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January 11[edit]

January 18[edit]

January 19[edit]

You vs. Sir Isaac Newton[edit]

Suppose you had an hour to convince Isaac Newton that you come from the future (or you are a super-genius), and you could only do math (no experiments or other physics). What would you show him? My assumptions:

  • Please don't look things up - just use what you know how to demonstrate from memory.
  • If you don't know what math Newton is likely to know, please don't look it up either - it's part of the challenge to come up with something that would work regardless.
  • Newton probably knew way more stuff than he ever published, so incremental improvements over his work might not be that impressive to him.
  • Newton is not a nice guy and not known to be patient, so you better get to the point quickly or he'll just kick you out. I.e. you better not tell him things that "everyone knows".
  • Newton wouldn't recognize most modern math as math, so I think topics like Abstract Algebra, Topology, etc - are out. Unless you can convince me otherwise.
  • I think most mathematicians of the period thought complex numbers were not actual numbers (not sure about Newton specifically, but Leibniz thought so). If you want to do Complex Analysis, you probably need to have a very strong motivation for it.
  • If you do Calculus-type stuff, it has to be done in the Newton's style, because he hates Leibniz's guts. So no infinitesimals or differentials, just derivatives.
  • Assume you meet Newton after he already published all his work.
  • Update: you are meeting with him on his turf, and you can't bring anything along with you.

I do have my own ideas, but I am curious what you would pick. --Ornil (talk) 04:11, 19 January 2015 (UTC)

I think fractals might be on interest to him, since they describe many things in nature, and were somehow overlooked until quite recently. StuRat (talk) 04:18, 19 January 2015 (UTC)
Were you planning to draw them? Also, would he agree it's math? Ornil (talk) 04:27, 19 January 2015 (UTC)
Sure, you could start by drawing a Koch snowflake, much like our animation, but as a series of steps, then show the math behind it, and point out the similarity to real snowflakes, and many other fractal structures, like the veins in a leaf. StuRat (talk) 04:32, 19 January 2015 (UTC)
Vectors. They were only formalised in the 19th century, and towards the end of that. They are elementary by modern standards and taught to school students, so are easy to explain, easy to understand. Not just the basics but things like dot, cross and triple product, the geometric product, applications and identities. You can lead into complex numbers and so give them a strong geometric basis. Polyhedra might be another thing; easily recognised and understood but with vectors you can write down the vertices and then do calculations, show how they are related etc..
Fractals would be hard as they only really took off with computers, which could both calculate and draw them. Other than that drawings of them have been done by skilled artists. Without those it would be very hard to get across the appeal and elegance of them. I'm also unsure anything would persuade him you were from the future – mathematics is timeless and you might just come across as a genius, or eccentric, or both.--JohnBlackburnewordsdeeds 07:20, 19 January 2015 (UTC)
No, Stu's right. Simple fractal structures like the Koch snowflake are easily understood. It's only for drawing the shapes in their full complexity that you need a computer.
For that matter, I wonder if Newton might not be interested in the idea of automated computation. You wouldn't be able to introduce him to electronics, but you could present the Turing machine as a theoretical concept. He'd know about the Pascaline and perhaps his rival Leibniz's Stepped Reckoner; you could tell him about Babbage's design for the difference engine and go on to the analytical engine design, which would have been comparable to the Turing machine in its generality. You might need more than an hour, though. -- (talk) 11:53, 19 January 2015 (UTC)

It's irrelevant to the question since we aren't allowed to bring anything with us, but I'm also reminded of a Witness to Yesterday episode. That was a Canadian show where historical figures were interviewed in a modern TV studio by host Patrick Watson. On the episode I'm thinking of, he's interviewing Shakespeare. At one point in the interview, Shakespeare says something like "that gives me an idea" and starts reciting some lines. Watson invites him to write them down, handing him a writing pad and a ballpoint pen. "This is a pen". he explains. "The ink is inside." Shakespeare is startled and delighted, saying how much faster he could work with such a pen. He experimentally writes his name a few times on the paper. Watson asks "What about your idea?" and Shakespeare says never mind, it was nothing. He returns the supplies to Watson — who quickly pockets the sheet of paper before continuing to the next topic. -- (talk) 11:53, 19 January 2015 (UTC)

