Wikipedia:Reference desk/Archives/Mathematics/2007 May 16

= May 16 =

infinity
can infinitey be equal to infinitey —The preceding unsigned comment was added by Sivad4991 (talk • contribs) 00:10, 16 May 2007 (UTC).


 * Inasmuch as infinity is some uniquely defined mathematical object, it is (like any other such object) equal to itself. As you can read in the article, there are actually many kinds of infinity in mathematics, such as the cardinal number ℵ0. Indeed, ℵ0 = ℵ0. Another infinity is c, the cardinality of the continuum. This is different: c ≠ ℵ0. --Lambiam Talk  01:04, 16 May 2007 (UTC)


 * (edit conflict) Yes, and they can be unequal, too. More correctly, there is no single number "infinity", as I hint at in the answer to your next question. There are many "numbers" (note that they are not real numbers, but usually part of some extension of the set of real numbers) that posses infinite properties, and what you mean when talking about infinity differs depending on the circumstances. For example, if you are talking about the $$infinity$$ in an infinite sum or limit such as
 * $$\lim_{x\to\infty}\frac{1}{x}=0$$
 * we're talking about a convenient notation to represent a definition that involves lots of upside-down As and back-to-front Es, without technically invoking any properties of "infinity". On the other hand, if you talk about the number of integers being infinite, you're talking about the cardinality of a set, and you use the symbol known as aleph null, which is the starting point for a whole bunch of big numbers. There are many more infinities, and it takes some work to formally define them, and define what it means for one of them to "equal" another (in at least one case, we even know that there are two infinities that may or may not be equal, but that which we can never prove one way or the other - see continuum hypothesis). Confusing Manifestation 01:07, 16 May 2007 (UTC)


 * Actually, Confusing, that's a somewhat, um, confusing way to put it. What we actually know is that certain specific axiomatizations of set theory, such as ZFC, can never prove it one way or the other. We don't know that it can't be settled by other axioms that can intuitively be seen to be true of the objects of discourse of set theory (the von Neumann universe). --Trovatore 01:16, 16 May 2007 (UTC)


 * Fair enough. It seemed to me that the poster probably wasn't at a point where they were likely to follow the finer points of set theory like that, so I was trying to keep it simple. Unfortunately, with infinity it's hard to keep things simple. Confusing Manifestation 02:14, 16 May 2007 (UTC)


 * (The correct English spelling is "infinity", dropping the "e" of "infinite" before adding the "y".) Mathematicians found a clever way to "count" sets even if the sets are not finite. If we are given a basket of apples and a basket of oranges, we can try to match each apple with an orange; if we succeed the numbers are equal, and if not we have shown that we have a greater number of one kind of fruit. Just so, we can reason in more abstract settings, though perhaps with surprising conclusions. We learn in our earliest years about the natural numbers: 1, 2, 3, and so on. We know there is not a greatest natural number, for if n is proposed we can simply increment it by 1, forcing a contradiction. But we can use the "match game" to show that there are exactly as many even natural numbers (multiples of two) as odd and even together. These are two equal infinities — in this approach (which we call cardinality). But we also have other infinities, and other approaches. It can be fun to explore, but most of the territory requires mathematical sophistication.
 * Another version of infinity is popular in geometry. We give points on a line using a coordinate, x, which measures distance from a fixed origin point. But sometimes we find it preferable to use ratios instead. Thus we take (x : w) to mean the point x&frasl;w, so long as w is not zero. All our original points can be written (x : 1); but now we have a new point, (1 : 0), which is a kind of geometric infinity. Curiously, if we negate this ratio, obtaining (−1 : 0), we have the same point. Thus, in this model, +&infin; = −&infin;, and our "real line" has become strangely like a circle! We can extend this same idea to points in a plane, with homogeneous coordinates (x : y : w). Now, instead of a single point at infinity, we have a line at infinity, (x : y : 0). In this model, the point at infinity along the x axis, (1 : 0 : 0), is different from the point at infinity along the y axis, (0 : 1 : 0). These infinities have proved themselves to be extraordinarily useful in practical computer graphics; they are an essential part of the design of the graphics card used in your computer. --KSmrqT 05:03, 16 May 2007 (UTC)

black hole
Ive heard that black holes are objects that have an infinite mass but is it even possible for an abject to have an infinite mass. And my teacher said that negative nubers arent number that arent shown in nacher is that true for infinate to. thanks --Sivad4991 00:22, 16 May 2007 (UTC)


