Search found 555 matches

by skullturf
Tue Feb 22, 2011 10:55 pm UTC
Forum: Mathematics
Topic: Problems that seem difficult but aren't
Replies: 75
Views: 9098

Re: Problems that seem difficult but aren't

Prove that, for any n>0, the numbers n+2 and n^2+n+1 cannot simultaneously be perfect cubes. I like this! My spoiler doesn't contain an explicit solution, but it gives a big hint. I can get a contradiction from the hypothesis that both n+2 and n^2+n+1 are perfect cubes. At first I thought something...
by skullturf
Tue Feb 22, 2011 7:56 pm UTC
Forum: Mathematics
Topic: Is this true?
Replies: 8
Views: 913

Re: Is this true?

So strictly speaking, the original question as phrased is trivial. However, there are closely related questions that are far from trivial. The first time I considered the question "Can the sum of two irrational numbers be rational?" I either noticed or was told, "Yes; for instance, pi...
by skullturf
Tue Feb 22, 2011 7:34 pm UTC
Forum: Mathematics
Topic: Is this true?
Replies: 8
Views: 913

Re: Is this true?

gorcee wrote:However, I am fairly certain that it is not known if [imath]\pi + e[/imath] is rational or not. So I don't think that what you're proposing is true.


That just means it's not known whether e is, in fact, a rational number minus pi.
by skullturf
Tue Feb 22, 2011 7:31 pm UTC
Forum: Mathematics
Topic: Is this true?
Replies: 8
Views: 913

Re: Is this true?

Do you mean exactly what you typed?

Carefully formulate your conjecture in the form "If P, then Q." Using only some simple logic, does Q follow from P "right away"?
by skullturf
Tue Feb 22, 2011 3:29 pm UTC
Forum: Mathematics
Topic: Problems that seem difficult but aren't
Replies: 75
Views: 9098

Re: Problems that seem difficult but aren't

There are two different versions of the nine dots problem. One asks for four lines, and one asks for three lines. (In both versions, you can't lift your pen from the page.) I agree with your point about 169ishness but I think it applies just to the three line version. Solution to the four line versi...
by skullturf
Tue Feb 22, 2011 2:11 am UTC
Forum: Mathematics
Topic: Complex integration- Not quite "getting" it.
Replies: 7
Views: 2000

Re: Complex integration- Not quite "getting" it.

Even though x+ix is complex, the "small increment" in the integral is dx, which is real. So our integral is not "really" a complex integral. If you have an integral that's a sum of "infinitely many" things of the form f(z)dz where f(z) and dz are both complex, then you'...
by skullturf
Mon Feb 21, 2011 1:59 pm UTC
Forum: Mathematics
Topic: [Homework] Subgroups of U(n)
Replies: 3
Views: 678

Re: [Homework] Subgroups of U(n)

A possible source of confusion is that the notation "mod" can have slightly different meanings. Sometimes it denotes an operation, and sometimes it denotes part of an equivalence relation. In some contexts (especially in some programming languages) "n mod k" means the UNIQUE numb...
by skullturf
Fri Feb 18, 2011 3:10 am UTC
Forum: Mathematics
Topic: The Shortest String Containing all Permutations of n Symbols
Replies: 29
Views: 28854

Re: The Shortest String Containing all Permutations of n Sym

undecim wrote:http://en.wikipedia.org/wiki/De_Bruijn_sequence


Note that this includes all words of specified length, including those where symbols are repeated.
by skullturf
Thu Feb 17, 2011 3:56 am UTC
Forum: Mathematics
Topic: How is l'hopital useful here?
Replies: 15
Views: 2224

Re: How is l'hopital useful here?

(Latex questions: Is there a better way to avoid having "for x" run together other than giving "for" its own column? Can I make the exponent of e look nicer in its present form?) My answers to those questions (others may have different tastes): --I am a strong believer in a^{b/c...
by skullturf
Wed Feb 16, 2011 10:45 pm UTC
Forum: Mathematics
Topic: Is this a meaningless math question?
Replies: 24
Views: 2857

Re: Is this a meaningless math question?