Had I time to prepare in advance of our meeting, I would learn (the base-10 digits of) some Mersenne primes, and mention these to Newton. Newton would certainly have been aware of Mersenne's work, and for some of the smaller claims I made would be able to verify their truth. This would support my claim that I had arrived from a future where computing machines existed. RomanSpa (talk) 12:16, 19 January 2015 (UTC)
Unfortunately, verifying by hand that a large number is prime is a lot of work. It might be better to provide the factorization of some large composite Mersenne numbers. (See the reference in that article to "a famous talk by Cole".) Not just one, which you might have discovered by luck, but more than one. -- (talk) 13:54, 19 January 2015 (UTC)
What a wonderful question. While I think that Newton was open-minded like all true successful geniuses, one hour might be on the tight side. A super-genius convincing a (merely) genius takes both luck and time, because the genius likely has had experience with cranks. I'd say this is harder than it was for Ramanujan to convince his initial audience. And he was very real and had some backup. If you recall the details of his story, his now famous 100 were put in the dumpster by more than one able mathematician when he sent his application letters. Strangely though, the more brilliant and successful people are (G. H. Hardy in this particular case), the more they tend to be apt to take the less immortal seriously (a sign of greatness). So there is a chance. But Newton will not be impressed by a single particular result, however impressing. A general approach is needed. YohanN7 (talk) 13:05, 19 January 2015 (UTC)
If you think about what Ramanujan did, he managed to show lots of small results rather than one big one, and in the end that proved succesful. Ornil (talk) 21:55, 19 January 2015 (UTC)
  • I'd try to show him some stuff that amounts to foundations of real analysis - it's in a sense his legacy, but he never really knew how to e.g. construct the real numbers formally. So I'd try to get him to admit that he couldn't say exactly what the continuum was. Then I'd construct whole numbers from the empty set, integers with a sign convention, rationals by some slight hand waving about equivalence classes, and the real numbers via Dedekind cut. But now I suspect he wouldn't necessarily appreciate the rigor, he was after all a very application minded fellow. If I recall correctly, he wanted the calculus for physics, it was Leibniz who had more of a "pure math" approach to it. SemanticMantis (talk) 14:54, 19 January 2015 (UTC)
  • Oh one more that would be easy - show him that there are more real numbers than integers, via Cantor's_diagonal_argument. SemanticMantis (talk) 14:54, 19 January 2015 (UTC)
    I think both your answers are a little optimistic in terms of using set theory for anything. It was only invented two hundred years later, once math got a lot more formal. Newton might just say that your definitions are ridiculous. After all, you'd claim that there are as many natural numbers as rationals, and that any segment of the real line has the same size. Well, if you claim that, then clearly cardinality is nonsense, as far as any practical person is concerned. Ornil (talk) 21:55, 19 January 2015 (UTC)
Well, I take your point that he might not see the value of the formality. And to a point, he's right to not care, after all he did develop the calculus without really defining what a real number is. But I'm pretty sure he could follow the assessment of infinite sizes of number systems through bijection, and that, while (0,1) is shorter than (0,2), it has the same number of elements - actually that statement might suit his ideas of the continuum rather well! Basically I think he'd understand, I just don't know if he'd care :) If nothing else it would be highly interesting to see how he'd react to the Cantor diagonal. I was mostly thinking of profound things that I could explain in a short time off the top of my head. I think he'd care a lot about dissipative systems and self organization, reaction diffusion equation, and similar topics, but I don't think I could say much about those in a short time and without prior rehearsal! SemanticMantis (talk) 22:23, 19 January 2015 (UTC)
The basics of abstract algebra: symmetry groups of polyhedra and their properties (subgroups, etc.); the fundamental theorem of algebra; possibly (assuming his interest can be sufficiently piqued) the fundamental theorem of Galois theory. Basic complex variables (perhaps not including very rigorous proofs, at least not to begin with). The hyperbolic plane, Fucshian groups, and modular forms. Some ordinary differential equations: (harmonic oscillator, damped harmonic oscillator, description of solutions using complex numbers). Elliptic functions and applications (e.g., solution of the nonlinear pendulum, arclength of an elliptic curve, arithmetic-geometric mean, geodesics on an ellipsoid). Number theory: The prime number theorem, L-functions and Dirichlet's theorem on the distribution of primes. Calculus in three Euclidean dimensions: Stokes' theorem, the divergence theorem. Derivation of the heat equation from Fourier's law. Derivation of Newton's law of heating and cooling :-). Basic probability theory: the law of large numbers, the central limit theorem (derivation from the Fourier transform). Basic algebraic geometry: the Nullstellensatz, Bezout's theorem, existence of the Hilbert polynomial, perhaps basic sheaf cohomology. Basic aspects of Teichmuller theory as applied to something Newton might have been interested; e.g., something on Igusa invariants of hyperelliptic curves. Basic topology. Failure of simple-connectivity of the group of rotations in three dimensions. Spinors. Sławomir Biały (talk) 22:55, 19 January 2015 (UTC)

