Wikipedia:Reference desk/Archives/Mathematics/2008 October 4

= October 4 =

request for solution for maths question
Upon solving the system of equation x15y17=r and x2y7=s for x and y, the answers always turns out to be of the form x=r^a/s^b and y=s^c/r^d where a, b, c and d are positive integers. Calculate the sum of a + b + e + d.

Ans: 31 Invisiblebug590 (talk) 10:24, 4 October 2008 (UTC)


 * Assuming you mean:
 * $$x^{15}y^{17}=r$$
 * $$x^2y^7=s$$
 * $$x=r^as^{-b}$$
 * $$y=s^cr^{-d}$$
 * then I get the following simultaneous equations:
 * $$15a-17d=1$$
 * $$17c-15b=0$$
 * $$2a-7d=0$$
 * $$7c-2b=1$$
 * which has a unique solution, but not one where a, b, c and d are integers, neither is their sum 31. Gandalf61 (talk) 11:00, 4 October 2008 (UTC)

maths question
the 2 squares in the diagram have side length 1cm, find the area of the slanted rectangle ABCD.


 * Sounds like homework which is a no no here. But as a hint your teacher was probably recently moving triangles around or else was doing similar triangles or else was doing the Pythagorean theorem. Dmcq (talk) 12:01, 4 October 2008 (UTC)


 * You'll find that the little right-angled triangle at the right is similar to the one whose hypotenuse is the diagonal of the original (two-square) rectangle, which should enable you to find the length of the sides of the slanted rectangle.…81.132.235.170 (talk) 14:24, 4 October 2008 (UTC)

There's a quicker way: (1) snip off the triangle with vertices B and C and the third point on the vertical line downward from B, and paste it on the left edge of the figure, getting a triangle below the original rectangle having one vertical side on the left and one horizontal side along the bottom of the original rectangle. (2) Now take the latter-mentioned triangle and turn it over and place it on top of the triangle that is the part of the original rectangle that is above the depicted diagonal. You see that the new rectangle has the SAME area as the original rectangle. Michael Hardy (talk) 21:57, 4 October 2008 (UTC)


 * ...or yet another way: Consider the triangle where the two rectangles overlap. The area of that triangle is exactly half of the area of either of the two rectangles; therefore the two rectangles have equal areas. Michael Hardy (talk) 23:12, 4 October 2008 (UTC)


 * You can add more and more rectangles going round in a spiral like in the Spiral of Theodorus which approximates the Archimedean spiral. Any idea what this one would approximate or does it stop going round? Just daydreaming, thought that was better than doing homework. Dmcq (talk) 09:38, 5 October 2008 (UTC)


 * A nice problem. Translate the geometry into algebra. Take the rectangle as unit of area. Represent the sides by complex numbers, a and b. The direction of a is â and the length of a is |a|, so a = â&middot;|a|. The direction of b is iâ and the length is |a|&minus;1, so b = iâ&middot;|a|&minus;1 = ia&middot;|a|&minus;2 = ia&middot;(aa*)&minus;1 = i&middot;(a*)&minus;1 where a* is the complex conjugate of a. The proces of replacing a rectangle by the new one is a &rarr; a+b = a+ i(a*)&minus;1. A numeric experiment shows that the first turn is after replacement number 28, the second turn is after replacement number 96, and the third turn is after replacement number 203. It doesn't look like if it stops turning, but it is too early to make a conclusion. Bo Jacoby (talk) 22:16, 5 October 2008 (UTC).
 * Continuing. If an =10, then an+1 = an+  i(an*)&minus;1 = 10+0.1i = 10&middot;e0.00005+i&middot;0.01. One turn is made after about 2&pi;/0.01 = 628 steps and an+628 = 10&middot;(e0.00005+i&middot;0.01)628 = 10.32. So the spiral is much more tightly wound than the Archimedian spiral. I doubt if |an| &rarr; ∞. Bo Jacoby (talk) 12:06, 7 October 2008 (UTC).