 * A black hole doesn't have infinite mass, but it does have a theoretically infinite density at its centre - note that the word infinity is used in several different forms in mathematics and science, and in this case it's referring to the "kind of infinite" you get when you divide a non-zero number by zero, also known as a singularity, although in the case of black holes it's more a case of "inside the event horizon we can't really see what's going on, but mathematically we can describe things outside the event horizon as though there were a singularity at the centre, so that's what we do".
 * For the same reason, you could argue for there being no numbers at all in nature, just human mathematics using them to describe what nature does. The argument is strongest for "infinity", and to a certain extent for negative numbers - but there are things that seem to need negative numbers, such as electric charge, although the choice of which charge is negative and which is positive was fairly arbitrary (and probably would have been better the other way around for some applications). Confusing Manifestation 00:58, 16 May 2007 (UTC)


 * Yes, black holes have finite mass, although a singularity is believed to exist at the center than defines a 'hole' in spacetime, at least theories tell us so. Nothing can have infinite mass, unless the universe is infinite (disputed) and you regard our universe as an 'object'. Negative numbers and infinities are both concepts, and their existence is often manifested within nature. Negative numbers can be used for quantum numbers, or just to mean the opposite direction (e.g. move -5m left, i.e., move 5m right). Infinity can exist too. The size of our universe may be infinite. The conductance, for example, of a superconductor is infinite. -- Freiddie 19:15, 17 May 2007 (UTC)

Interesting math topic for a presentation
I'm in high school and I have to give a math presentation very soon. Thus, I want to know of some interesting math topics. My presentation can be on any mathematical topic. An example of an interesting topic would be Pascal's triangle. —The preceding unsigned comment was added by Metroman (talk • contribs) 02:28, 16 May 2007 (UTC).

For the love of everything self-referential, please use Interesting number paradox.--Kirby♥time 02:48, 16 May 2007 (UTC)

"Interesting" is a very subjective word - just about anything in maths could be the subject of an "interesting" presentation, if well-presented and pitched at the right audience. That said, for a high school class many of the really amazing topics are probably a bit too complicated right now (also, given the amount of maths taught in high school there would be a big difference between what you could present to a class in the first year of high school and in the last year). If you want something that can be examined at many different levels (pun not intended) and allows for some really pretty pictures, consider fractals. Confusing Manifestation 03:50, 16 May 2007 (UTC)


 * Well, there are plenty of topics to choose from. What math class are you in? Algebra? Pre-cal? Cal? It'd help to know. For a very basic presentation, you could do a report on how .999... = 1. It would be interesting, especially considering most people don't believe that it's true until you prove it to them mathematically. N3rday 04:24, 16 May 2007 (UTC)
 * Even after you prove it to them, half of 'em don't believe it. (A second proof will convince half of the remaining skeptics, etc., etc.)  You could also teach 'em how to play tic-tac-toe in an arbitrary number of dimensions.  That's pretty fun. -GTBacchus(talk) 05:16, 16 May 2007 (UTC)


 * Perhaps some very basic topology? Everyone likes working with clay and turning donuts into coffee mugs. A bit harder conceptually, but also fascinating is cardinal number: most people don't recognize that there are different sizes of infinity.  nadav 04:45, 16 May 2007 (UTC)


 * I prefer ordinals to cardinals. Or, based on their article, the p-adics look pretty nifty. Like doing arithmetic in a warped mirror. In a totally different direction, how about that sand-on-a-vibrating-plate thing? Waves can be fun. (I'd recommend showing a video from YouTube, rather than an actual vibrating plate.) Combinatorial game theory is cool too, especially if you could demonstrate the solution to a game all your classmates have played before. Black Carrot 05:27, 16 May 2007 (UTC)


 * If you think Pascal's triangle is interesting, then one approach is to see where it can take you. For example, it describes the coefficients for expansions of (a+b)n for natural numbers, n, according to the binomial theorem. So a natural pursuit is that theorem (due to Isaac Newton), its use in differential calculus, or its extension to more general exponents. Or you could explore the combinatorial connection, such as why a binomial coefficient is also pronounced "n choose k" — written $$\tbinom{n}{k}$$, or why the numbers along a row sum to 2n, or why a number on one row is the sum of two numbers on the row above it, or various other patterns. Another connection is with Bézier curves, and the Bernstein polynomials used to construct them. Specifically, the four polynomials for a cubic Bézier curve are
 * $$ \tbinom30 (1-t)^3, \quad \tbinom31 (1-t)^2 t, \quad \tbinom32 (1-t) t^2, \quad \tbinom33 t^3, $$
 * and these curves are the most heavily used in all of computer graphics (except for line segments, I suppose), such as in drawing the shapes of letters in a PostScript font.
 * It is hard for us to say what you and your classmates will find interesting, and difficult to recommend topics without knowing your resources and time constraints and mathematical background. If you know how to search the Web, you may find that an excellent way to "free associate" to find ideas. --KSmrqT 06:03, 16 May 2007 (UTC)
 * PS: More ideas can be found at The Math Forum. --KSmrqT 11:25, 17 May 2007 (UTC)