This is a very good question, except for one thing. Newton was a physicist. Why are we restricted to just pure mathematics? Anyway I would make this claim [WITHIN the conditions, but I'd have a better answer if I could use physics instead of math]: "In the future people communicate publicly over vast distances, not in private chambers. Giving everyone large but not infinite computational powers, it is possible to enter a vast space full of strangers, tell them what you are doing, read a number out loud for the whole room (with every stranger overhearing), and have one stranger in particular use what you have spoken out loud to perform his own calculations, then read to you the result with everyone overhearing, with the effect that you will learn a secret he has communicated to you - and you alone - and which nobody else can learn, despite communicating in a throng. Although he was the same class of stranger as everyone else - you have never met - heard the same numbers from you as everyone else, and has had his answers heard by everyone, nobody other than you can know what he has stated. For example, you could enter a room, read numbers, and have each person communicate their age to you by replying with a number they have calculated, with everyone overhearing. Still, you will know all of their ages and nobody will know each other's. This is vastly useful in the future, as we communicate in public using machines rather than in private chambers." That is an extraordinary claim that I can then prove (with or without reference to the computers that make this possible and interesting, which I think would interest Isaac) via public key encryption. He would certainly immediately understand the difficulty of factoring primes, as well as RSA quite easily. Given my claims about the future, threading the rest of modern society around this, and computers, is going to be a walk in the park. (talk) 00:57, 20 January 2015 (UTC)

