Wikipedia:Reference desk/Archives/Mathematics/2007 November 15

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November 15

Limits question

My memory of limits is rusty. My calculator is basically telling me that the limit as $x$ approaches infinity of $3x$/sqrt($x^{2}+4$) = 3, but I don't see why this is. Help!! Thanks, anon. —Preceding unsigned comment added by 71.249.151.43 (talk) 00:48, 15 November 2007 (UTC)

Well, $x/({\sqrt {x^{2}+4}})$ (where x is >0) for all intensive purposes equals 1. So basically you're left with $3*1$, or just 3 -- MacAddct  1984 ^{(talk • contribs)} 01:02, 15 November 2007 (UTC)

And to make that just a fraction clearer, x^2 + 4 grows like x^2 for large x, so sqrt(x^2+4) behaves like sqrt(x^2) = x, so x/sqrt(x^2+4) goes to 1 as a limit for large x. Confusing Manifestation(Say hi!) 01:08, 15 November 2007 (UTC)

Yeah, I think we were taught in high school that when dealing with limits and very large values of x, you can basically treat any instance of +/- a constant to be zero. -- MacAddct  1984 ^{(talk • contribs)} 01:31, 15 November 2007 (UTC)

The key here is the comparison of the powers of x. Since on the bottom x^2 is under a root it's power is effectively halved. So ${\frac {x}{x}}$ is basically what your looking at for arbitrarily large x so as $x\to \infty$ your function w/out the 3 goes to 1, so your get 3. A math-wiki 05:00, 15 November 2007 (UTC)

Take $1={\frac {\left({\tfrac {1}{x}}\right)}{\left({\tfrac {1}{x}}\right)}}$ (which is true for x not equal zero, so is true for $x\to \infty$) and get ${\frac {3x}{\sqrt {x^{2}+4}}}\cdot 1={\frac {3\cdot {\tfrac {x}{x}}}{\left({\tfrac {\sqrt {x^{2}+4}}{x}}\right)}}={\frac {3}{\sqrt {\tfrac {x^{2}+4}{x^{2}}}}}={\frac {3}{\sqrt {1+{\tfrac {4}{x^{2}}}}}}$. Can you see the limit now? --CiaPan 07:38, 15 November 2007 (UTC)

Note that all the "lower powers are insignificant when $x\to \infty$"-type solutions are good rules of thumb if you know what you're doing, but in some cases can lead to incorrect results if you're not careful (e.g., $\lim _{x\to \infty }(x({\sqrt {x^{2}+4}}-x))$). If you really want to be sure you have it right, you need to do a calculation along the lines suggested by CiaPan. -- Meni Rosenfeld (talk) 09:28, 15 November 2007 (UTC)

Here is another approach: under the assumption that L is nonnegative, and with all limits for x going to +infinity, lim 3x/(x²+4)^1/2 = L ⇔ lim 9x²/(x²+4) = L² ⇔ lim (x²+4)/x² = 9/L² ⇔ lim 1 + 4/x² = 9/L². Since the last limit clearly equals 1, we find 9/L² = 1, or L = 3.  --Lambiam 09:53, 15 November 2007 (UTC)

What's next?

Since I have a lot of free time right now and I'm not in school, I was wondering what mathematics courses would follow AP Calculus, AP Statistics, High School Discrete Math, and High School Algebra 2. So I might be able to do more self study before I go to College. Thanks in advance! A math-wiki 05:06, 15 November 2007 (UTC)

Linear algebra and diffEQs, from there on it is really you choice what you want to take. —Cronholm¹⁴⁴ 05:15, 15 November 2007 (UTC)

There are literally hundreds of books out there you could use for self study; I am sure that some of the regulars here can recommend some of the better ones.—Cronholm¹⁴⁴ 05:19, 15 November 2007 (UTC)

I often suggest those who haven't done so yet take an excursion into number theory. (I was surprised, when I first studied it in college, that it was often in the past taught in high school.) --jpgordon^∇∆∇∆ 07:10, 15 November 2007 (UTC)

