Wikipedia:Reference desk/Archives/Mathematics/2009 January 24

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January 24

would you give credit?

for a math problem such as: "a and b are each two-digit positive integers such that ... What are the values of A and B?"

would you give credit for an answer such as: "since the brain isn't sequential, but very, very parallel, and there are only 90*90 = 8100 possible values of a and b, I decided to think of every combination of values at once and see if any fit the criteria. I immediately saw that only a=76, b=39 fit the criteria", circling the given answer (which is correct)

thank you. —Preceding unsigned comment added by 82.120.227.136 (talk) 12:00, 24 January 2009 (UTC)

No, if it is understood that students are supposed to justify their answers in general. "Proofs" involving an appeal to superior mental abilities not shared by other people are invalid, since the point of a proof is to convince others. Joeldl (talk) 12:54, 24 January 2009 (UTC)

okay, what if the stdent then proceeds to write out the combinations and the wrong results, all the way up to the right one. In fact, something like this happened to me: instead of 76, 39, the answers were very small (3,5 or something) and I actually saw the answer as I was reading the problem. So what I then did was write out the mental steps I tried, all the way up to the right answer. But I got 0 points for the question, because the teacher wrote that this was just "guesswork". It was emphatically not guesswork, but quite scientific. —Preceding unsigned comment added by 82.120.227.136 (talk) 14:05, 24 January 2009 (UTC)

If, having seen the answer, you were able to write down a proof that it was in fact the answer, then I would give credit. Otherwise I would not. Algebraist 14:12, 24 January 2009 (UTC)

because the question asked for the single answer, would showing that the answer arrived at fit the (simple arithmetical) criteria be enough? (the reason I ask is because we don't normally call doing arithmetic a "proof" -- as you just called it -- that the equality is true for those values). —Preceding unsigned comment added by 82.120.227.136 (talk) 14:27, 24 January 2009 (UTC)

I would take a request for the single answer as an instruction that you find the answer and demonstrate that it is the only answer. Algebraist 14:34, 24 January 2009 (UTC)

There are two issues here.

1. As I understand it, you are saying that the question was of such a nature that once the numbers 76 and 39 had been found, verification that they satisfied the required condition was straightforward. In that case, it would be typical at the university level not to hold it against the student that they had used scrap work to find the answer. What matters is that it works. (That being said, taunting the marker isn't good.) On the other hand, especially at a lower level, it might be customary to require work to be shown as a measure to prevent cheating.

2. The question clearly implies that there is only one choice of a and b that works. At the university level, this would ordinarily be part of what needed to be proved by the student if the question was stated as you've indicated. The fact that the question says this is so doesn't mean it can be taken for granted. If the question is interpreted as being more like a puzzle, then perhaps this would not be understood to be part of the job. That wouldn't be typical language on a math exam, though. Joeldl (talk) 14:58, 24 January 2009 (UTC)

I am utterly confused that you would say "the fact that the question says this is so doesn't mean it can be taken for granted". If the question says "a and b are positive integers" I don't have to prove they're positive. If they ask for THE SINGLE POSSIBLE combination that satisfies their criteria, I don't have to prove that there is only a single possible combination. —Preceding unsigned comment added by 82.120.227.136 (talk) 15:27, 24 January 2009 (UTC)

It would be easier to discuss the question if you gave us its actual wording, rather than giving a different paraphrase every time you post. Algebraist 15:47, 24 January 2009 (UTC)

Agree with Algebraist. Actually, I would not see "a and b are each two-digit positive integers such that ... What are the values of A and B?" as providing the reader with the premise that there is a unique solution (although of course I would strongly suspect that it's unique—they should really have phrased it differently otherwise). If your question actually explicitly read "There is a single pair of two-digit positive integers a and b such that ...", it would be an entirely different matter. -- Jao (talk) 15:54, 24 January 2009 (UTC)

thanks for your responses. I don't have the text in front of me anymore, you are all right: what is an acceptable response depends entirely on the phrasing of the question. my questions to you are resolved for now... —Preceding unsigned comment added by 82.120.227.136 (talk) 16:12, 24 January 2009 (UTC)

why is mathematics a requirement

I mean beyond arithmetic —Preceding unsigned comment added by 82.120.227.136 (talk) 14:07, 24 January 2009 (UTC)

