of n upto nth position defined

Following are the steps for the "Anupam's Formula"

Step 1

Let
a = xⁿ - (x-1)ⁿ

b = (x-2)ⁿ - (x-1)ⁿ

c = (x-3)ⁿ - (x-2)ⁿ ...

p = (x-n)ⁿ - (x-n-1)ⁿ

Step 2

a1 = a - b - c - .. - z

a2 = b -c - ...- z

a3 = c- .. - z

...

p1 = a1 - a2 - a3 - ..

Step 3

Follow Step 2 repeatedly until there is only one amount left

Step 4

This amount is equal to n!

Example :

Take

n=2

12² - 11² = 144 - 121 = 23

11² - 10² = 121 - 100 = 21

So, 23 -21 = 2 = 2!

Again for n=3

Step 1

16³ - 15³ = 4096 - 3375 = 721

15³ - 14³ = 3375 - 2744 = 631

14³ - 13³ = 2744 - 2197 = 547

13³ - 12³ = 2197 - 1728 = 469

Step 2

721 - 631 = 90

631 - 547 = 84

Step 3

90 - 84 = 6

Step 4

6 = 3!

This is have tested upto 10 and have found to be correct.

Anupamdutta (talk) 10:00, 8 March 2008 (UTC)Anupam Dutta <anupamdutta@rediffmail.com>

Anupamdutta (talk) 08:48, 8 March 2008 (UTC) Anupam Dutta <anupamdutta@rediffmail.com>

User:Anupamdutta|Anupamdutta]] (talk) 07:36, 1 March 2008 (UTC) Anupam Dutta

Well Done! but when n=2 x²-(x-1)²-(x-2)² = x²-(x²-2x+1)-(x²-4x+4) has x terms left over.87.102.83.246 (talk) 14:48, 1 March 2008 (UTC)

How did you verify it? There is always going to be a (1-n)xⁿ term on the LHS, so it certainly can't be constant, which the RHS is. --Tango (talk) 14:58, 1 March 2008 (UTC)

Well, I suppose it works in the case of n=1, but that's not very helpful... --Tango (talk) 14:59, 1 March 2008 (UTC)

All I can say is I wish it did work.. then I'd have all the fun of showing that fn(x,n)/fn(x,n-1) = n etc I was almost 'excited' about it, then saw it didn't work.. I feel cheated.87.102.83.246 (talk) 15:29, 1 March 2008 (UTC)

The leading term of the polynomial

xⁿ – (x – 1)ⁿ – (x – 2)ⁿ – … – (x – n)ⁿ

is (1 – n)xⁿ, so the formula is incorrect (one side of the equation is constant while the other is not). This may explain why you have not found it elsewhere. On the other hand, it is well-known that the identity

$\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x-k)^{n}=n!$

holds for any real number x and natural number n. This can be proved using properties of the forward difference operator. Michael Slone (talk) 19:30, 1 March 2008 (UTC)

question

explain the steps involved in solving an algebraic expression that contains a variable. —Preceding unsigned comment added by 124.13.112.127 (talk) 15:46, 1 March 2008 (UTC)

That depends on the expression. What is it you're trying to solve? --Tango (talk) 16:02, 1 March 2008 (UTC)

Assume the variable is x. I assume the expression is a sum of one or more polynomials and possibly fractions with polynomials for numerator and denominator. If the whole expression is a single polynomial P(x), we'll work with that. If there are fractions, first bring everything into a single fraction P(x)/Q(x) in which P(x) and Q(x) are polynomials. For example, x + 1 − 2/x = (x² + x − 2) / x. See if there are common factors of P(x) and Q(x) that you can cross out against each other. Then discard (what is left of) Q(x). We now want to solve P(x) = 0. Write P(x) in standard polynomial form, and consider the degree. If the degree is 1, you have a linear equation; just apply the solution method. If the degree is 2, you have a quadratic equation; just apply the solution method. If the degree is larger, you may be out of luck; although there are algebraic methods for solving the general cubic and quadratic equation, I don't recommend even trying to apply them. What you can do is see if there is some simple root, like x = 0 or x = 1, using trial and error. If you succeed in finding some value r such that P(r) = 0, you have found a root of P, but you also know that P(x) is then evenly divisible by x−r, and the quotient is a polynomial of lower degree that you may hope to solve. Test all solutions found for being roots of the original algebraic expression, since some solution steps may have introduced extraneous solutions.  --Lambiam 21:43, 1 March 2008 (UTC)

If you are stuck trying to solve P(x)=0, I recommend the Durand-Kerner method. Bo Jacoby (talk) 22:53, 1 March 2008 (UTC).