I don't know about "courses", but I can suggest subjects. Mathematical logic, at least the basics of it, can work wonders for one's understanding of what it is exactly we are doing when we prove theorems etc. Set theory (at least to the level of understanding cardinality well) is used extensively in any sort of serious mathematics. Some abstract algebra (groups, rings, etc.) would be good if it was not covered already. I suspect that you have been taught calculus at only an intuitive level (without δ-ε definitions of limits) - if so, you should try learning the rigorous foundations. If you were only taught vector calculus at a basic level, extending your knowledge would be helpful. Some graph theory can also be useful. There are some fairly fundamental subjects which are considered more advanced, such as general topology and complex analysis.

Regarding some of the topics suggested by others - Linear algebra is extremely important, though you may be able to survive for a while with only a basic knowledge of ODEs and number theory.

It will also be very helpful to read some books about recreational math, history of mathematics and the like. I especially recommend books by William Dunham (not the songwriter).

You should also consider branching out to subjects which do not fall directly within the realm of mathematics, such as physics, chemistry, computer science (mostly the basics of complexity theory, and complexity analysis of some famous algorithms), and programming.

I think these should get you well on the way to studying mathematics at an academic level. I guess for most of these there will also be courses in college, so you don't have to study absolutely all of them right now. -- Meni Rosenfeld (talk) 10:01, 15 November 2007 (UTC)

I absolutely agree with Cronholm. The "natural" progression is linear algebra and ordinary differential equations. If you want to do some studying on your own, the absolute best and my favorite text book is the "Advanced Engineering Mathematics" by Erwin Kreyszig. I love that book. It is very well written, easy to read, with plenty of examples, illustrations, and exercises. This book is a perfect transition to upper mathematics. It includes all sorts of topic (including linear algebra and differential equations) all in one. And since he wrote it as a perfect reference book also, it reviews everything else that you have supposed to have learned before also (in case you forget). A Real Kaiser 16:30, 15 November 2007 (UTC)

I think we're getting a bit ahead of ourselves. There's a lot of mathematics before you, but there's also plenty of time to get through it as well... From my experience teaching high school and college mathematics, here's what I'd say. Make sure that you are rock solid with algebra (look for a text with the words "College Algebra" in the title and learn all of that). Know your trig identities cold so that you can do trig substitutions for integrals in your sleep. You don't indicate which AP Calc you took. Was it AB (focusing on derivatives) or BC (adding in integrals and series). Make sure that you know your series cold. It's possible to get a 4 on the BC calc exam without knowing series anywhere as much as you need to do. I'm guessing that you're looking at going into science/math/engineering in college so that influences things. The very next step that you'll be taking in college will be either Calc II (integrals and series) if your 5 on the AP Calc test as on the AB test or Calc III (multi-variable calculus) if your 5 was on the BC test.

I would recommend as a good point for progressing, beyond making sure that your algebra and trig are rock solid, would be to get What is Mathematics? by Courant and Robbins. This is a good overview of the world of mathematics and a improvement of your fundamentals. It will leave you better prepared for college mathematics than virtually all of your peers. If you're interested in number theory, I would suggest coming at it from Oystein Ore's Number Theory and its History which again will force you to be rock-solid on your algebra while giving you a good overview of problems in number theory. As an added bonus, you could get the pair of books for under 30 bucks and have more than enough to keep you occupied for the next few months depending on what else you're doing until next fall. Donald Hosek 17:29, 15 November 2007 (UTC)

Thanks for all the advice. To help narrow the field, I am completely comfortable with formalized proofs and have written my own before. I have a thorough understanding of all Algebra 1 and 2 topics and some more advanced ones like the Cubic and Quartic Formula. I have been introduced to graph theory (in Discrete Math). I am very comfortable with Derivatives and Integrals, I took the AB test and got a 5 quite easily (BC level Calculus was not offered at my school). I did selfstudy most of the latter chapters we didn't cover in Calculus, I am familiar with Double and Triple Integrals though my understanding of them is still rather vague. We also did use the delta-epsilon definitions of a limit quite a bit. A math-wiki 22:41, 15 November 2007 (UTC)

