Search found 555 matches

by skullturf
Mon Apr 25, 2011 9:33 pm UTC
Forum: Mathematics
Topic: What is this law/identity/thing called?
Replies: 10
Views: 1906

Re: What is this law/identity/thing called?

Plus, I think the fact that every vector space has a basis might actually require the axiom of choice.
by skullturf
Mon Apr 25, 2011 4:16 pm UTC
Forum: Mathematics
Topic: What is this law/identity/thing called?
Replies: 10
Views: 1906

Re: What is this law/identity/thing called?

Hey, this thread reminds me: Is there a special name for the fact that in a vector space, we have c(d v ) = (cd) v where v is a vector, and c and d are scalars? Probably, people just call this "associativity". But note that technically, there are two different operations here (multiplying ...
by skullturf
Sat Apr 23, 2011 12:07 am UTC
Forum: Mathematics
Topic: Compound Interest
Replies: 14
Views: 2337

Re: Compound Interest

My immediate reaction was (1) I don't like the wording of the question (too "talky", which can make things imprecise or ambiguous), and (2) I interpret the answer to be

Spoiler:
r. The variable r determines the doubling time; the variable t is the doubling time (if e^{rt}=2).
by skullturf
Fri Apr 22, 2011 4:55 am UTC
Forum: Mathematics
Topic: Math Olympiad Problems
Replies: 8
Views: 2045

Re: Math Olympiad Problems

The way that I did #2 was to do long division! Draw the usual picture for long division, where you're dividing 5 into a multi-digit number (number of digits to be determined) whose digits begin a,b,c,... The result of the division has to be a number whose digits are b,c,... There are only so many po...
by skullturf
Thu Apr 21, 2011 4:30 pm UTC
Forum: Mathematics
Topic: Math Olympiad Problems
Replies: 8
Views: 2045

Re: Math Olympiad Problems

I don't know where and how to start with this kind of problem As general advice, play around the question a bit, and consider specific examples. For #1, what are some examples of perfect squares where you can do this? (i.e. delete the last two digits and get a perfect square.) Perhaps even slightly...
by skullturf
Wed Apr 20, 2011 12:44 am UTC
Forum: Mathematics
Topic: Little probability problem (additional tries for successes)
Replies: 10
Views: 1601

Re: Little probability problem (additional tries for success

Here's another way to approach it: You are hoping to get 12 successes, in which case you are going to get 11 "extra" attempts Is that correct? It's late in the day here, but I thought the OP meant that for each success in the original 20 rolls, you get one more roll. So for example, it co...
by skullturf
Tue Apr 19, 2011 2:27 am UTC
Forum: Mathematics
Topic: Hyperbola problem help.
Replies: 8
Views: 1162

Re: Hyperbola problem help.

chewey wrote:Yes I know that the distance between any two points would be square root((x-a)^2+(y-b)^2).


And similarly, the length of any vector (u,v) is sqrt(u^2 + v^2).
by skullturf
Tue Apr 19, 2011 1:30 am UTC
Forum: Mathematics
Topic: Hyperbola problem help.
Replies: 8
Views: 1162

Re: Hyperbola problem help.

Do you know an algebraic expression for the distance between two points, and/or for the length of a vector?
by skullturf
Tue Apr 19, 2011 1:27 am UTC
Forum: Mathematics
Topic: If you learned statistics would you never buy insurance?
Replies: 92
Views: 10993

Re: If you learned statistics would you never buy insurance?

OP: Would you be happy with the following toy example as an illustration? You live in a town of 10,000 people. Each year, each resident of the town has a 1 in 10,000 chance of experiencing a serious accident or disaster that costs that person 100,000 dollars. (It's as though once a year, "Fate&...
by skullturf
Mon Apr 18, 2011 1:59 am UTC
Forum: Mathematics
Topic: Hyperbola problem help.
Replies: 8
Views: 1162

Re: Hyperbola problem help.