I certainly wouldn't recommend that method. It's very confusing for someone not familiar with the concepts and only gives approximate solutions. --Tango (talk) 23:28, 1 March 2008 (UTC)

What would you recommend? Bo Jacoby (talk) 00:47, 2 March 2008 (UTC).

The Factor theorem (ie. plug in numbers until you find a root), combined with polynomial long division. If you're working over the rationals, you can narrow down possible roots using the Rational root theorem. --Tango (talk) 13:08, 2 March 2008 (UTC)

And when these methods fail because the polynomial has no rational roots, what then? Bo Jacoby (talk) 23:07, 2 March 2008 (UTC).

The the answer is "The polynomial has no rational roots." If you're working over the reals, chances are it's a real world problem, in which case you should just use a computer and be done with it. --Tango (talk) 01:07, 3 March 2008 (UTC)

In school, if you're solving a polynomial over the reals, odds are best that you just learned the quadratic formula. Black Carrot (talk) 20:16, 5 March 2008 (UTC)

The request was: explain the steps. Using a computer does not explain the steps. Bo Jacoby (talk) 15:25, 7 March 2008 (UTC).

Grading

I have an ongoing problem with a course I grade, but do not teach. Specifically I grade student’s calculus homework. Since the beginning of the semester I’ve noticed a trend that many times students do completely wrong work, and then at the end put “=[answer in back of the book]”

Anybody know of a good way of grading such things? The method I’ve been using has been to make a rubric for each problem, and assign points for both their work and answer, but it’s really getting to me when I end up giving students half or more of the points for something that looks to me like they don’t really know how to do. —Preceding unsigned comment added by 130.127.186.122 (talk) 16:39, 1 March 2008 (UTC)

I assume you mean they get the answer right, but have copied it??

I'm not a teacher but I can remember what at least one teacher told us at school and that was that you get neglible points for actually getting the answer right.

Surely the point system should favour very heavily using the right method for solving that problem.

Points deducted for each type of mistake made, eg numerical, algebraic, errors relating to rounding and accuracy.

In other words 1/10 for the answer.87.102.83.246 (talk) 17:09, 1 March 2008 (UTC)

I remember at A-level we got "Method marks" and "Accuracy marks". There were a certain number of each available for each question. If you got 0 for the method marks, you automatically got 0 for the accuracy marks regardless of whether or not you got the right answer. So, for a short question worth 2 marks you would get 2 for correct method and correct answer, 1 for correct method but wrong answer and nothing for incorrect method with or without the correct answer (longer questions are a little more complicated, but the principle is the same - a correct answer without evidence of where it came from is worth nothing). That's the system I would use if I were you. --Tango (talk) 17:48, 1 March 2008 (UTC)

I agree with the previous posters, but allow me to play devil's advocate for a moment. I know a fifth-grade mathematics teacher who is in his mid-twenties and not long out of college. I'm a friend of his parents, and his mother told me his story. When the young man was growing up, he was found to have a learning or cognitive disability of some kind. When he would do math homework or take a math test, he was usually able to come up with the correct answer, but he had difficulty proving how he obtained it. He was getting very low marks and was becoming very frustrated. Eventually it was discovered that he had a problem, and that problem was addressed properly. I believe one approach that was used was to give him an oral exam rather than a written one.