Given that, and keeping my assumption about your educational goals, I would say that you want to really learn integration and series. You might find a BC AP Calculus prep book to be helpful, and also a GRE Math (Subject) prep book (I recommend the Princeton book). It's worth noting that most of the GRE Math test is Calculus, with little depth in other subject areas so you can actually get a fair amount profitably from the book. I also would still recommend the Courant and Robbins book I mentioned above as a good overview of mathematical study and a way to fill in any gaps you might have. Donald Hosek 00:48, 16 November 2007 (UTC)

And just to add a bit on that: I took AB Calc well before you were born, but then took the BC test. I did well enough (4), to get placed into Calc III, but as a consequence, my knowledge of series was a bit weak, and I still could use some improvement in some of my subtler integration skills. I wouldn't mind getting a chance to teach Calc II at some point in the future to really learn it. Donald Hosek 00:56, 16 November 2007 (UTC)

I must say I am amazed at how much you have been taught before ever having been to college - it's quite different from what we have around here (some of these I've only studied during my second academic year).

Mentioning What is mathematics has reminded me of this, which could be a good read for you, and I maintain my expressed dissatisfaction with What is mathematics and Stewart's revision, as well as my recommendation of Dunham's The Mathematical Universe. -- Meni Rosenfeld (talk) 13:03, 16 November 2007 (UTC)

There are quite a few electives in many subject areas at my high school, so most of the math classes there are electives, and furthermore I was already a year ahead in math when I began attending high school I also have done a number of self studies of particular problems and such ever since my Sophomore year when I noticed my affinity and liking for math. A math-wiki (talk) 09:56, 17 November 2007 (UTC)

Reducing a set (matrix?) to all elements being 0 or 1

I have a vague recollection of this from my college linear algebra class, but can't remember what it is called. I write software now, and have had a few occasions recently where this idea would have made a good function name. This is more of a request for a reference pointer than a how to. Thanks Nihmrat 16:23, 15 November 2007 (UTC)

It's called Gaussian elimination officially I think, but I call it Row reduction, which redirects to the same page. Gscshoyru 16:28, 15 November 2007 (UTC)

I believe you mean Reduced row echelon form.A Real Kaiser 16:29, 15 November 2007 (UTC)

Reduced row echelon form is what I was thinking of, and the related Gaussian elimination seems to be a close corollary to the process I am writing, and therefore a suitable name for the method in my code. Thanks. Nihmrat 16:44, 15 November 2007 (UTC)

set theory

If 46% student like physics and 58% like maths than how much like both, and how much do not like both subject 123.236.61.117 18:21, 15 November 2007 (UTC)

You should do your own homework. In any event, you don't have enough information to answer the question, although you can set minimum and maximum bounds for these. Suppose the numbers were 40% and 70%. Then you could think of it as 100 people with, say people 1-40 liking physics and people 31-100 liking math. That gives no people hating both (0%), and people 31-40 liking both (10%). That's one extreme where we minimize the overlap. We can also maximize the overlap so that it's people 1-70 liking math. Now people 1-40 like both (40%), while people 71-100 like neither (30%). Does that give you enough information to do your homework now? Donald Hosek 19:33, 15 November 2007 (UTC)

In other words, see pigeonhole principle. —Tamfang (talk) 01:00, 22 November 2007 (UTC)

lines

What's the difference of a secant line and a tan line? Please? —Preceding unsigned comment added by 24.76.248.193 (talk) 22:52, 15 November 2007 (UTC)

A secant joins two points on a curve, a tangent touches just one. If you start with a secant, and move the two points closer together, the secant will become a tangent. Confusing Manifestation(Say hi!) 22:59, 15 November 2007 (UTC)

The importance of this compairson is that it lead to a solution for the Tangent Line problem via Limits, which in turn results in the Definition of the Derivative (for f(x)). A math-wiki 23:06, 15 November 2007 (UTC)

Oh, okay. Both their equations look practically the same. —Preceding unsigned comment added by 24.76.248.193 (talk) 23:08, 15 November 2007 (UTC)

