Search found 825 matches

by Torn Apart By Dingos
Wed Mar 16, 2011 2:17 am UTC
Forum: Mathematics
Topic: 0÷0
Replies: 22
Views: 3122

Re: 0÷0

Arguing on the internet, how did I fall into this trap again?
by Torn Apart By Dingos
Wed Mar 16, 2011 1:51 am UTC
Forum: Mathematics
Topic: 0÷0
Replies: 22
Views: 3122

Re: 0÷0

The square root thing you demonstrate doesn't require us to modify any fundamental axioms, nor does it break most of higher mathematics. It doesn't require us to drop any of the field axioms, but a priori, there's no reason they should be our only axioms. One could imagine a different set of axioms...
by Torn Apart By Dingos
Tue Mar 15, 2011 4:07 pm UTC
Forum: Mathematics
Topic: Matrix multiplication seems so arbitrary
Replies: 19
Views: 4094

Re: Matrix multiplication seems so arbitrary

Linear algebra is really about linear transformations. By the properties of linearity, a linear transformation is uniquely defined by what it does to the unit vectors, because if you know that f((1,0))=u and f((0,1))=v, then you know that f((a,b))=f((a,0))+f((0,b))=af((1,0))+bf((0,1))=au+bv (I reall...
by Torn Apart By Dingos
Tue Mar 15, 2011 2:02 pm UTC
Forum: Mathematics
Topic: 0÷0
Replies: 22
Views: 3122

Re: 0÷0

I disagree with your first idea, that what I did wasn't really division. The thing is that it's really similar to calling i the square root of -1, No. It is not similar at all. The existence of a square root of -1 does not violate any of the laws of algebra. The existence of 1/0 would do so. Yes it...
by Torn Apart By Dingos
Mon Mar 14, 2011 11:08 pm UTC
Forum: Mathematics
Topic: [0,1] ~ [0,1) (Cardinality)
Replies: 5
Views: 3754

Re: [0,1] ~ [0,1) (Cardinality)

It's easy to find an explicit bijection. All you want to do is to remove a single point from your set and show that your new set has the same cardinality. This should be familiar to you: how do you show that {0,1,2,3,...} has the same size cardinality as {1,2,3,...}? A bijection in this case is f(n)...
by Torn Apart By Dingos
Sat Aug 14, 2010 10:03 am UTC
Forum: Mathematics
Topic: Does anyone else create formulae on the spot?
Replies: 21
Views: 2546

Re: Does anyone else create formulae on the spot?

I don't. I consider linear formulopodes too trivial and non-linear ones too difficult.
by Torn Apart By Dingos
Tue Aug 10, 2010 7:16 pm UTC
Forum: Mathematics
Topic: Lectures for my mp3 player
Replies: 3
Views: 843

Re: Lectures for my mp3 player

I don't think that's possible or that such a thing exists. You can't listen to a lecture the same way you'd listen to an audiobook. You need time to reflect on the ideas, and also it's much easier if you can read the notation on a blackboard. What I suggest instead is that you find a mathematical pr...
by Torn Apart By Dingos
Mon Aug 09, 2010 5:09 pm UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

Alright, that (almost) works. I'd be interested in if an actual tiling can be made, but your construction works if we massage the definitions a little: In my opinion, a tiling should be a partition of a set. So no overlapping of edges. Thus, we should ask if we can tile the "half-open" imp...
by Torn Apart By Dingos
Mon Aug 09, 2010 3:30 pm UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

Qaanol wrote:
Spoiler:
then rectangles can be made arbitrarily small in both dimensions simultaneously, and can thus tile any arbitrary rectangle.

Why should this be true? You can't tile a square with disks (no disk, no matter how small, can ever cover a corner), so why should these rectangles be able to do the job?
by Torn Apart By Dingos
Mon Aug 09, 2010 2:07 am UTC
Forum: Logic Puzzles
Topic: A new kind of hat puzzle!
Replies: 2
Views: 1825

A new kind of hat puzzle!

I'm loving the recent surge in puzzle threads here! I'll contribute one of my most recent favorite puzzles. I'm posting it in this forum because I think it requires more than just a trick to solve. During the years I've seen lots of hat puzzles on these forums, but I haven't seen this one. I saw thi...
by Torn Apart By Dingos
Mon Aug 09, 2010 1:13 am UTC
Forum: Logic Puzzles
Topic: Colliding balls
Replies: 7
Views: 1750

Re: Colliding balls

Oh, you're right. Yes, that was what I was thinking of, but I realize now that's not what I wrote in the problem statement. :)
by Torn Apart By Dingos
Mon Aug 09, 2010 1:09 am UTC
Forum: Logic Puzzles
Topic: Colliding balls
Replies: 7
Views: 1750

Re: Colliding balls

I mean the left one starts out by going right and the right one starts out by going left, so they will collide with eachother and both go off the table. You need to contract the interval in (2) just as you did in (3). Yay for edits: Well, take the radius as r/2 minus epsilon, and you...
by Torn Apart By Dingos
Mon Aug 09, 2010 1:02 am UTC
Forum: Logic Puzzles
Topic: Colliding balls
Replies: 7
Views: 1750