This shouldn't change how you grade your students' homework, but all educators should keep in the back of their minds the fact that a student may have special needs. Realize that those needs might not have been identified, and be keen to determining when a special need might exist. Jonneroo (talk) 18:06, 1 March 2008 (UTC)

I'd personally say that unless they show working, any answer should score very little. When it comes down to it, the actual answer is not often that useful, the point is that other people should be able to follow how you got it. It would be pointless to publish a paper and say "this is true" without providing any supporting proof. If I were teaching, I would award maybe 10% of the marks of a question for the right answer, all the rest for working (allowing errors to be carried forward). -mattbuck (Talk) 19:16, 1 March 2008 (UTC)

Usually the parameters of the grading method I use for a single given question depend somewhat on the question. I set a maximum score for an answer that does not actually solve the problem, but for which all steps are correct and show progress. For an easy problem that maximum is lower (say 50%) than for a difficult problem (where it is, say, 75%). What I reward are meaningful steps, with minor deductions for trivial errors. If these steps get the student halfway to the solution, that may mean a score of 50%. If the final answer is copied but not a reasonable conclusion of the preceding steps, I would not count that as positive. It is the steps taken towards a solution that count, not the formulas between the steps. For steps that are completely bogus I apply a large deduction, making sure that I don't keep applying it when one same fundamental misunderstanding mars several answers. I also may award (unannounced) "bonus points" for things that strike me as elegant or insightful beyond the call of duty. I try to select the questions so that they are uniformly spread from very easy to fairly difficult, and then – this is unusual – give all equal weight for determining the combined grade. So a student who solves the easier half perfectly and the more difficult half not at all ends up with a 50% combined grade (and not much less, as would be the case if easier problems have a substantially smaller weight).  --Lambiam 20:58, 1 March 2008 (UTC)

What's normal is to weight questions by length, rather than difficulty. A question that can be solved in one line is worth one or two marks regardless of difficulty. A question with takes a couple of pages is worth a lot of marks, again, regardless of difficulty. I think that works pretty well. Oh, and personally, I wouldn't give bonus marks like that - it means people can end up getting more than 100% and that just confuses things. You could have a couple of marks for "Elegance and style" included in the mark scheme along with everything else (and made public knowledge, otherwise people will get confused). --Tango (talk) 21:13, 1 March 2008 (UTC)

In math, do you have to do exercises so that your algorithm works well in general??

I got almost no points for a correct answer to something like "there are two numbers" and I just tried

0, 1 too small

0, 2 too small

0, 3 too small

0, 4 too big

1, 3 too small

2, 3 too small

3, 3 too small

4, 3 Bingo

which I had done in my head, but since there's no guarantee the answers are integers, it was just "lucky" that I came up with (4,3) in my head.

The teacher gave me almost 0 of the 12 points and said that that's just guessing. I disagree. I think it's good policy to think a problem through, and if you happen to come upon the answer that way, why not.

it's like plotting points on a graph. although I didn't do that, if I had, to see what the functions look like, and one of the points I pick happens to solve the system, hey, good job me, right. Not, bummer dude, no points, you just guessed... (especially since you can see I clearly picked progressive numbers...)

What do you guys think? I say I should have gotten 12 out of 12 for being smart enough to calculate the answer in my head... And lose 0 points for not giving a general approach first. It didn't ask for one! —Preceding unsigned comment added by 79.122.66.157 (talk) 20:49, 1 March 2008 (UTC)

That kind of thing used to annoy me to. Trial and error is a valid method for solving problems, it's just not practical in many cases. The question was intended to test your ability to come up a general method for solving that type of problem. In my opinion, it's the fault of whoever set the question not being careful to make sure they came up with a question which actually required you to use an advanced method, and you should have got full marks. The only thing you have to be careful with is writing down enough to prove that the answer you've found is the answer. Just writing down "4, 3" isn't enough, you need to write down the calculation to show that 4, 3 is a solution to the problem. If you did that, I would have given you full marks and changed the question for next year. --Tango (talk) 21:06, 1 March 2008 (UTC)

(after edit conflict) Normally guess-and-check is by itself a legitimate solution method (if it works – a big if), but I assume that part of what you are being taught, and expected to be able to apply, are certain solution methods that work in general, and (I can't know for sure, but) perhaps it was your mistake that you did not grasp that the instructor wanted to test your mastery of that solution method.