They are very similar, the distinction is that in defining the Secant line of a partular curve THROUGH $(x_{1},f(x_{1}))$ and $(x_{2},f(x_{2}))$ we can say that $x_{1}<x_{2}$. With the Tangent Line, we take the limit as $x_{1}$ approaches $x_{2}$ or in symbols $\lim _{x_{1}\rightarrow x_{2}}$ and that Tangent Line has the same slope as f(x) AT the point $(x_{2},f(x_{2}))$. A math-wiki 23:20, 15 November 2007 (UTC)

Rubik's Cubes

I've been trying to apply some of my mathematical knowledge to finding sequences that do specific things to the Rubiks Cubes, but I'm running into difficulty finding a good system of notation for the individual Cubelets themselves. The problems seems to be that everything is relative with Rubik's Cubes. For example there are 24 different ways to orient the Cube but orienting the Cube doesn't change it current position. And in having solved the 2x2x2, 3x3x3, and a few times the 4x4x4 I have, I have noticed that you can apply sequences in many different orientations as well. I need a system of notation that can accurately describe the changes that a particular move or sequence will have on the cube, in such a manner as to allow easy comparison across many different orientations. Thanks A math-wiki 23:00, 15 November 2007 (UTC)

Does Rubik's Cube group help?  --Lambiam 23:41, 15 November 2007 (UTC)

Not really, I'm not familiar with groups, rings, etc. I was looking for a more naive approach to it. Namely notation that would support such a more naive approach. A math-wiki 00:45, 16 November 2007 (UTC)

To give some more information, I am attempting to create a way to keep track of indivicual Cubelets so that I can 'write' 'equations' for the translations that certain sequences do, and then use these equations to derive more of them, much like the formation of Trigonometric Identities. A math-wiki 01:08, 16 November 2007 (UTC)

My sense is that Wikipedia's style is generally not that helpful for learning a lot of mathematical topics. What you're looking for to do this is a permutation group. Joseph Gallien's textbook Contemporary Abstract Algebra actually treats the Rubik's cube case directly, and you should be able to work at least through that section without any more background than you have already. It is, however, a painfully expensive book, especially for your purposes. Donald Hosek 01:12, 16 November 2007 (UTC)

Also consider David Joyner's Adventures in Group Theory which more heavily emphasizes using permutation groups in Rubik's cube like puzzles. The equations you are looking to write are very likely expressed in books such as this. It is also more likely to be at a nearby library. JackSchmidt 01:38, 16 November 2007 (UTC)

From looking at the Permutation Group article, I definantle don't know enough about it to make any utility for now, other suggestions?? A math-wiki 03:29, 16 November 2007 (UTC)

I suggest starting with simple permutation groups. I would start with S₃ which is the simplest non-trivial case. We have a total of 3! possible permutations of {1, 2, 3}. We can look at cycles of 2 elements which we traditionally write in the form, e.g., (1 2) which means take the first element and move it to the second element and the second and move it to the first. How many possible 2-cycles are there remembering that (2 1) = (1 2)? Note that 1-cycles are identies. How about 3-cycles? How many are there and what do they do? Explore how these cycles combine. Does the commutative law apply for all permutations? How about some subset? Work out the answers and see what you come up with. Donald Hosek 06:17, 16 November 2007 (UTC)

Very interesting (did some investigating) how does one notate 3-cycles, 4-cycles (for $S_{4}$ or larger) etc. A math-wiki 08:00, 16 November 2007 (UTC)

Never mind, found the answer on wiki. :) A math-wiki 10:32, 16 November 2007 (UTC)

