Search found 555 matches

by skullturf
Thu Dec 02, 2010 3:27 pm UTC
Forum: Mathematics
Topic: Fermi-type problem: coins and probability
Replies: 3
Views: 681

Fermi-type problem: coins and probability

I live in Delaware. I always have some coins in my possession -- the amount and composition fluctuates as I go shopping, do laundry, use vending machines, etc. About how often do I own a coin that was once owned by Hillary Clinton? (Obviously this isn't a pure math problem, but I am interested to se...
by skullturf
Thu Dec 02, 2010 3:02 pm UTC
Forum: Mathematics
Topic: Forbidden subwords -- logic vs intuition
Replies: 13
Views: 1859

Re: Forbidden subwords -- logic vs intuition

The point is that the situation in which you know that one child is a boy, but not which one, never comes up in real life. People do not go around saying "I have two children and at least one is a girl." If making such statements were a daily ritual among people with two children, at leas...
by skullturf
Thu Dec 02, 2010 3:58 am UTC
Forum: Mathematics
Topic: Forbidden subwords -- logic vs intuition
Replies: 13
Views: 1859

Re: Forbidden subwords -- logic vs intuition

Yeah, that's similar to how I eventually managed to convince myself. Here's an analogy I thought of: Suppose your city has exactly the same number of cats as dogs. So if you throw a dart at a random spot in your city, you're just as likely to hit a cat as a dog. (Let's pretend cats and dogs are the ...
by skullturf
Thu Dec 02, 2010 2:05 am UTC
Forum: Mathematics
Topic: Forbidden subwords -- logic vs intuition
Replies: 13
Views: 1859

Forbidden subwords -- logic vs intuition

One thing in mathematics that can be frustrating, but also very interesting, goes something as follows. You're trying to decide which of two competing statements, P or Q, is true. At first, your intuition tells you the correct answer is probably P. Later, you discover a proof (yours or someone else'...
by skullturf
Thu Dec 02, 2010 1:31 am UTC
Forum: Mathematics
Topic: Forming a proof about a triangle
Replies: 10
Views: 1674

Re: Forming a proof about a triangle

but I'm taking a course on discrete math that's supposed to serve as an introduction to proofs, among other things, and all the direct proofs I've seen, and the explanation in my text, are such that you take one expression (not an equality) and transform it to another. So in this case, I would take...
by skullturf
Wed Dec 01, 2010 3:39 am UTC
Forum: Mathematics
Topic: Dice probability question (some number appears at most once)
Replies: 6
Views: 777

Re: Dice probability question (some number appears at most o

It's not homework. I'm not a student. My formula for the probability that at least one number comes up at most once is: sum((-1)^(m-1)*binomial(k,m)*sum(binomial(m,j)*(k-m)^(n-j)*product(n-i,i=0..j-1),j=0..m),m=1..k)/k^n I'm using Maple syntax. As in the original post, n is the number of rolls and k...
by skullturf
Tue Nov 30, 2010 6:40 pm UTC
Forum: Mathematics
Topic: To what does this series evaluate?
Replies: 12
Views: 2389

Re: To what does this series evaluate?

I wonder if it's possible to prove that \sum_{k=1}^{\infty} k^{-k} is irrational. An admittedly vague meta-mathematical guideline is that sums that converge "too fast" are often irrational or even transcendental. The denominator k^k is of course pretty large. There's a simple proof that e ...
by skullturf
Mon Nov 29, 2010 2:19 am UTC
Forum: Mathematics
Topic: Dice probability question (some number appears at most once)
Replies: 6
Views: 777

Re: Dice probability question (some number appears at most o

This is 1 - P(each side shows up at least twice). Which is actually pretty easy, I think. You just do counting: the total number of possibilities is k^n, while the number of ways to have each side at least twice is, I believe, k^(n-2*k) * n!/((n-2*k)!*(2^k)). 2*k rol...
by skullturf
Sun Nov 28, 2010 11:16 pm UTC
Forum: Mathematics
Topic: A little homework help (mean value theorem)
Replies: 11
Views: 890

Re: A little homework help (mean value theorem)

Doesn't the MVT apply to all functions which are "curved" and have a continuous line going between point A and B. Or am I missing something? Almost. For an informal description of the types of functions to which MVT can be applied, one might say something like "the graph of the funct...
by skullturf
Sun Nov 28, 2010 10:10 pm UTC
Forum: Mathematics
Topic: A little homework help (mean value theorem)
Replies: 11
Views: 890

Re: A little homework help (mean value theorem)

Those look like the right calculations for finding c. Note that if we're being precise, there's a little bit more to the question: your first task is to show that f satisfies the hypotheses of the mean value theorem. If you refer back to the statement of the mean value theorem that you learned, it'l...
by skullturf
Sun Nov 28, 2010 5:48 pm UTC
Forum: Mathematics
Topic: Dice probability question (some number appears at most once)
Replies: 6
Views: 777