Re: Colliding balls

Sure! I solved them the same way. I thought the trick needed for (3) was neat. About your (2): Actually this will depend on the radius of the balls. Suppose you only have two balls, one on each edge of the table, precisely so large that they touch. They will both fall off instantaneously (not ta...
by Torn Apart By Dingos
Sun Aug 08, 2010 11:07 pm UTC
Forum: Logic Puzzles
Topic: Colliding balls
Replies: 7
Views: 1750

Colliding balls

Suppose you have n identical balls on a 1 meter long (one-dimensional) table, each with radius r and speed 1 m/s, but with different orientations (some go left, some go right). The table is frictionless and collisions are elastic. 1) At most how many collisions will there be before they all fall off...
by Torn Apart By Dingos
Sun Aug 08, 2010 10:40 pm UTC
Forum: Mathematics
Topic: Pairing off points: More surprising trickiness
Replies: 18
Views: 2219

Re: Pairing off points: More surprising trickiness

antonfire & Talith: Very elegant variant of OverBored's solution.
by Torn Apart By Dingos
Sun Aug 08, 2010 10:36 pm UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

Torn Apart By Dingos: There checkerboard argument works to rule out an infinite tiling. The path argument carries through as well, with a bit of thought. Sure, there are infinite number of vertices, but the ones you can reach can only have whole-number displacements, of which there can be a finite ...
by Torn Apart By Dingos
Sun Aug 08, 2010 12:12 am UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

aleph_one: Nice trick! Talith: Thank you for making to reexamine my counterexample, I had given up on that approach. :) (Though I thought that you could do it by only going right and up from the bottom left corner, which I had a counterexample to - but when I examined it, I realized that it might wo...
by Torn Apart By Dingos
Sat Aug 07, 2010 10:56 am UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

OverBored: A 3x3 rectangle can be decomposed to nine 1x1 rectangles, and each of these have equal amounts of white and black no matter how we offset them. I think that proof is fine, and the upside of that one is that it doesn't need to assume a finite tiling (it assumes there are no rotated rectang...
by Torn Apart By Dingos
Fri Aug 06, 2010 6:47 pm UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

Talith: I had that idea too, but it doesn't seem to be true. NEW EDIT: That approach seems to work! Proof: Assume that a big rectangle is tiled by finitely many proper rectangles. I'll prove that we can move along a path from one corner to another along full integer sides of proper rectangle...
by Torn Apart By Dingos
Fri Aug 06, 2010 1:22 pm UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

What do you mean by "done the same way"? Yes, a hypothetical tiling using rotated rectangles.
by Torn Apart By Dingos
Fri Aug 06, 2010 1:00 pm UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

Syrin wrote:No, because an infinite case is identical to a finite case (as it must be tiled by proper rectangles, and at least one side of a proper rectangle is at least 1)
I don't understand your reasoning.
by Torn Apart By Dingos
Fri Aug 06, 2010 11:44 am UTC
Forum: Mathematics
Topic: Tiling a rectangle: Another surprisingly tricky problem
Replies: 45
Views: 5677

Re: Tiling a rectangle: Another surprisingly tricky problem

Do we need to worry about infinite tilings and/or rotated rectangles?
by Torn Apart By Dingos
Thu Jul 29, 2010 4:05 am UTC
Forum: Mathematics
Topic: Polygon inside a polygon: A surprisingly tricky problem
Replies: 20
Views: 3280

Re: Polygon inside a polygon: A surprisingly tricky problem

I really like this puzzle and I look forward to seeing the second solution from you, aleph_one. :) Post more puzzles!
by Torn Apart By Dingos
Thu Jul 29, 2010 4:03 am UTC
Forum: Logic Puzzles
Topic: Cat and mouse
Replies: 29
Views: 12991

Re: Cat and mouse

I read another wording of this puzzle on a thread on mathoverflow and got exactly the same solution as aleph_one. The wording of the puzzle in that thread seems to imply that 2(N-2) is the best we can do. How could we prove this?
by Torn Apart By Dingos
Sun Jul 04, 2010 10:59 am UTC
Forum: Mathematics
Topic: Is it just me, or does the average guy really suck at math?
Replies: 57
Views: 7459

Re: Is it just me, or does the average guy really suck at ma

I was amused when one of the characters pondered for a moment before deciding 645 was not a prime.
by Torn Apart By Dingos
Wed Jun 09, 2010 9:29 am UTC
Forum: Mathematics
Topic: Probability Question
Replies: 14
Views: 2145

Re: Probability Question

Fix m=4, the length of the shorter string. Let x(n) be the number of strings of length n containing the shorter string as a substring (for now, we don't allow it be re-ordered). The first occurrence of the shorter string in a string of length n is after at some position i such that 0<=i<=n-m (so tha...
by Torn Apart By Dingos
Mon Jun 07, 2010 4:12 pm UTC
Forum: Mathematics
Topic: Probability Question
Replies: 14
Views: 2145

Re: Probability Question

Probably pretty small anyway. This can be calculated, but you are asking the wrong question, for many reasons. 1. The probability depends on the given string. "11" will appear less often than "12" in a longer string. 2. The four digits don't appear in her phone number, or social ...
by Torn Apart By Dingos
Wed Jun 02, 2010 10:22 am UTC
Forum: Mathematics
Topic: Math discovered or invented?
Replies: 110
Views: 15773

Re: Math discovered or invented?