I don't know why you have to bring physics into it. RSA is indeed a perfectly reasonable thing to show even under my conditions. Note that it does involve non-trivial abstract algebra. I don't think you could necessarily provide proofs for those in an hour. Ornil (talk) 02:59, 20 January 2015 (UTC)
Sorry, my answer was first an objection about the conditions, then I answer within the conditions! I've added an edit in brackets to the beginning of my answer. Note that your question had started "Suppose you had an hour to convince Isaac Newton that you come from the future (or you are a super-genius), and you could only do math (no experiments or other physics)." Why can't you have said "only do math and physics without experiments (i.e. thought experiments or appeals to existing experience only)" - my answers would have been much more interesting. (Such as relativity, electricity, magnetism, what-have-you, all different things. Quantum mechanics. Atomic theory.) Just, way more interesting than pure math. (As I have answered.) Also atomic theory is a bit of HUGE one, because actually as far as I understand about half of Newton's body of work was in physics, and half in chemistry (alchemy). But since he was an alchemist, his experiments and conclusions using alchemical symbols is worthless. We can follow them and see why he got the results he got, but he gained zero insight without John Dalton, Atomic theory, etc. So they're worthless. However, personally I would stick to physics rather than explaining every one of his chemical reactions through modern chemistry, since I didn't learn the latter well, and the former is easy to explain in popular terms without very detailed calculations in many cases. (talk) 03:36, 20 January 2015 (UTC)
Well, this is WP:RD/math here. Talking with Newton about theoretical physics is an interesting, separate question, which would fit nicely in WP:RD/Science. -- Meni Rosenfeld (talk) 11:51, 20 January 2015 (UTC)
PS. I think you are mistaken if you think Newton would understand your explanation above about cryptography. Your description is too abstract and, I believe, would only make sense to someone who already knows what you are talking about. It's also lots of inferential steps away from what Newton would be familiar with. You'd lose him long before you make a point that would impress him.
If we do want to explain cryptography, I'd forgo all this "enter a vast space full of strangers" talk, and focus on sending messages in envelopes facing the risk of interception. I'll also start with symmetric cryptography and only then move on to explain the merits of public key cryptography. -- Meni Rosenfeld (talk) 21:59, 20 January 2015 (UTC)
Basel problem. Stirling's approximation. Asymptotic expansions. Karatsuba algorithm. Jordan normal form. -- Meni Rosenfeld (talk) 15:19, 20 January 2015 (UTC)
Special relativity. Simple enough to explain to a physics genius in a short time, easy to demonstrate consistency (when he quizzes about various implications), and it is in his field: physics. You'd catch his interest enough that he'd want to know more. Radical enough that it is clearly not from his contemporaries. —Quondum 22:46, 20 January 2015 (UTC)
Given the time allotment, how about Maxwell theory, with special relativity left as a rather straightforward homework exercise. It is Newton after all. Sławomir Biały (talk) 22:59, 20 January 2015 (UTC)
If the tools are available, this would be an excellent topic, since light was of intense interest to Newton. One would have to weigh carefully whether differential calculus and vector algebra would be sufficiently familiar and mutually understood building blocks. —Quondum 23:18, 20 January 2015 (UTC)
Informing Newton of the propositions inferred from the Michelson–Morley type experiments would be doing physics not math contrary to the OP's request, however since Newton was keen on both I would be remiss to not show him the simplicity (using only grade-school algebra) of the rather different propositions I've inferred per his philosophy of deduction or Hypotheses non fingo. -Modocc (talk) 04:54, 21 January 2015 (UTC)
I feel that making deductions from premises is in the spirit of the OP's framing of the requirements. One is not performing experiments, nor is one demonstrating that the premises or conclusions apply to the real world; one is only demonstration that one has access to a body of consistent reasoning of a mathematical nature; that this may plausibly be a (to him) radical interpretation of real physics makes it nonabstract and interesting. For any individual to develop the amount of theory that we learn in undergraduate physics today would in Newton's day mark them as a super-genius. Physics has the advantage of being simple yet based on mathematical reasoning; pure mathematics could suffice, but has the disadvantage that one would spend days simply defining and explaining the myriad terms and concepts that we use. —Quondum 05:16, 21 January 2015 (UTC)
OK, then if it's in the spirit of this request to report on mathematical deductions based on thought experiments then I'll likely be elaborating upon my answer later. --Modocc (talk) 05:37, 21 January 2015 (UTC)
I think I must have a very different impression of Newton than you guys have. I feel like if I told Newton that his whole life's work is inaccurate, I'd be kicked out before I finish a sentence. They guy was super-jealous of his (well-deserved) greatness. Besides, most people even at the time Einstein couldn't conceive of non-flat space, and that was way after non-Euclidean geometry was well known in math. Sure, you could describe Michelson-Morley to him, and he'd say you are a lying crackpot. The conditions of the experiment mean that you have to convince him you are from the future, which is something he'd be extremely skeptical about, he just wouldn't assume you are telling him the truth. This is why math is so much more promising than physics, unless you get to do convincing experiments. Math is something that is intrinsically undeniable, whereas physics requires a proper experimental base. --Ornil (talk) 06:18, 21 January 2015 (UTC)
I pretty much agree with that, so my first task would be to demonstrate, mathematically, mass-energy equivalence with classical physics which is trivial given that photons have momentum and that takes about fifteen minutes. That should impress him and he should be able to grasp its implications. My second task would be to show my propositions (which are not relativistic therefore don't lead into the paradigm that would give him pause) and conclusions. That would take a bit longer, and you're right, without a firm experimental basis he's like to be skeptical, but at least I'd have a better chance of not getting kicked out. --Modocc (talk) 07:16, 21 January 2015 (UTC)
Maybe I am not following what you are proposing, but you'd have to at least assume the formula for the energy and momentum of a photon for which you have no experimental basis. In general, Newton's theory of mechanics is far more logical and beautiful than Einstein's - you'd never propose the latter unless you knew that there was a problem with the former. How could you convince Newton that there was such a problem? Mathematically, it's perfect. Sure, if you had Maxwell's equations, you might suspect that the speed of light is doing something strange, disobeying Galileo's relativity, but you don't have those, and you couldn't even write them without explaining tons of math first, not to mention that Newton's doesn't even know about electric current, let alone about magnetism. And he doesn't believe that light is a wave. --Ornil (talk) 07:38, 21 January 2015 (UTC)
Your question is framed as "You vs. Sir Isaac Newton" thus I am approaching this inquiry on the things that I happen to know, which often differs substantially from what anyone else here can "look-up" when it's unpublished. Interestingly enough, my work aligns substantially with what Newton knew because I maintain, via my modeling, that photons, like other quanta and particles of matter actually do obey the velocity addition of Galileo relativity. I won't need Maxwell's equations nor wave theory, and I don't even need to mention atoms or electrons! Regarding mass-energy, I'm referring to what's called the center-of-mass argument (I think). I'll have to return to this later though, because I need to retire.-Modocc (talk) 08:50, 21 January 2015 (UTC)