Let L be the line x + y = 1. Let P = (x,y) be a general point whose distance from the origin is twice its distance from L. The tricky part would probably be coming up with a non-messy expression for the distance from P to L. Let (a,b) be the point on L closest to P. POSSIBLY HELPFUL OBSERVATION: Sin...
by skullturf
Sun Apr 17, 2011 11:35 pm UTC
Forum: Mathematics
Topic: Guess numbers by knowing their sum and product
Replies: 11
Views: 1885

Re: Guess numbers by knowing their sum and product

{2,6,6} and {3,3,8} have the same sum and the same product, but I don't know if you can find an example with no repeated elements. ...and after playing around for a while, I found {4,9,10} and {5,6,12}. How I found it: Inspired by jaap, I looked for solutions of the form {ax,by,cz} and {bx,cy,az}. F...
by skullturf
Sun Apr 17, 2011 9:04 pm UTC
Forum: Mathematics
Topic: Basic sin(x)cos(x)dx integral question
Replies: 4
Views: 2828

Re: Basic sin(x)cos(x)dx integral question

The derivative of cos is -sin, not sin.
by skullturf
Sun Apr 17, 2011 9:02 pm UTC
Forum: Mathematics
Topic: Free Will & Reimann Hypothesis?
Replies: 11
Views: 1780

Re: Free Will & Reimann Hypothesis?

...Also, humans are by nature finite creatures, so our set of beliefs at any given moment (or over the course of our lives) is finite. Just thought I'd have a wonder about this: I believe "n+1 is greater than n" with any integer in place of n. This means that I have an unbounded set of be...
by skullturf
Sun Apr 17, 2011 12:41 am UTC
Forum: Logic Puzzles
Topic: The surprise exam paradox
Replies: 217
Views: 66494

Re: The surprise exam paradox

I mean, clearly one can make a smaller set of premises that's obviously inconsistent. If there is only one day in the problem, and our premises include "You know for sure that there will be an exam tomorrow" and "You know for sure that you do not know what day the exam will be."...
by skullturf
Fri Apr 15, 2011 2:19 pm UTC
Forum: Logic Puzzles
Topic: The surprise exam paradox
Replies: 217
Views: 66494

Re: The surprise exam paradox

Certainly, induction in general is a valid method of proof. Induction can sometimes lead to counterintuitive results (cf. the blue eyes on an island problem), but with problems like that one or the surprise exam one, if you state your premises carefully, you can use induction to prove *something*. (...
by skullturf
Thu Apr 14, 2011 5:01 pm UTC
Forum: Mathematics
Topic: |i|
Replies: 34
Views: 5864

Re: |i|

I'm still unsure why anyone would object to calling |a+bi| the "absolute value" of a+bi.
by skullturf
Wed Apr 13, 2011 10:29 pm UTC
Forum: Mathematics
Topic: I don't get the Riemann hypothesis
Replies: 16
Views: 3181

Re: I don't get the Riemann hypothesis

Suppose we define zeta(s) to be the sum of 1/n^s over all positive integers n, and suppose for the moment that all you know are real numbers. So for now, you only know zeta(s) as a function of a real variable s. Then zeta(s) is defined for s>1 (because the series converges), and undefined for s <= 1...
by skullturf
Wed Apr 13, 2011 3:54 pm UTC
Forum: Mathematics
Topic: I don't get the Riemann hypothesis
Replies: 16
Views: 3181

Re: I don't get the Riemann hypothesis

This may possibly be helpful to people trying to get an informal idea of what the Riemann Hypothesis is about. http://bentilly.blogspot.com/2011/03/elementary-explanation-of-riemann.html (Irrelevant side note: the author went to my high school many years ago, and is the brother of poker player Jenni...
by skullturf
Tue Apr 12, 2011 2:17 pm UTC
Forum: Mathematics
Topic: Substitution
Replies: 18
Views: 2743