An important part of being "smart" and successful in life is understanding what it is other people want to hear from you – although it remains your call what you do with that understanding. If this was an oral test, and you have mastered the general method, you could have said: "but I can also show you how to solve this methodically". If you were however unable to apply the general method because you have not mastered it, then (in my opinion) the low score was not entirely undeserved.  --Lambiam 21:12, 1 March 2008 (UTC)

I'm not an educator, but I'd like to comment. It depends on how the instructor worded the question and what methods the student had been expected to learn up to this point. Let's say the actual problem is:

Solve for x and y where x + y = 7 and x * y = 12.

If students had been taught how to solve for two equations with two unknowns, then I don't feel the trial-and-error approach is satisfactory, even if he did reach the correct answer. If he was not expected to know how to use an accepted method (such as the substitution method), then trial and error should be permitted. Jonneroo (talk) 21:18, 1 March 2008 (UTC)

To clarify, there was an edit conflict, and I basically ended up saying what Lambiam said, but hadn't seen his post until now. Jonneroo (talk) 21:20, 1 March 2008 (UTC)

(edited) Duh, sorry; a more correct example for the substitution method would have been

Solve for x and y where x - y = 1 and x + y = 7.

- Jonneroo (talk) 21:30, 1 March 2008 (UTC)

I can tell you guys what the problem was. It didn't say "solve for". It said this:

"There is a series blah blah blah, with property blah blah blah. There is also another series blah blah blah, with property blah blah blah. Give the next number in the second series." It didn't say anything about giving any kind of general solution, and it was part of a test that tested many different things, like a final. Apparently the grader was irked that I didn't set up equations and solve them, and said that this is just guessing, and there is no guarantee the results would be integers. That's true enough, but then he proceeded to give me almost 0 of the 12 points the problem was worth, which would have been okay if I had just written the sentence "I guessed (answer), and look, it's right, because: _____", but instead I did what I showed at the top of this post. I wrote down the mental guesses I made before coming up with the correct answer. This is very scientific, the same as if I had plotted experimental points on a graph, only I'm smart enough not to need a graph to see whether answers are decreasing or increasing. I consider it akin to punishing someone for being too smart... If you want a general answer, or a particular methodology, you have to say "give a formula" or use a word like "solve". You can't ask "what is the number?" and then punish someone for being scientific enough to rigorously find it without introducing a bunch of unnecessary abstractions. I mentioned this was like a final, so time constraints meant that once I had an answer from first blush, why should I also write down how I would solve a complex system? It's unnecessary and a waste of time. I hope you guys will back me up.

—Preceding unsigned comment added by 79.122.66.157 (talk) 13:20, 2 March 2008 (UTC)

What exactly were those properties? I'm curious.

As was stated earlier, did you check your answer? Or was all that was on your page just the list of numbers? –King Bee ^{(τ • γ)} 13:42, 2 March 2008 (UTC)

As long as you showed why your answer was right, you should have got full marks. Correct answer with correct method (regardless of whether it's the intended method or not) should always get full marks, you only look at the mark scheme if there's a mistake. --Tango (talk) 14:00, 2 March 2008 (UTC)

You should get zero. You tried only integers and got lucky, the method was incorrect and faulty.--Dacium (talk) 23:46, 4 March 2008 (UTC)

Dude, blunt version: You lost a few points on one test (or maybe just a problems set?) for not really having learned the material. Shake it off and do better next time. That will be much more valuable to you in the long run than lawyering yourself a few extra points, that probably won't even change your final grade, on the grounds that the teacher didn't explicitly close off the trial-and-error possibility in the question. --Trovatore (talk) 03:25, 6 March 2008 (UTC)

Exactly. The questioner misses the purpose of examination. It isn't too see if you get the right answer, as though it's important somehow that (4,3) solves the problem, but to check if you've understood the material. An answer like the above does not demonstrate that, so you should not get a good score for it.--Fangz (talk) 16:57, 6 March 2008 (UTC)