As the OP, I'd like to summarize a little and give my own ideas. I really like Meni Rosenfeld's ideas, since I think they are the right sort of problems to convince Newton. My own issue with them is that I couldn't do most of them without looking things up (or maybe rederiving, but that's iffy), so they mostly wouldn't work for me. Sławomir Biały's list is way too long for me, and I don't know how to do most of them. In general, it's nice to pick something to do with series, because Newton would recognize them and yet there were many things not yet known. I am surprised nobody brought up Euler's formula - it can be reasonably explained and used to some effect. But more generally, I feel like geometry would be most basic, and demonstrating some models that show that denying Euclid's parallel postulate can lead to a consistent geometry would be very impressive. Also, it would be pretty nice to show a formula for the n'th Fibonacci number and the general method for solving recurrences. This sort of thing which is easy to grasp and yet unknown is basically ideal - the issue is that he may well know something like that. Somebody brought up Babbage's engine - I'd rather do Boolean algebra and lambda calculus. Maybe he'd like Peano's axioms to build arithmetic on a sound basis, like geometry. Maybe Gaussian elimination in systems of linear equations (unless he knows that) and some ideas about matrices and determinants. Maybe quaternions - although I don't know what applications they have (without looking it up), so maybe he wouldn't find them interesting. There are lots of Computer Science-y things that may well work (like graph theory) and I am very comfortable with those. --Ornil (talk) 07:22, 21 January 2015 (UTC)

Would he not likely wonder what is to be gained by denying Euclid's fifth postulate? -Modocc (talk) 07:50, 21 January 2015 (UTC)
Of course he would. The answer is that it constitutes a proof that it's independent, which is something lots of people tried to disprove. And of course surface geometry of a sphere is immensely practical - we live on one. Ornil (talk) 16:20, 21 January 2015 (UTC)
It would be interesting, so I see your point as it was to Gauss for instance, but probably not Earth shattering. Even so, showing that the postulate is independent may simply affirm to Newton its utility as a postulate, especially with regards to the abstract absolute space that he believed in and assumed. In broader terms, I'm certain he recognized that such consistency within paradigms, although important with respect to logic, is also a major element of fiction, storytelling and the like. -Modocc (talk) 17:45, 21 January 2015 (UTC)

IMO, one should just talk of general ideas and statements of great theorems (but not their proofs) so as to maximize the number of ideas presented. Well, there are many great ideas directly related to his interests... From Gauss: On non-Euclidean geometries and differential geometry. From Poincaré: Dynamical systems and chaos. From Klein: The Erlangen program. And if there was time left, also talk about some ideas of Hamilton, Darboux, Liouville, Gödel, Kolmogorov, maybe also of Cantor and Abel. There are so many things that would interest the young Newton from the time when he was most interested in mathematics (before the age of 30?), but it would be hard to expose it without preparation, so many notation shifts and terminology with different meanings from what they had in the past! (talk) 08:35, 21 January 2015 (UTC)

The theory of determinants would be something I think anyone could probably talk about. Since Newton was a pioneer in (among other things) what we would now consider to be "classical invariant theory", it would be easy to get him interested in them. Determinants, minors, Cramer's rule, etc. There are many small miracles that I think would be sufficient to pique Newton's interest. Classical invariants of binary forms (a la Sylvester) is something else worth considering, but nowadays is a rather obscure branch of knowledge, even though it is not difficult or deep. Sławomir Biały (talk) 13:13, 22 January 2015 (UTC)

January 20[edit]

What is % of world coastline represent?[edit]

This wikipedia page (List_of_countries_by_length_of_coastline) says that Coast/area ratio(m/km²) of the world is 7.80. How would this be represented in percentage??
What % of the world is coastline? (talk) 11:57, 20 January 2015 (UTC)