 * By any chance should it be $$\tbinom32 (1-t) t^2$$ above?
 * Anyway, perhaps your classmates will be interested in Platonic solids, especially if they play D&D :) -- Meni Rosenfeld (talk) 17:38, 16 May 2007 (UTC)
 * (Good catch; fixed. --KSmrqT 21:01, 16 May 2007 (UTC))


 * Minimal surfaces can be modelled by soap films. Chuck 18:06, 16 May 2007 (UTC)


 * What about the Monty Hall Problem? →Ollie (talk • contribs) 18:17, 16 May 2007 (UTC)


 * Don't know how long your presentation needs to be, but if it's short, the proof that the square root of 2 is irrational is surprisingly simple to understand. Chuck 18:28, 16 May 2007 (UTC)
 * Use local links when possible. Irrationality of √2 is also described in Wikipedia — see Square root of 2 page, section Proof of irrationality. --CiaPan 07:54, 17 May 2007 (UTC)


 * Nomograms is a cool subject which doesn't have to be too difficult for the audience to understand. In case you aren't familiar with them, they are basically papers with specially crafted scales on them that allows you to make calculations by drawing lines and such. I'd say they went mostly out of fashion when we got pocket calculators. The Smith chart (which is useful to this day) is my favourite, but more complicated than the others and prehaps only interesting if you are already familiar with transmission lines, for which it is designed. A related subject is analogue computers. Finally, I suggest map projections, which would teach your audience something that could be useful for everyone to know about. —Bromskloss 19:27, 16 May 2007 (UTC)


 * A slide rule is a form of nomogram, and predated the calculator.


 * Prime numbers could be tackled on many levels. e.g. there is no greatest one, use in public-key cryptography.86.132.166.226 10:52, 17 May 2007 (UTC)
 * And if you go in the right direction fast enough, you reach the p-adic numbers, which, as mentioned above, are very cool. 3 + 9 + 27 + 81 + 243 + ... = -1/2. Algebraist 14:56, 17 May 2007 (UTC)


 * I think graph theory can excite even the most mathematically unenthusiastic. Show those "draw this figure without lifting your pen" puzzles, and lead into a discussion of eulerian circuits. Or maybe a simple proof showing there are only 5 Platonic solids using an argument by symmetry and Euler's formula. iames 17:14, 17 May 2007 (UTC)

Thanks for all the suggestions. I ended up doing a presentation on the birthday paradox. Metroman 00:51, 19 May 2007 (UTC)

negative numbers
what are some ease (or special tricks you mite say) ways divide negative numbers --Sivad4991 20:49, 16 May 2007 (UTC)


 * ...dividing negative real numbers is no different than dividing positive real numbers - just make sure you keep on top of your positive and negative signs - two negatives make a positive, and all that. Does this question have a depth I'm not seeing? Icthyos 21:18, 16 May 2007 (UTC)

Ignore the signs when you do the division, then use this chart to figure out the sign of the result:

Starting numbers        Result ======================= ======== Both positive            Positive Both negative           Positive 1 positive, 1 negative  Negative

Or, another way to put that is:

Starting signs          Result ======================= ======== Both the same            Positive Each different          Negative

StuRat 21:49, 16 May 2007 (UTC)


 * What if one of them was different but the other one wasn't? :)  JackofOz 08:37, 17 May 2007 (UTC)


 * In other words, $$(\mathbb{R}\setminus\{0\},\cdot) \cong (\mathbb{R}^{+},\cdot)\times\{+,-\}$$. Sorry, couldn't resist it. Yes I know my soul is now lost to category theory. Algebraist 15:53, 17 May 2007 (UTC)


 * What happened to zero? --Lambiam Talk  16:19, 17 May 2007 (UTC)
 * 0 doesn't count (hastily edits) Algebraist 16:31, 17 May 2007 (UTC)
 * As in: 1 counts, 0 doesn't count, and −1 counts backwards. --Lambiam Talk  16:39, 17 May 2007 (UTC)