We can say that the one-element set {10697} has natural density zero, and we could loosely paraphrase that by saying something like "zero percent of natural numbers are equal to 10697". But yes, we should be careful about saying "choose a random integer" -- it's impossible to ass...
by skullturf
Tue Feb 15, 2011 11:28 pm UTC
Forum: Mathematics
Topic: Is this a meaningless math question?
Replies: 24
Views: 2857

Re: Is this a meaningless math question?

For a "typical" continuous random variable, the probability of any one particular value is 0. This may seem strange, but as others have said, "probability 0" does not mean "impossible to occur". Here's one attempt to make it a little more intuitive. Consider a very long...
by skullturf
Tue Feb 15, 2011 9:27 pm UTC
Forum: Mathematics
Topic: "nontrivial subgroups" (terminology question)
Replies: 7
Views: 2282

Re: "nontrivial subgroups" (terminology question)

My cursory searching seemed to indicate that it's almost standard to use "trivial subgroup" to refer only to the one-element group. But I agree with Token: the words "trivial subgroup" are close to "trivially a subgroup", in which case it's not that far-fetched to inter...
by skullturf
Tue Feb 15, 2011 6:18 pm UTC
Forum: Mathematics
Topic: "Oh no! We forgot how to say... math... stuff!"
Replies: 294
Views: 93052

Re: "Oh no! We forgot how to say... math... stuff!"

wait, I've always learned to write intervals as [a;b], [a;b[, ]a;b] or ]a;b[. With pointy part facing toward being inclusive. I always thought that was the default way of doing it... My impression is that the outward-pointing square brackets are European and the round brackets are North American?
by skullturf
Sun Feb 13, 2011 3:20 pm UTC
Forum: Mathematics
Topic: Poisson vs. Binomial
Replies: 8
Views: 3974

Re: Poisson vs. Binomial

You use the binomial when you have a given sample size. In my opinion, that's the key. Notice that on the page linked to earlier, the typist problem DOESN'T tell you how many words there are on a page, and the computer problem DOESN'T tell you how many operations the computer performs in a day. Ano...
by skullturf
Sat Feb 12, 2011 8:22 pm UTC
Forum: Mathematics
Topic: How difficult is a Real Analysis class?
Replies: 79
Views: 20784

Re: How difficult is a Real Analysis class?

My two cents on that issue: I agree that a calculus student who gets close to 100% on the computational problems can probably handle a relatively simple epsilon-delta proof, but I think there are also many decent-enough students who maybe tend to score in the B range and for whom epsilon-delta proof...
by skullturf
Fri Feb 11, 2011 10:05 pm UTC
Forum: Mathematics
Topic: "nontrivial subgroups" (terminology question)
Replies: 7
Views: 2282

"nontrivial subgroups" (terminology question)

If I have a group G and I say "consider its nontrivial subgroups" with no further explanation, what do I mean? What are the "trivial subgroups" I'm ignoring? (a) the group containing only the identity (b) the group containing only the identity, and G itself. A rather cursory Goog...
by skullturf
Fri Feb 11, 2011 7:47 pm UTC
Forum: Mathematics
Topic: Invinity * 0 =/= 0?
Replies: 11
Views: 1514

Re: Invinity * 0 =/= 0?

This is definitely a digression, but am I remembering correctly that there's a very specific context in measure theory where one might define zero times infinity to be zero?
by skullturf
Thu Feb 10, 2011 2:23 am UTC
Forum: Mathematics
Topic: Big O Questions
Replies: 16
Views: 1408

Re: Big O Questions

If all functions involved are (eventually) monotone, does it follow that any two must be comparable in a big-O sense? Or do we need a bit more than that? (And if so, then maybe if we restrict ourselves to, say, functions built out of powers, exponentials, and logs, will it then follow that any two f...
by skullturf
Wed Feb 09, 2011 10:07 pm UTC
Forum: Mathematics
Topic: Big O Questions
Replies: 16
Views: 1408

Re: Big O Questions

I think #1 is easier than #2. I'll start with a vague hint for #1: Wouldn't it be nice if the b wasn't there?
by skullturf
Wed Feb 09, 2011 9:36 pm UTC
Forum: Mathematics
Topic: The Tau Manifesto
Replies: 165
Views: 44840