The units are at the top of the column; m/(km^2). -- Q Chris (talk) 12:14, 20 January 2015 (UTC)
Which means that you can't represent it as a percentage. A percentage is a ratio of two quantities of the same type. Rojomoke (talk) 12:52, 20 January 2015 (UTC)
You could just change the meters to km. Or not?? (talk) 12:56, 20 January 2015 (UTC)
It's not a percentage, or cannot be converted into one, as you're comparing two things, length in metres and area in square kilometres. To make it a percentage you can add an extra dimension.
E.g. assume the coast is 100m wide, so it has an area length * 100m. Taking the ratio of 7.8, that's 7.8 metres [per sq km]. So for every square km (1 000 000 sq m) there's 7.8 x 100 = 780 sq m. This gives
780 / 1 000 000 = 0.00078 = 0.078%
--JohnBlackburnewordsdeeds 12:57, 20 January 2015 (UTC)
But inst X squared km, just X * X km? (talk) 13:05, 20 January 2015 (UTC)
No, X square km is an area of X squares 1km by 1km. On the other hand, X km squared is an area measure of a square X km by X km, which makes X2 square kilometers. --CiaPan (talk) 13:57, 20 January 2015 (UTC)
Note that the length of the coastline is always indeterminate, as each coastline has a different length, depending on how closely it is examined. See coastline paradox. Therefore, the ratio of that number to any other is also meaningless. StuRat (talk) 14:44, 20 January 2015 (UTC)
Since this is the Maths desk I should probably point out that dimensional analysis is (i.e. "you cannot compare km to km2") is not necessarily the right concept here, but measure theory is. The (2d) measure of a one dimensional object (like the coastline) is zero, so the ratio mu(costline)/mu(all land) is zero. I.e. coastline is 0% of all land. Now as StuRat has pointed out people argue that the costline has length (i.e. 1d measure) infinity. In this setup the coastline is really a fractal, that is it has (or can be assigned) a dimension d, with 1<d<2. Googling gives estimates of the dimension of coastlines between d=1.25 and d=1.5 (Norway). In any case the 2-d measure of a 1.5-dimensional object is still zero, so the result remains that the coastline is 0% of all land. (talk) 23:55, 20 January 2015 (UTC)
If the coastline length were well defined, we could say that the ratio of land area to coastline is N km. For an intuitive sense of what this means, imagine a planet that has the same land area and the same coastline length, but where all the land is in strips running around lines of latitude; the ratio is half the average width of the strips. —Tamfang (talk) 08:48, 21 January 2015 (UTC)

January 21[edit]

Triangles and circumcenters[edit]

My points are (-3,1) (-1,-1) (4,-2)

I graphed the triangle and found all the perpendicular bisectors. I got them all to connect at a point and that's the circumcenter.

How do you solve for this algebraically? — Preceding unsigned comment added by (talk) 00:20, 21 January 2015 (UTC)

Homework? There's probably a nicer way to do this, but a brute force approach should work... Select two points, work out the gradient of the line between them, then find their midpoint, and write down the equation for the perpendicular running through that midpoint (recall that the product of the gradients of two perpendicular lines is -1). Repeat for another pair of points. Now you have a pair of simultaneous linear equations. Solve for their intersection. RomanSpa (talk) 01:22, 21 January 2015 (UTC)
P.S. I hope there's a more elegant approach, and that someone here will remind me what it is! Thanks. RomanSpa (talk) 01:22, 21 January 2015 (UTC)
That's how I'd do it, too. StuRat (talk) 06:06, 21 January 2015 (UTC)
I would plug vertices' (x,y) coordinates into the general circle equation (x-a)^2+(y-b)^2=r^2, eliminate r^2, then expand squares and reduce a^2 and b^2 to obtain a 2×2 linear equation system with a,b unknown. --CiaPan (talk) 06:20, 21 January 2015 (UTC)
That's much neater than my way. Interestingly, your approach requires us to know that the intersection of the perpendiculars is the circumcenter (presumably from the geometric proof), while mine/StuRat's doesn't require this knowledge. Once we have the intersection of the perpendiculars it's easy to prove algebraically that this is the circumcenter, of course (hint for the original questioner: it's just Pythagoras...). I suspect this was someone's homework question, and now we can see why: we can prove that the circumcenter is the intersection of the perpendiculars geometrically, and I suspect our original questioner has already seen this proof; now we see that we can prove the same thing algebraically. That is, one area of mathematics can be mapped onto another area - here we're mapping geometry to algebra - and proving something in one area can (with a suitable choice of mapping) prove something in another area. This is an important technique in more advanced mathematics, and I suspect our original questioner is just encountering this idea for the first time. RomanSpa (talk) 13:19, 21 January 2015 (UTC)
Given a triangle with vertices (x1, y1), (x2, y2), (x3, y3), the equation of the circumcircle is given by the determinant
x^2+y^2 & x & y & 1 \\
x1^2+y1^2 & x1 & y1 & 1 \\
x2^2+y2^2 & x2 & y2 & 1 \\
x3^2+y3^2 & x3 & y3 & 1 
You can then find the coordinates of the center by completing squares. The final expression is a bit messy when you expand everything out, a fraction with 12 term numerator and 6 term denominator for each coordinate. --RDBury (talk) 10:36, 21 January 2015 (UTC)

January 22[edit]

January 23[edit]


What do you call a bipyramid with shaved base? In other words, make two identical Egyptian pyramids. Make cuts to create identical vertical walls. The structures' footprints/bases are now identical smaller squares. Glue the squares together so it's a 12-sided solid.