 * I thought it would it would go to infinity since there seems no reason for it stop at any value - the rectangle can get thinner and thinner. I must admit I'm surprised the spiral gets tighter and tighter so quickly, perhaps it's going down like the sum of 1/n which goes to infinity very slowly. It must be close to being smooth after a few turns so it could be approximated by a differential equation - or even two to bound it the way Archimedes did to find strict limits for pi. Dmcq (talk) 09:12, 8 October 2008 (UTC)
 * If a = |a|&middot;ei&middot;v, then the next point is a+ i&middot;a*&minus;1 = |a|&middot;ei&middot;v + i&middot;(|a|&middot;ei&middot;v)*&minus;1 = |a|&middot;ei&middot;v + i&middot;(|a|&middot;e&minus;i&middot;v)&minus;1 = |a|&middot;ei&middot;v + i&middot;|a|&minus;1&middot;ei&middot;v = |a|&middot;(1+ i&middot;|a|&minus;2)&middot;ei&middot;v = |a|&middot;(1+ |a|&minus;4)1/2&middot;ei&middot;(v+arctan ≈ |a|&middot;(1+|a|&minus;4/2)&middot;ei&middot;(v+) ≈ |a|&middot;eundefined/2+i&middot;(v+|a|&minus;2) . After k steps further we reach b ≈ |a|&middot;ek&middot;/2+i&middot;(v+k&middot;|a|&minus;2) . One turn is made when k&middot;|a|&minus;2 = 2&pi;. So k = 2&pi;&middot;|a|2, and b ≈ |a|&middot;e2&pi;&middot;&middot;|a|&minus;4/2+i&middot;(v+2&pi;&middot;|a|2&middot;|a|&minus;2) = |a|&middot;e&pi;&middot;+i&middot;v  = a&middot;e&pi;&middot; ≈ a&middot;(1+&pi;&middot;|a|&minus;2). The distance to the next winding of the spiral is |a|&middot;(1+&pi;&middot;|a|&minus;2)&minus;|a| = &pi;&middot;|a|&minus;1. After no more than (|a|+1)/&pi; turns the radius has exceeded |a|+1. So Dmcq is right: the radius of the spiral is unlimited. Bo Jacoby (talk) 21:43, 8 October 2008 (UTC).


 * Gosh, well, glad to have provided you something to think about. Looks like a very inefficient way of computing &pi; ;) Dmcq (talk) 22:54, 8 October 2008 (UTC)

The logics of significance and insignificance
So my question is not as interesting as the headline might suggest, but here goes:

Is it correct to say that the statements "X is not insignificant" and "x is significant" are not logically equivalent? And would it be correct to say that this is so because the opposite of "significant" is "not significant", and not "insignificant"? Also: is there a way to show this using predicate logic? (Don't worry: this is not a homework, I just had a discussion about it today with a lawyer-friend). —Preceding unsigned comment added by 83.250.105.19 (talk) 17:58, 4 October 2008 (UTC)


 * Yes, I agree with your conclusion ... and I also agree with your reasoning behind it. Perhaps a more intuitive example helps us to understand the question you have raised.  Statement A ("The odor is pleasant.") means something different than Statement B ("The odor is not unpleasant.").  I think that the main distinction is that there can exist some middle neutral ground ... a continuum somewhat along the lines of: unpleasant odor -> neutral odor (or non-odorous at all, even) -> pleasant odor.  Thus, an odor that is "pleasant" is pleasant.  However, an odor that is "not unpleasant" may either be (a) pleasant ... or ... (b) neutral.  If an odor is "not unpleasant", that does not necessarily mean that it must belong to category "a" (pleasant odors), since it can also belong to category "b" (neutral odors).  And items in category "b" are items different than those called "pleasant odors" (namely, they are neutral odors).  Thanks.   (Joseph A. Spadaro (talk) 18:15, 4 October 2008 (UTC))