Re: The Tau Manifesto

Well, I'd write A = \frac 12 \tau r^2 much like I write \int x dx = \frac 12 x^2 which makes sense to me, because area is an integral over perimeter. Also, formulas for the area of a circular sector "naturally" contain 1/2. For example, \frac12 r^2 \theta or maybe \frac12 r L where L is t...
by skullturf
Wed Feb 09, 2011 7:58 pm UTC
Forum: Mathematics
Topic: The Tau Manifesto
Replies: 165
Views: 44840

Re: The Tau Manifesto

So you want me to use (t/2)r^2 for the area of a circle and e^i(t/2) = -1? I can understand your first objection. But I have no aesthetic objection to the second formula you write. Half a turn is multiplication by negative 1! I agree that practically speaking, we're probably stuck with pi and are n...
by skullturf
Wed Feb 09, 2011 3:56 pm UTC
Forum: Mathematics
Topic: proofwiki
Replies: 11
Views: 1654

Re: proofwiki

As others have alluded to, this problem is, in a sense, not "just" about high-school-level algebraic manipulations, but it can be fruitful to look at it from a more general or abstract perspective. There's a neat article by Gowers that I think is relevant. http://www.dpmms.cam.ac.uk/~wtg10...
by skullturf
Wed Feb 09, 2011 1:57 am UTC
Forum: Mathematics
Topic: proofwiki
Replies: 11
Views: 1654

Re: proofwiki

It's late in the day, so I could be not seeing something, but it seems to me that even a specific case, like rationalizing the denominator in

1/( 2^{1/2} + 2^{1/3} )

would be nontrivial. At the very least, not the type of problem one typically does in high school algebra.
by skullturf
Tue Feb 08, 2011 5:32 pm UTC
Forum: Mathematics
Topic: Problems that seem difficult but aren't
Replies: 75
Views: 9098

Re: Problems that seem difficult but aren't

That's a neat proof, Jyrki. But personally, my favorite so far is mike-l's with the goodbyes and handshakes. My proof was more boring. First I conjectured the correct answer (straightforward enough to guess it by playing with specific small cases), and then I made an argument by strong induction rel...
by skullturf
Tue Feb 08, 2011 3:07 pm UTC
Forum: Mathematics
Topic: Problems that seem difficult but aren't
Replies: 75
Views: 9098

Re: Problems that seem difficult but aren't

This problem probably is pretty difficult. But there is a beautiful solution. The sort of solution when you know it, it becomes hard to imagine how it can be done differently. You have a pile of N stones. You do the following: you take a pile and separate it into two smaller piles, multiply the num...
by skullturf
Mon Feb 07, 2011 12:50 pm UTC
Forum: Mathematics
Topic: Question about a solid of revolution
Replies: 7
Views: 1220

Re: Question about a solid of revolution

To give a complete general answer to this question might be subtle. Naturally, I'd recommend drawing a picture in each case. But I'm pretty sure that in many natural cases, there's no need to treat the situation differently or break the problem up into pieces. For example, let R be the region bounde...
by skullturf
Fri Feb 04, 2011 3:33 pm UTC
Forum: Mathematics
Topic: Fun Math Riddles
Replies: 54
Views: 6420

Re: Fun Math Riddles

I never had a problem with the St Ives riddle. I thought it was really obvious. And I don't think it's akin to the comic posted above. In the comic, it's using language terribly to mean something (bad communication), while the St Ives riddle is fine in communication. My issue may have been more wit...
by skullturf
Thu Feb 03, 2011 10:47 pm UTC
Forum: Mathematics
Topic: Fun Math Riddles
Replies: 54
Views: 6420

Re: Fun Math Riddles

When I encountered the St. Ives riddle as a child, the answer bugged me. (I suppose I'll put the rest of this post as a spoiler, even though "it's an old nursery rhyme, come on.") The answer I was given as a kid was: "Only one. The rest were all going away from St. Ives." I didn&...
by skullturf
Thu Feb 03, 2011 9:28 pm UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 494633

Re: Favorite math jokes

Serge Lang is known for being an incredibly prolific author of textbooks, despite being only one person. I think the idea of the joke is that the members of Bourbaki were like, "Geez, even though we're a collection of people, we can't compete with that guy Lang!"
by skullturf
Thu Feb 03, 2011 8:58 pm UTC
Forum: Mathematics
Topic: Probability (marksmen hitting a target)
Replies: 10
Views: 2639