What's the Washington Monument's geometric shape? Is it different if the visible ground touching faces were vertical? Sagittarian Milky Way (talk) 06:32, 23 January 2015 (UTC)

Have any pics of the objects you have in mind ? StuRat (talk) 06:37, 23 January 2015 (UTC)
I've made it clearer. Sagittarian Milky Way (talk) 16:33, 23 January 2015 (UTC)
With regular faces, I think you're describing the elongated square bipyramid. If the Washington Monument had regular faces, it would be an elongated square pyramid. Its actual shape is sometimes called an obelisk, though MathWorld seems to disagree about the meaning of that term. (It's definitely an obelisk in the architectural sense.) -- BenRG (talk) 22:35, 23 January 2015 (UTC)

How do you solve this equation? (find all possible values of x)[edit]

18(-3^\frac{x}{2})^x - (-3^\frac{x}{2})^{2x} = 81
The Transcendent One 17:10, 23 January 2015 (UTC)

Step 1. Simplify the equation

18*(((-3)^(x/2))^x)-(((-3)^(x/2))^(2*x)) == 81

18 * (-3)^((x^2)/2) - ((-3)^(x^2)) == 81

Step 2. Substitute x^2 with B

18 (-3)^(B/2) - (-3)^B == 81

step 3. Substitute (-3)^B with y

18 Sqrt[y] - y == 81

step 4. solve for all possible value of y (remembering that square root of y has two possible solutions

Step 5. solve for B

step 6. solve for x


oh yeah! I almost forgot, you got to know if you want to work in the REAL domain or in the COMPLEX domain. In other words, do you want only the results in real numbers or in complex numbers. (talk) 17:58, 23 January 2015 (UTC)

The foregoing answer reads -3^\frac{x}{2} as meaning (-3)^\frac{x}{2}; it's a different problem if it means -(3^\frac{x}{2}). —Tamfang (talk) 08:05, 25 January 2015 (UTC)
Yes, I believe exponentation has a higher precedence than negation.
Also, finding all possible solutions (which can be complex numbers in general) would be preferable. — Preceding unsigned comment added by The Transcendent One (talkcontribs) 15:17, 25 January 2015 (UTC)

A Maths Problem[edit]

Whats 9+10 and 6x6? — Preceding unsigned comment added by (talk) 19:34, 23 January 2015 (UTC)

Since I can't imagine that you are simply asking for 9+10=19 and 6x6=36, which your could discover via Google, Wolfram Alpha, a calculator, or doing it in your head, I figure it must be a trick question and your "and" refers to a binary AND. Since 19 = 16 + 2 + 1 and 36 = 32 + 4, the answer to your question is zero. -- ToE 23:06, 23 January 2015 (UTC)

January 24[edit]

Why does lineal algebra have such a central role in many, if not all, math degrees?[edit]

Math degree programs seems to include invariably Calculus (single and multivariate), probability, and lineal algebra. Then, you can find more applied fields like game theory, analysis of algorithms. Why not universal algebra, or, abstract algebra? Wouldn't these be more basic than lineal algebra? --Senteni (talk) 14:46, 24 January 2015 (UTC)