 * It's just a matter of definitions. If you strictly define insignificant as "not significant", then it's trivial. If it's not binary, however -- if there's a neutral choice, neither significant nor insignificant -- then it's a different issue. --jpgordon&#8711;&#8710;&#8711;&#8710; 18:06, 4 October 2008 (UTC)
 * Not insignificant is an example of a rhetorical figure known as litotes. Ask about it on the language desk. --Trovatore (talk) 18:16, 4 October 2008 (UTC)
 * I don't see what the neutral choice between significance and insignificance could be... --Tango (talk) 19:12, 4 October 2008 (UTC)
 * What if "significant" means "known to be significant" and "insignificant" means "known to be not significant" etc? —Preceding unsigned comment added by 65.110.174.74 (talk) 21:09, 4 October 2008 (UTC)
 * That's a bit philosophical for me... "If the probability of falsely rejecting the null hypothesis is less than 5% but nobody knows it, is it significant?" (cf. If a tree falls in a forest). --Tango (talk) 21:16, 4 October 2008 (UTC)


 * I took the original question to be generic for any adjective ... not specific to the adjective "significant". I assumed that the original poster was asking whether or not the adjectives "X" and "not un-X" (or "not in-X") are logically equivalent ... regardless of which adjective "X" is considered.  Granted, some adjectives are more prone to having a neutral intermediate than others.   (Joseph A. Spadaro (talk) 23:55, 4 October 2008 (UTC))
 * Yes, that was exactly my question. Thanks a lot for everybody's input. —Preceding unsigned comment added by 83.250.105.19 (talk) 07:24, 5 October 2008 (UTC)


 * Don't know if you're interested in this side of it, but a fair bit of research has been done in logical systems with more than two truth values. One of the results of this research is fuzzy logic. Black Carrot (talk) 07:27, 5 October 2008 (UTC)


 * Incidentally, I'm not sure I'd say that the "unpleasant", "not unpleasant", "pleasant", and "not pleasant" lie on any kind of a continuum. Unpleasantness describes the existence of something disagreeable. Pleasantness describes the existence of something agreeable. Each lies on a continuum of its own, from "very X" to "not very X". I would say that "not significant" and "insignificant", however, are describing different parts of the same range from "very significant" to "not very significant". So that, for instance, in designing a fuzzy logic description these three would each have a rating from 0 to 1 attached to them. Black Carrot (talk) 07:51, 5 October 2008 (UTC)

Definite matrix
I am reading a paper on the work of Wilhelm Cauer which uses the phrase "positive definite matrices" in regard to the n x n matrices L, R and D in this equation,


 * $$\boldsymbol{A}=\lambda^2 \boldsymbol{L} + \lambda \boldsymbol{R} + \boldsymbol{D} \,\!$$

I get the "positive" bit (I think) but am not so sure about "definite". The paper is written in English by a German so there may possibly be a language issue. I thought at first it just means "positive real" but later on we have -

"A key problem is the simultaneous principle axis transformation of two quadratic semi-definite forms (as opposed to the standard case where at least one form is non-degenerate)."

Which implies that I actually haven't understood a thing and "definite" means something else. Anyone know what?  Sp in ni ng  Spark  18:29, 4 October 2008 (UTC)
 * You can't separate "positive" from "definite"; it's a single term, positive definite. See positive-definite matrix. --Trovatore (talk) 18:43, 4 October 2008 (UTC)


 * Ta very much, quick answer, slow to look at it.  Sp in ni ng  Spark  21:36, 4 October 2008 (UTC)

"Undiscovered integer"?
In one of his books(I think The Demon-Haunted World) Carl Sagan suggests the thought experiment: "What if there is an undiscovered integer between six and seven?"(followed up by a question about the possibility of an undiscovered chemical element whose atomic number is that of the new integer). Does the idea of an undiscovered integer have any sort of meaning at all, or is this more likely intended to suggest, like many paradoxical thought experiments, that our grip on reality may not be as good as we think? 69.107.248.106 (talk) 20:52, 4 October 2008 (UTC)