Is it safe to say the axioms are invented and the theorems and proofs are discovered? I dont think natural numbers exist outside of human experience (or the experiences other highly intelligent beings that can define numbers). Do other forms of life experience the natural numbers? What about jelly ...
by Torn Apart By Dingos
Mon May 10, 2010 11:56 am UTC
Forum: Mathematics
Topic: Taylor Series
Replies: 15
Views: 2694

Re: Taylor Series

This is an important point that has to be stressed. People often complain about Wikipedia being awful for learning math, but that's not the point of Wikipedia. Wikipedia, as its name suggests, is meant to be an encyclopedia. It's a reference. You wouldn't try to learn French from Wikipedia (even th...
by Torn Apart By Dingos
Mon May 10, 2010 5:27 am UTC
Forum: Mathematics
Topic: Taylor Series
Replies: 15
Views: 2694

Re: Taylor Series

Taylor series are essentially polynomials of infinite degree. Polynomials are the simplest functions to deal with, and therefore it is convienient to write functions as power series if possible. For one thing, it allows us to calculate numerically functions to arbitrary precision: it is unclear, for...
by Torn Apart By Dingos
Sun May 09, 2010 6:00 pm UTC
Forum: Mathematics
Topic: Quaternions and graphics programming
Replies: 11
Views: 1535

Re: Quaternions and graphics programming

Sorry, I was wrong, your vectors were already normalized.
by Torn Apart By Dingos
Sun May 09, 2010 1:57 pm UTC
Forum: Mathematics
Topic: Quaternions and graphics programming
Replies: 11
Views: 1535

Re: Quaternions and graphics programming

Thanks, I had read most of the wiki stuff but hadn't really been able to gather much off it. This makes much more sense now. So the euler rotations of 90 degrees about each axis would be represented with the quaternions (cos(\pi/2) + sin(\pi/2)(1, 0, 0), cos(\pi/2...
by Torn Apart By Dingos
Fri May 07, 2010 12:21 pm UTC
Forum: Mathematics
Topic: Can we truly prove anything?
Replies: 86
Views: 10201

Re: Can we truly prove anything?

wouldn't that make any proof we've ever made completely useless? Not really. All that means is that ZFC is a bad foundation and we'd find another one. This got me thinking. Would it be possible that all sufficiently powerful systems (in which we can do that math we want to do; Euclidean geometry do...
by Torn Apart By Dingos
Fri Apr 02, 2010 10:14 pm UTC
Forum: Mathematics
Topic: A Graph of gravity + friction motion
Replies: 7
Views: 1298

Re: A Graph of gravity + friction motion

BasV wrote:Can I rewrite

Code: Select all

LINEAR_DAMPING^0 .. LINEAR_DAMPING^n


to anything 'direct' ?
Yes, this is a geometric series (look this up on wikipedia, there's a very simple derivation for the formula). If d=LINEAR_DAMPING, then [math]d^0+...+d^n=\dfrac{1-d^{n+1}}{1-d}.[/math]
by Torn Apart By Dingos
Fri Mar 26, 2010 7:15 pm UTC
Forum: Mathematics
Topic: amsthm
Replies: 2
Views: 859

Re: amsthm

Is this what you're looking for? I'm using the following. Apart from those below, you can use the built-in \begin{proof}...\end{proof}. \usepackage{amssymb, amsmath, amsfonts, amsthm} \theoremstyle{plain} \newtheorem{Theorem}[equation]{Theorem} \newtheorem{Lemma}[equation]{Lemma} \newtheorem{Proposi...
by Torn Apart By Dingos
Fri Mar 12, 2010 5:22 pm UTC
Forum: Mathematics
Topic: Four color theorem
Replies: 82
Views: 18901

Re: Four color theorem

The surrounding ones don't have to be different colors. They can be colored A,B,A,C.

The innermost five parts of the following diagram is the same as your example, and it's colored with four colors.

Image
by Torn Apart By Dingos
Sat Mar 06, 2010 9:27 am UTC
Forum: Mathematics
Topic: Chain email
Replies: 28
Views: 3183

Re: Chain email

Actually that was what I was referring to. Writing "a=b=2a=2b" (equality sign between equations) or "a=>2a/2" (arrow where there should be an equality sign). I've seen worse, but admittedly it's rare. You're right that it doesn't mean they don't understand equality, but I'd like ...
by Torn Apart By Dingos
Fri Mar 05, 2010 11:41 pm UTC
Forum: Mathematics
Topic: Chain email
Replies: 28
Views: 3183

Re: Chain email

This is not an equivalence relation, and it's not assignment. Writing it this way doesn't help the reader, it is just confusing and perpetuates a bad understanding of the equality sign. Seriously? Do you think anyone is confused about what equality is because of an unusual use in a puzzle? Do you r...

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