A requirement for what? Algebraist 14:12, 24 January 2009 (UTC)

I'm guessing the poster means at school. One of the main purposes of school is to prepare children so they can be useful members of society. Being able to do some maths is very useful in modern society and having some people able to do maths well is pretty much essential. Parents like it because children with some maths earn more money. Also it is considered a good idea to educate people so they have some conception of how to think and how the material world works. Dmcq (talk) 15:36, 24 January 2009 (UTC)

YOu might be interested in Mathematics education and Philosophy of education though they seem to just assume it is necessary. Even the ancient Egyptians taught elementary maths. Dmcq (talk) 15:46, 24 January 2009 (UTC)

You can answer your question with a question: Why do you analyze poetry in Language Arts? —Preceding unsigned comment added by 99.255.228.5 (talk) 16:24, 24 January 2009 (UTC)

that's an easy one - poetry gets you laid. This has been proven to me time and time again, and I suspect if I hadn't actually known what the memorized poem meant, I wouldn't have even made it as far as second base. —Preceding unsigned comment added by 82.120.227.136 (talk) 17:07, 24 January 2009 (UTC)

Ah, but math is also important in that respect: "Let's see, body weight of X divided by Y beers over Z hours yields desired result, now solve for Y". :-) StuRat (talk) 19:16, 24 January 2009 (UTC)

First, everyone has a need for some basic math, like arithmetic, percentages, decimals, fractions, time math, and some basic 2D geometry, like the perimeter and area of a circle or rectangle. However, many schools have requirements that go far beyond what everybody needs. They do this just in case you want to become a scientist or engineer. You might say you don't want those careers, but kids often change their minds, so isn't it better to be prepared for whatever career you pick ? StuRat (talk) 17:25, 24 January 2009 (UTC)

Maths is not just needed for scientists and engineers. Many humanities (Sociology, Psychology) need Statistics. Business Studies also needs a lot of Maths. Pilots need to understand Aerodynamics for which again quite a but of higher maths is necessary. You'd be surprised how many careers are closed if you turn your back on maths. 195.128.250.35 (talk) 22:50, 24 January 2009 (UTC)

The OP might be interested that making lots of money is also a very successful way of getting laid. And compared to poetry it can get a lot else besides. Maths is much better than poetry at getting money. Personally I'm sorry that so few people seem to see the beauty of the world but extol poetry or music. Dmcq (talk) 21:43, 24 January 2009 (UTC)

Are you suggesting that maths has something to do with the world? --Tango (talk) 22:08, 24 January 2009 (UTC)

Of course its got something to do with the real world. Unfortunately it doesn't have a lot to do with making money. Having an IQ below 90 and no morals gives you much more mileage.195.128.250.35 (talk) 22:50, 24 January 2009 (UTC)

Doing anything well requires intelligence. Dance for instance is a reliable indicator of intelligence. To Tango's question. The world supports mathematics. Also mathematics is the appropriate language for theories about the world. By the way tomorrow being the 250th anniversary of Robert Burns birth and us talking about poetry and love can I quote a bit by him? ;-) Dmcq (talk) 22:58, 24 January 2009 (UTC)

"The attraction of love, I find, is in an inverse proportion to the attraction of the Newtonian philosophy. In the system of Sir Isaac, the nearer objects are to one another, the stronger is the attractive force; in my system, every mile-stone that marked my progress from Clarinda, awakened a keener pang of attachment to her."