 * Seven is usually defined to be the next integer after six (see Peano axioms), so in that sense there can't be an integer inbetween. However, there is a difference between the mathematical concept of integers and the real life concept of collections of discrete objects. We model those discrete objects as integers, but that model could conceivably be flawed (well, I can't conceive how that would work myself, but the point stands that mathematical models are not absolute in the same way that mathematics is [if you ignore Gödel, anyway...]). --Tango (talk) 21:13, 4 October 2008 (UTC)


 * You mean like the extra hours in a day that only managers know about? :) Dmcq (talk) 21:17, 4 October 2008 (UTC)
 * More seriously for people with Dyscalculia often really wouldn't notice if a few numbers were missing. In fact I think most probably all people have blindnesses of one sort or another. Some people just can't give a greater weight to something that's bigger or see that a lot of small bits can add to a large amount. Probabilities are a complete haze. Ask what shape you get if you stick a corner of a cube into some plasticine and many people will say four. Personally I can see colours perfectly well but the colour of my front door is something I have explicitly learned because it seemed so silly not to know it when asked. Dmcq (talk) 21:40, 4 October 2008 (UTC)


 * Microsoft Supercomputer Discovers New Integer Between 5 and 6 - Fredrik Johansson 22:18, 4 October 2008 (UTC)


 * In the Internal Set Theory approach to non-standard analysis the natural numbers turn out to contain "non-standard" elements, even though the set of naturals is exactly the same as we usually think of it. In some sense these non-standard naturals are undiscovered and undiscoverable - it's not possible for 379274, or any other integer you might think of, to be non-standard, because it is "uniquely defined by a classical formula".  To quote Robert's book Nonstandard Analysis (in Dover, everyone should get one), "These nonstandard integers have a certain charm that prevents us from really grasping them!  Their existence is axiomatic". 163.1.148.158 (talk) 11:03, 5 October 2008 (UTC)


 * In first-order Peano arithmetic, there exist models in which nonstandard elements exist. They have the properties of natural numbers required by the axioms, but they aren't natural numbers (generally, they live 'above' the natural numbers, and succeed each other in various patterns). They're sometimes called alien intruders, because "they live among us". Maelin (Talk | Contribs) 11:36, 5 October 2008 (UTC)
 * According to our article (and my limited understanding), there can't be any nonstandard elements between six and seven—they always come after all the standard natural numbers. Of course, no one ever said you have to believe the Peano axioms in the first place. People have from time to time expressed doubts that a function with the properties ascribed to S actually exists, or that Peano induction is valid. -- BenRG (talk) 12:50, 5 October 2008 (UTC)


 * There certainly isn't a non-standard integer between 6 and 7 --- such a number wouldn't be an integer. I brought up nsa in response to the second part of the question. 163.1.148.158 (talk) 13:49, 5 October 2008 (UTC)


 * Oh, agreed. I didn't mean to suggest that those nonstandard elements could lie between the standard natural numbers. I was just mentioning them out of interest. It would be a pretty crummy arithmetic system indeed if it allowed elements to lie between six and seven. Maelin (Talk | Contribs) 01:16, 6 October 2008 (UTC)


 * What if there's an undiscovered (or forgotten) letter between C and D and nutrition deficient in the vitamin for that letter is the cause of lots of illnesses in humans? &#x2013; b_jonas 13:26, 5 October 2008 (UTC)
 * Then we're all doomed, doomed! --Tango (talk) 16:31, 5 October 2008 (UTC)
 * And since we're not doomed . . . Zain Ebrahim (talk) 13:20, 6 October 2008 (UTC)


 * Black Carrot (talk) 03:22, 6 October 2008 (UTC)


 * If you like this sort of thing you might like reading some of Gregory Chaitin's stuff, like his number that is defined but can't be computed or some theorems being random. I think there's probably some really simple things which are inaccessible to our way of thinking and imagination, perhaps computers will tell us some of them. Dmcq (talk)