Re: Probability (marksmen hitting a target)

Failure to interpret something as conditional probability is perhaps also at play in the boy-girl problem. If I ask "At least one of Pat's two children is a girl. What's the probability that the two children are different genders?" I suppose it could be argued that I'm asking "Here's ...
by skullturf
Thu Feb 03, 2011 6:41 pm UTC
Forum: Mathematics
Topic: Probability (marksmen hitting a target)
Replies: 10
Views: 2639

Re: Probability (marksmen hitting a target)

Seconded. I instantly interpreted it as a conditional probability problem, which would mean your instructor was wrong. True, the question maybe doesn't contain the words "given" or "conditional", but I think it's pretty standard to interpret problems of the form "[Descriptio...
by skullturf
Wed Feb 02, 2011 10:04 pm UTC
Forum: Mathematics
Topic: Fun Math Riddles
Replies: 54
Views: 6420

Re: Fun Math Riddles

If you have the 100% version for the coins, and if you describe your strategy in just the right way, the "right" generalization to the dice may come to you. (It might also help to reread Jyrki's comment above.)
by skullturf
Wed Feb 02, 2011 6:37 am UTC
Forum: Mathematics
Topic: Fun Math Riddles
Replies: 54
Views: 6420

Re: Fun Math Riddles

...For the coins example, there are only 4 deterministic strategies for each player. Run through them, see what happens. That's actually how I got it, rather than having a "flash of insight" right away. I thought: Base your guess on previous tosses? doesn't seem helpful Randomize your gue...
by skullturf
Wed Feb 02, 2011 5:43 am UTC
Forum: Mathematics
Topic: Help! Make x the subject!
Replies: 8
Views: 1999

Re: Help! Make x the subject!

If you can get away with fewer cases, it's tidier. Maybe try to do as much as you can in the general case -- when are you forced to consider different cases?

(edit: ninja'd)
by skullturf
Tue Feb 01, 2011 8:11 pm UTC
Forum: Mathematics
Topic: Moving around the 8 cardinal directions
Replies: 9
Views: 1359

Re: Moving around the 8 cardinal directions

(I'm envisioning your vectors as column vectors.)

To rotate 45 degrees counterclockwise, you could left-multiply by the matrix whose rows are [1 -1] and [1 1], and then change any 2's to 1's and any -2's to -1's.
by skullturf
Tue Feb 01, 2011 3:47 pm UTC
Forum: Mathematics
Topic: Fun Math Riddles
Replies: 54
Views: 6420

Re: Fun Math Riddles

This was really confounding me for a while last night, but I think I know the answer now. And yeah, I figured out the coin example first, then thought about how to generalize that to the dice.

Nice puzzle.
by skullturf
Mon Jan 31, 2011 1:23 am UTC
Forum: Mathematics
Topic: Probability (chances of winning a game)
Replies: 6
Views: 1282

Re: Probability Problem

Also, suppose the number of people is a bit larger -- say 10. If they played 10 games, it seems unlikely that we have 10 different winners. (If, say, the first 7 games had 7 different winners -- which certainly would not always happen -- then to "hit everyone", you'd need to hit a 30% chan...
by skullturf
Thu Jan 27, 2011 6:44 pm UTC
Forum: Mathematics
Topic: Does this series converge!?
Replies: 32
Views: 4369

Re: Does this series converge!?

As already noted, the best way to see that this series converges is to use the definition. If you feel some need to have it covered by one of the standard "tests", just note that it is a geometric series with first term 0 and ratio 0. Heh, that's kind of funny, because I've told students ...
by skullturf
Fri Jan 21, 2011 7:15 pm UTC
Forum: Mathematics
Topic: Measuring performance in a league table
Replies: 5
Views: 609

Re: Measuring performance in a league table

If the Elo system gives you a lower rating than you would have liked, you can say "Don't bring me down."

(Sorry.)
by skullturf
Fri Jan 21, 2011 7:08 pm UTC
Forum: Mathematics
Topic: N sided regular solids
Replies: 21
Views: 1887

Re: N sided regular solids

Since there only are the five platonic solids (stopping at 20 faces), they don't approach a sphere in the sense of having the number of faces approach infinity.

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