A grounding in mathematics is a requirement for enjoying the Bard's romantic citing of proportionality functions. Cuddlyable3 (talk) 23:13, 24 January 2009 (UTC)

By the way you might like to read this story in the Wall Street Journal Doing the Math to Find the Good Jobs Dmcq (talk) 23:09, 24 January 2009 (UTC)

And somebody showed a while ago,I can't find the reference, why the "attraction of love" according to the Bard meant he needed at least two lovers if he wanted to come close enough to them for all practical purposes. You can't do that just with poetry. Reminder to self - must write a Java program where the users can place where they think they all were :) Dmcq (talk) 01:17, 25 January 2009 (UTC)

That will be a useful program for anyone in a manage å trois fraught with 3-body problems.Cuddlyable3 (talk) 21:19, 25 January 2009 (UTC)

Alternative inequality symbols and logic

I may have posted this previously. If so, my apologies.

I've never really been happy with < and >. First, let's say I want to list the solution to X² > 4. I end up with something like:

(X < -2) OR (X > 2)

Similarly, for X² < 4 I get an answer

(X > -2) AND (X < 2)

This could also be expressed as:

-2 < X < 2

The first case, however, can't be expressed the same way. So, my idea was to replace > and < with my proposed symbols "ʘ" for "zero-toward" and "¤" for "zero-fro". In this case, the first answer becomes

X ¤ ±2 ("X is zero-fro from plus or minus two.")

And the second answer becomes:

X ʘ ±2 ("X is zero-toward from plus or minus two.")

This would also help in cases where the sign isn't clear, like DEPTH > -500 feet. Does this include depths of 501 feet or 499 feet ? DEPTH ¤ 500 feet makes it absolutely clear that we mean depths farther from zero, so 501, but not 499, feet. We could also replace ≤ and ≥ with ʘ and ¤. I've just chosen those symbols because they are readily available, but an arrow pointing into a zero, with an optional underline, and an arrow pointing out of a zero, with an optional underline, would be even better. You could also generalize the symbols, as needed. So, if X is less than 1 or greater than 2, you could write:

 1.5
X ¤ 1,2 ("X is one-point-five-fro from 1 and 2.")

To be sure, there would still be cases where greater than/less than notation would be used, but my notations seems better for certain cases.

So, my question is, has this ever been proposed ? StuRat (talk) 19:10, 24 January 2009 (UTC)

You may find absolute values easier to work with. Your first example can be written |x|>2. I think anything you can do with your symbols you can do with the regular symbols and absolute values, eg. your last example can be written as |x-1.5|>0.5. This approach has the advantage of being easy to manipulate, I'm not sure how one would do algebraic manipulations with your symbols (although that may just be because I'm not used to them). --Tango (talk) 20:22, 24 January 2009 (UTC)

As an alternative to using absolute values as suggested by Tango, I think a solution might be to use set notation, for example x ∉ [-2,2]. Does this do the trick? Joeldl (talk) 20:33, 24 January 2009 (UTC)

Or $x\in \mathbb {R} \setminus [-2,2]$. --Tango (talk) 20:46, 24 January 2009 (UTC)

Answer: in fact, no, it has never been proposed before. Does this suggest anything ? ; --84.221.199.249 (talk) 22:25, 24 January 2009 (UTC)

How do you know it's never been proposed? Have you read everything ever written on the subject of inequalities? --Tango (talk) 22:43, 24 January 2009 (UTC)

You seem to be saying that nothing which has not yet been proposed has any possible value, so we should never propose anything different. I'm glad everyone doesn't think that way, or that would be the end to all progress. StuRat (talk) 05:59, 25 January 2009 (UTC)

Symbols > and < and the way the unhappy questioner uses them are standard in major computer languages. Two barriers to any new system of inequality symbols are A) the inertia of millions of programmers and their mega-mass of software, and B) the inconvenience of using symbols that are not instantly available on every ASCII code keyboard.

An unclear sign in the example of DEPTH does not arise for programmers because declaring a variable named DEPTH implies no physical significance for the value of the variable.