Dice probability question (some number appears at most once)

If a fair die, having k sides, is rolled n times, what's the probability that there's at least one side of the die that comes up at most once? As a starting point, consider k = 6. For instance, say you roll 12 times. If you get 314156226354, then there's no side that came up at most once, but i...
by skullturf
Sun Nov 28, 2010 2:28 pm UTC
Forum: Mathematics
Topic: Dumb arithmetic errors
Replies: 5
Views: 1092

Re: Dumb arithmatic errors

One piece of advice that I, and others, have told students: If you realize somehow that you must have made a mistake somewhere (because your final answer makes no sense, or because it disagrees with an authority), then work it out again , as opposed to reading your (now known to be faulty) work and ...
by skullturf
Sun Nov 28, 2010 2:24 pm UTC
Forum: Mathematics
Topic: Do these polynomials have names?
Replies: 5
Views: 1007

Re: Do these polynomials have names?

Since the G_n are essentially Chebyshev polynomials with a change of variable, wouldn't they actually be orthogonal if you choose the interval and the weight function correctly? Edit: In general, if a sequence of polynomials is generated by the "right" kind of recurrence, must it be an ort...
by skullturf
Sun Nov 28, 2010 3:08 am UTC
Forum: Mathematics
Topic: Do these polynomials have names?
Replies: 5
Views: 1007

Do these polynomials have names?

(1) Is there an equivalent of the online encyclopedia of integer sequences for polynomials? In particular, for polynomials in one variable where the nth polynomial in the sequence has degree n? (2) Failing that, does anyone happen to recognize the following sequences of polynomials, or the recurrenc...
by skullturf
Wed Nov 24, 2010 12:57 am UTC
Forum: Mathematics
Topic: Alternate Definitions of the Gamma Function
Replies: 9
Views: 1157

Re: Alternate Definitions of the Gamma Function

Does your text anywhere mention that Gamma is the unique analytic function which satisfies the recurrence formula and \Gamma(1)=1 ? Its true, even if it doesn't prove it, though I suspect it does. Strictly speaking, is this correct as written, if you don't also say something about log-conve...
by skullturf
Tue Nov 23, 2010 4:01 pm UTC
Forum: Mathematics
Topic: Have You Been Taught Things Which Aren't True?
Replies: 97
Views: 16335

Re: Have You Been Taught Things Which Aren't True?

Darryl wrote:I talked to someone whose college professor taught her that you include 1 in prime factorizations


How many times? :D
by skullturf
Tue Nov 23, 2010 3:05 am UTC
Forum: Mathematics
Topic: Have You Been Taught Things Which Aren't True?
Replies: 97
Views: 16335

Re: Have You Been Taught Things Which Aren't True?

I actually use ÷ when I teach university students, believe it or not! I often use it when I teach the ratio test, and I have one moderately large fraction divided by another. In such situations, writing one large fraction above another sometimes looks too cluttered, and putting a / between them is s...
by skullturf
Thu Nov 18, 2010 4:12 am UTC
Forum: Mathematics
Topic: Have You Been Taught Things Which Aren't True?
Replies: 97
Views: 16335

Re: Have You Been Taught Things Which Aren't True?

and the other is that there is an interesting set of numbers called "whole numbers" that differ from natural numbers in that 0 is not a natural number. I was under the impression that the term "natural numbers" was a notorious example of a term where there isn't much of a consen...
by skullturf
Wed Nov 17, 2010 3:18 pm UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

cameron432 wrote:
lim ( (nlog(n-1)) / (nlog(n)) )

lim ( (log(n)/log(1)) / (log(n)) )




Also, log(a-b) is not the same as log(a)/log(b), if that's one thing you were trying to do there.
by skullturf
Wed Nov 17, 2010 2:23 am UTC
Forum: Mathematics
Topic: Multiplying by Zero. Need help understanding.
Replies: 20
Views: 3028

Re: Multiplying by Zero. Need help understanding.

OK, that makes way more sense. I learn through language so if I don't have something presented in a certain way, I may not pick up the whole meaning. But your explanation makes a lot of sense. Cool, I'm glad. :) And you're certainly not alone in your confusion. I remember times when one of my stude...
by skullturf
Wed Nov 17, 2010 2:12 am UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

@ Eebster -- That's fascinating! Thanks!

I was wondering whether or not the "straightforward" way, where you just chug through using your Calc I techniques, would work here at all. It looks like it does, but very few humans would have the patience to go through the steps.
by skullturf
Wed Nov 17, 2010 2:07 am UTC
Forum: Mathematics
Topic: Multiplying by Zero. Need help understanding.
Replies: 20
Views: 3028

Re: Multiplying by Zero. Need help understanding.