If you just want a notation, then the standard notation for 3x3x3 Rubik's cube (as popularised by David Singmaster, I think) is described here. You start by attaching a label to each face of the cube - with 3x3x3 cube this is well defined, as centre piece of each face does not move. Usual face labels are F(ront), B(ack), L(eft), R(ight), U(p) and D(own) - alternatively, you could label each face by the colour of its centre cube, but then you have a notation that is specific to one cube. Then X means rotate face X by a quarter turn clockwise (i.e. clockwise when face X is facing you), X' means rotate face X by a quarter turn anticlockwise, and X2 (or sometimes X²) means rotate face X by a half turn. Sequences of moves are written and read from left to right - so XYZ means move X followed by move Y followed by move Z. Brackets are used to indicate repeated sequences - so (XYZ)2 is short for XYZXYZ. I imagine someone somewhere has come up with a similar but extended notation for 4-cubes and 5-cubes. Gandalf61 11:21, 16 November 2007 (UTC)

For our original poster's sake, it's worth noting that the above notation (I would tend to prefer X³ to X' myself) is a specialized subset of the permutation notation that you're learning above. I think some of the implications of the notation will become apparent as you start exploring permutation groups. -- Donald Hosek (talk) 18:03, 16 November 2007 (UTC)

I am aware of that notation and it is a good way to notate sequences, but what it lacks is the ability to allow me to easily search for new sequences that do specific things, like swap two corners that are on the opposite vertices (diagonally through the Cube) of the Cube for instance. A math-wiki (talk) 01:28, 17 November 2007 (UTC)

The notation must be able to keep track of both position and orientation of each Cubelet in a relative manner. A math-wiki (talk) 01:29, 17 November 2007 (UTC)

This is very much true. What those notations indicate will be sets of 4-cycles in a larger permutation group. <aside>Note that we're talking about products of disjoint 4-cycles for each face rotation (that is, something along the lines of (1 2 3 4)(5 6 7 8)... where there are no overlapping elements in any of the 4-cycles).</aside> You end up with something that's simpler than $S_{\text{whatever the large number here is}}$, but still fairly complex. I've not looked into it in great detail, but I think that compositions of face rotations probably make the simplest way to describe a state of the cube with some effort involved in being able to simplify longer sequences of rotations. This is where a study of simpler permutation groups will be helpful (for example, in S₄, (1 2)(2 3) = (1 2 3). Donald Hosek (talk) 02:05, 17 November 2007 (UTC)

Oh, a P.S., I'd be willing to do a seminar in basic group theory (at wikiversity? I have no idea) with a goal of being able to approach the Rubik's cube problem if anyone is interested. Donald Hosek (talk) 02:06, 17 November 2007 (UTC)

I'd be interested, I don't know if I will be able follow it all, but I could try. Also Upon thinking about it somemore I have concluded that the key may lie in classifying the different types of Cubes. I came to this conclusion when think about large Cubes like 5x5x5, 6x6x6 etc. There are three basic types with 3, 2, and 1, colored sides. They are called in order Corner Cubes, Edge Cubes, and Face Cubes. But there is more that one can distinguish on large cubes. For example on a 5x5x5 You can distinguish between the Center Edge Cubes and The non-Center Edge Cubes (this is because a center edge cube can't ever occupy a position that isn't a Center Cube spot). There are even more subtle relationships too, for example it is always possible on a 4x4x4 to distinguish the "left edge" cube from the "right edge" cube on a PARTICULAR edge since it is impossible to create the translation of "sliding" the cube from left to right or vis versa. A math-wiki (talk) 10:11, 17 November 2007 (UTC)

I suppose what I am really looking for is a intuitive or analytical way of determine if and if so how to solve the Cube a certain way. I created the graph for an $S_{3}$ and drew the edges that represent 2-cycles ((1 2), (1 3), and (2 3)). The graph has 6 vertices (6 possible positions), and all vertices are of degree 3 (3 possible moves (1 2),(1 3), (2 3)). This has given me some ideas about a possible way to determine the minimum number of moves to go from one position two another. It's rather obvious from looking at the $S_{4}$ (which has 24 vertices) that inspection is not an effective technique for analyzing the Cube groups. But perhaps analysis of the Symmetries of those graphs may allow for an easier derivation of the minimum numbers of moves to solve as well as how to do a specific alteration without disturbing any other pieces (or the minimum number of other pieces). A math-wiki (talk) 04:15, 18 November 2007 (UTC)