I suspect that there is nothing so impractical that someone somewhere has not proposed it. IMHO what is proposed here is an unnecessary invention. Cuddlyable3 (talk) 00:00, 25 January 2009 (UTC)

In my opinion a shortcut notation for "(X < -2) OR (X > 2)" would not be useful because it would only work in this special case that the two bounds happen to be equally far away from zero. If it were a little bit off, like "(X < -1) OR (X > 3)", you notation wouldn't encapsulate it. Yet in general there is nothing more special about "(X < -2) OR (X > 2)" than "(X < -1) OR (X > 3)" or any other expression with bounds that are not negatives of each other. --76.167.241.238 (talk) 08:21, 25 January 2009 (UTC)

In my notation, "X ¤ -1,3" would represent "(X < -1) OR (X > 3)". It's still far more compact. StuRat (talk) 15:48, 25 January 2009 (UTC)

|X-1|>2 is just as compact, so I still don't see a benefit to your notation. --Tango (talk) 16:33, 25 January 2009 (UTC)

But that doesn't state the value of X, you must solve it to figure out what X is, while my way states it explicitly. StuRat (talk) 19:20, 25 January 2009 (UTC)

That says "X must be at least a distance of 2 away from 1", that seems pretty explicit to me. --Tango (talk) 20:18, 25 January 2009 (UTC)

Actually that's "X is farther than 2 away from 1". But, while it might be basic math, you still have to do it to figure out that this means X is outside of the -1 to 3 range. StuRat (talk) 01:41, 26 January 2009 (UTC)

Oh, sorry, yes, it was a strict inequality. Depending on what you want to do with X, my version may actually be more useful. Nevertheless, I think the conversion is so trivial that I would barely realise I was doing it, so your version may be more useful for school children, but once you reach a certain level of mathematics you stop caring about such simple stuff and just get on with it. --Tango (talk) 02:05, 26 January 2009 (UTC)

Not just school children, I'd say most adults would prefer to actually have the values listed than have to do a calculation to find them, even a trivial one. Maybe Lincoln would have been an exception, though, with his "four score and seven years ago". StuRat (talk) 16:32, 26 January 2009 (UTC)

Exact; I can imagine that this notation may be useful at most in some special local context (e.g. a paper) where for some reason one has a great number of inequalities of that kind (but then I would prefer the use of absolute value as other say, or otherwise a set-teoretic notation). Note that the standard order relations come together with a whole set of general rules (like x>y => x+z>y+z,... ): that's why they are so useful. By the way, if I have something to regret about the order notation, it is that "$\textstyle x\leq y$" is longer both to read and to write compared to "$\textstyle x<y$", while it is much more commonly used. I would propose the (vaguely grotesque) terms "subequal" and "superequal" for weak inequalities, but I know that they would have no better chance than the OP's one :-( pma (talk) 10:50, 25 January 2009 (UTC)

The advantage you mention of a notation like 1 ≤ x ≤ 4 over x ∈ [1,4] largely disappears when the implicit "and" is replaced with an "or." As for your comment that ≤ is more important than <, that is reflected in mathematical language in France, where in the latter half of the 20th century mathematicians began reading ≤ as "inférieur à" and < as "strictement inférieur à." This convention isn't universal, but it is quite common. "Positif" and "Strictement positif" are used similarly. Joeldl (talk) 20:51, 25 January 2009 (UTC)

If you find that ≤ and ≥ take longer to say because you read them as 'is less than or equal to' and 'is greater than or equal to', I have found that a convenient abbreviation for informal conversation (or reading to yourself) is to read them as "leequal" and "greequal" respectively, or if you find such neologisms awkward, perhaps "is at most" or "is at least". Maelin (Talk | Contribs) 06:19, 26 January 2009 (UTC)

Some people have used ]1,3( as the inverse of (1,3], see Interval (mathematics), however ISO in its wisdom says ]1,2[ is what I usually write as (1,2) which would cause problems with that. Dmcq (talk) 14:53, 25 January 2009 (UTC)