It has just always bothered me that if you have five items and multiply them zero times, that there should now be zero items. I still need a little clarification if thats ok. Zero is NOT the number of times you're performing the action. You are multiplying the five items by the NUMBER zero. It's li...
by skullturf
Wed Nov 17, 2010 1:54 am UTC
Forum: Mathematics
Topic: Multiplying by Zero. Need help understanding.
Replies: 20
Views: 3028

Re: Multiplying by Zero. Need help understanding.

Different people's intuitions will differ, but what always made the most sense for me with these types of questions was just continuing the pattern. If you believe 5 times 3 is 15 5 times 2 is 10 5 times 1 is 5 then surely 5 times 0 being 0 continues that pattern. OK, so we all know basic math rules...
by skullturf
Wed Nov 17, 2010 12:44 am UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

@ ++$_ I like it! There may be various ways to make precise the idea that log log n - log log (n-1) is "basically" the derivative of log log n. In a sense, you're comparing the slope of the secant line with the slope of the tangent line, so one could claim you're using the "same"...
by skullturf
Tue Nov 16, 2010 11:57 pm UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

Come to think of it, a proper proof of the limit of log n (n+1) as n-> infinity on its own (without the surrounding ^n) might be interesting. Hmm... Could probably throw taylor series at it, but ehh... That one succumbs to L'Hopital's rule very well. I initially thought my problem could probably be...
by skullturf
Tue Nov 16, 2010 10:43 pm UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

Minor quibble regarding skeptical scientist's answer: If we allow n to take on real values, then we can't prove (1-e)^n > 1-ne "by induction". However, we can still say that if k is a real number > 1 and if e is a real number between 0 and 1, then (1-e)^k > 1-ke. This can be done similarly...
by skullturf
Tue Nov 16, 2010 9:24 pm UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

You said real numbers, not real positive numbers. True, but I also said limit as n approaches infinity. skeptical scientist did something similar to what I did. My estimation of the difficulty of the question was similar, too -- I think it's probably too tricky for a typical first year calculus stu...
by skullturf
Tue Nov 16, 2010 8:52 pm UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

@ mike-l: Now that I think about it, I agree with you. L'Hopital's rule, applied once, implies that the answer (if it exists) is the same as the limit of [log(n-1)/log(n)]^(n-1). L'Hopital's rule applied k times implies that the answer, if it exists, is the same as the limit of [log(n-1)/log(n)]^(n-...
by skullturf
Tue Nov 16, 2010 8:00 pm UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

That looks correct, and is simpler than what I did. Very good.

Okay, say we modify the problem slightly: say n is a real number instead of being restricted to a positive integer.
by skullturf
Tue Nov 16, 2010 6:18 pm UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Re: Tricky limit problem for a calculus course

This is not homework. In fact, I am a college-level math instructor who's composing a test today, and I'm not including this particular problem because it appears too tricky. However, it's always possible I missed a clever shortcut. The problem: Find the limit of ( log(n-1)/log(n) )^n as n approach...
by skullturf
Tue Nov 16, 2010 5:50 pm UTC
Forum: Mathematics
Topic: Tricky limit problem for a calculus course
Replies: 32
Views: 2886

Tricky limit problem for a calculus course

This is not homework. In fact, I am a college-level math instructor who's composing a test today, and I'm not including this particular problem because it appears too tricky. However, it's always possible I missed a clever shortcut. The problem: Find the limit of ( log(n-1)/log(n) )^n as n approache...
by skullturf
Wed Jul 02, 2008 5:35 pm UTC
Forum: General
Topic: What is your name an anagram of?
Replies: 384
Views: 53850

Re: What is your name an anagram of?

My real name, including my middle name, can become

"Disc drive admirer"
by skullturf
Fri Jun 27, 2008 5:45 pm UTC
Forum: General
Topic: "idiot test" aka "read everything before doing anything"
Replies: 88
Views: 69458

Re: "idiot test" aka "read everything before doing anything"

To an extent I agree with those of you who cited xkcd comic #169. At the same time, though, I can see how being faced with a test like this in elementary school or junior high can be a neat little lesson. My suspicion is that when many people write the test, the first instruction (read everything be...
by skullturf
Fri Jun 27, 2008 4:13 pm UTC
Forum: General
Topic: "idiot test" aka "read everything before doing anything"
Replies: 88
Views: 69458

"idiot test" aka "read everything before doing anything"

Hi everyone, Most of you have probably come across something that might have been called a "reading comprehension test" or "idiot test" or something like that. As a newbie, I'm not supposed to post links here, so I've copied and pasted an example of the kind of test I mean. Maybe...
by skullturf
Thu Jun 26, 2008 4:07 pm UTC
Forum: General
Topic: Who the dickens are you?
Replies: 10922
Views: 2381421

Re: INTRO THREAD THE THIRD

Hi everyone. I'm 34 and I live in Toronto. I've been reading xkcd for a long time, and I registered a while ago, but yesterday I posted in the forums for the first time. My post then disappeared. What's going on? Was it because I didn't post in the intro thread first? Can my post be resurrected, or ...

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