From the US university programs I'm familiar with, the Calculus (and differential equations), probability, and linear algebra classes you mentioned are expected to be completed at the underclassman (freshman/sophomore) level, and may be all the math that is offered at a two-year junior college. Math majors will be expected to take a full year of abstract algebra (as well as topology and real analysis) as upperclassmen (juniors/seniors), as well as additional in-major electives. So I don't see linear algebra playing the the central role you describe, although it provides an introduction to some of the elements of abstract algebra, and is often the first class in which students are expected to be able to write formal proofs (aside from the delta epsilon proofs that may still be required in some calculus classes). This is the first roadblock in the studies of some math majors, and I know a few who sailed through calculus, but changed majors after having trouble in linear algebra and realizing that it was unlikely that they would make it through an abstract algebra class. -- ToE 18:07, 24 January 2015 (UTC)
(ec) "We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury." (Irving Kaplansky, writing about himself and Paul Halmos).
Why study linear algebra? Because it gets everywhere. On the applications side: quantum mechanics -- linear algebra; co-ordinate transformation locally -- linear algebra (globally too, if it's flat); basis function decompositions -- Fourier theory, Laplace theory, wavelets, functional analysis, orthogonal function theory -- all linear algebra. Big multidimensional simulations -- eg numerical weather forecasting, econometrics -- mostly (big) linear algebra. Engineering applications -- stability theory, signal processing -- mostly linear algebra. Linear calculations can be big and fast; even for nonlinear problems, the solution of choice will often go through solving a sequence of linearised steps. And the linearisation may be very revealing about the nature of the solution.
On the purer side, too: every group has a linear representation (group representation theory), leading to a lot of pure maths (as well as a fair number of applications), exploring group theory (and deeper extensions of it) in a systematic way. So there is a lot that you can represent with matrices. And even if you're interested in more abstract properties of algebra, it's useful to be able to illustrate them in a concrete way, or show phenomena in simple toy systems. Matrices can often provide that -- whether it's simply introducing the idea of non-commutativity, all the way to really quite hairy stuff.
So yeah, there's a reason that linear algebra is right up there with calculus and probability. Jheald (talk) 18:30, 24 January 2015 (UTC)
  • I clearly poorly expressed myself. I mean "central" in the sense that it's basic, not that it is the learning object. Do many things build upon it? Couldn't the programs have been organized like the Algebra book of Serge Lang,where Lineal Algebra just shows up on part III? His book puts Groups, Rings, Modules, Galois theory, Fields before the Matrix and the like. --Senteni (talk) 23:54, 24 January 2015 (UTC)
Well, not really. More general does not necessarily equate to "better". For example, the entire field of representation theory exists because linear algebra is "easy", whereas group theory is "hard". Basically all non-trivial problems in group theory involve realizing the groups involved as matrices in some fashion. But that's just in abstract algebra. You can work in other fields like analysis or dynamics, and never encounter a "ring" or "group", yet use linear algebra regularly. There one studies linearizations of problems, again because linear things are easy to do, but non-linear things are hard. In a sense, this is also what differential calculus is about too. So, yes, you need linear algebra to do just about anything in mathematics. Whereas group theory and ring theory, not so much. Sławomir Biały (talk) 00:41, 25 January 2015 (UTC)
It is linear with an r at the end. Yes lots of things build on it, in fact it has more direct applications around mathematics than anything else, probably the only places where you don't use it would be in areas like number theory or set theory. See special linear group for instance for just one application in group theory. Dmcq (talk) 01:06, 25 January 2015 (UTC)
In Spanish it is álgebra lineal. -- ToE 21:12, 25 January 2015 (UTC)

January 25[edit]

Factorization of \sum_{n=0}^M~(-1)^n~n^k[edit]

For k ≥ 2 and even values of M = 2N we have


For k ≥ 2 and odd values of M = 2N + 1 we have


where Pk and Qk are polynomials of degree k - 2 in N. My conjecture, based on computer aided verification for all values of k\le10^3, is that Pk and Qk are irreducible over the rationals. What are your opinions on the subject, and how might one prove (or disprove) such a conjecture ? Is there any literature or research on this particular topic ? Thank you. — (talk) 11:20, 25 January 2015 (UTC)

I don't know about irreducibility, but the polynomials seem to be closely related to the Euler polynomials; see Faulhaber's formula for a possible explanation. --RDBury (talk) 13:03, 25 January 2015 (UTC)
One can express them easily as the difference of two of those Faulhaber's formulae, for instance leaving out the N=0 case we get  F_{2N}(k) - 2^{1+k}F_N(k) for the first one where F_p(k)=\sum_{i=1}^k i^p so that gives an expression in terms of either Bernoulli polynomials or Bernoulli numbers as desired. Or one can get a generating function but I don't know if any of that advances towards the target. It might be interesting to see how the polynomials behave in modular arithmetic. Dmcq (talk) 18:50, 25 January 2015 (UTC)

Simplex coordinates starting with (0,0,0,...),(1,0,0,...)[edit]

I'm looking for a reasonably easy way to calculate the coordinates for an n dimensional simplex where the first two coordinates are at the origin and (1,0,0...) and all-coordinates are non-negative (essentially each time the next dimension is added on, it gets added on in the positive value of the next dimension). So after origin and (1,0,0,0...) the next is (1/2,sqrt(3)/2,....). and the next is (1/2,sqrt(3)/6,?,0,0,0,0), etc. I'd like a formula where I can put in d for dimension of the simplex and n for which coordinate in the the simplex. So for d=3, n=2, I'm getting the sqrt(3)/6.Naraht (talk) 16:07, 25 January 2015 (UTC)

Did you look over here? YohanN7 (talk) 16:15, 25 January 2015 (UTC)

Question in applied linear algebra[edit]

At the Science Desk, I asked a linear algebra question regarding how to best remove instabilities from a linear system. I thought the Science Desk was a slightly better fit because the instabilities are a consequence of measurement uncertainties; however, if anyone here has any insights, they would also be welcome. Please reply at the Science Desk. Dragons flight (talk) 19:50, 25 January 2015 (UTC)