Re: Substitution

People seem to be taught that certain letters/variables/pronumerals(I hate this word) have specific meaning (for example in the equation of the line mx + c, c is the intercept rather than the intercept is c), instead of a generic thing which we can do various things to. Heh, case in point: I was ta...
by skullturf
Tue Apr 12, 2011 4:44 am UTC
Forum: Mathematics
Topic: Substitution
Replies: 18
Views: 2743

Re: Substitution

It's a good question, and I wish I had a good answer. I've encountered students who, when given f(x)=x^2, will mistakenly write f(x+h) = x^2+h as opposed to f(x+h)^2. I find it hard to come up with a good way to explain why it's wrong. I've sometimes said things like "f of blah is blah squared&...
by skullturf
Tue Apr 12, 2011 3:36 am UTC
Forum: Mathematics
Topic: Linearity defined
Replies: 11
Views: 1680

Re: Linearity defined

What I want to know is that are those properties of an integral (or any transformation) true BECAUSE it is linear, or it is linear BECAUSE those conditions are true? I would say the latter. I was taught that some transformation T is said to be linear if T(f+g) = T(f) + T(g) AND T(cf) = c*T(f) Exact...
by skullturf
Mon Apr 11, 2011 8:21 pm UTC
Forum: Logic Puzzles
Topic: My write-up of the "Blue Eyes" solution (SPOILER A
Replies: 1368
Views: 424629

Re: My write-up of the "Blue Eyes" solution (SPOILER A

This may be helpful: Again, say Alice, Bob, Carol, and Doug are all islanders with blue eyes. The reality of the island is that there are 100 blue-eyed people and 100 brown-eyed people. In Alice's mental model of the island, there are 100 brown-eyed people, 99 blue-eyed people, and one person of unk...
by skullturf
Mon Apr 11, 2011 4:50 am UTC
Forum: Mathematics
Topic: Getting my undergrad in math. How about you?
Replies: 51
Views: 8498

Re: Getting my undergrad in math. How about you?

Me:

Age: 30s
Education: PhD in math
What I'm doing now: Assistant professor in mid-Atlantic US
by skullturf
Mon Apr 11, 2011 12:42 am UTC
Forum: Mathematics
Topic: Calc I Problem
Replies: 4
Views: 758

Re: Calc I Problem

Nice problem. I tried a blind alley or two before I hit upon something that worked. Here's what I did. (Note: t = theta.) Multiply the integrand by (sin t)/(sin t) = (sin t)/sqrt(sin^2 t). Next, replace 3 + sin^2 t with 4 - cos^2 t. Then, the substitution u = cos t is...
by skullturf
Sun Apr 10, 2011 11:10 pm UTC
Forum: Mathematics
Topic: Writing a Calculus Textbook
Replies: 6
Views: 2205

Re: Writing a Calculus Textbook

Something possibly relevant: Serge Lang once wrote a calculus textbook that differed from the typical textbook, in what sounds like similar ways to what you have in mind. His book was shorter and more conceptually oriented than the typical calculus textbook, and eschewed lengthy lists of problems. h...
by skullturf
Sat Apr 09, 2011 3:21 pm UTC
Forum: Mathematics
Topic: Help with equation
Replies: 4
Views: 1229

Re: Help with equation

Well, we do whatever's in parentheses first. So it's 48/2*12 or 48/2(12) I believe most verbal explanations of order of operations will say something like "perform multiplication and division from left to right in the order that they occur". Following those directions in the above, you wou...
by skullturf
Fri Apr 08, 2011 12:55 am UTC
Forum: Mathematics
Topic: A series whose convergence or divergence is unknown
Replies: 3
Views: 1037

Re: A series whose convergence or divergence is unknown

Hm, it seems to me that it would be relatively easy to come up with a series that no one currently knows the convergence of. Try it. I suspect non-artificial examples are harder to find than you'd think, but I'd welcome further examples. Also, when sin is close to 0, I'd say n^3*sin(n)^2 is close t...
by skullturf
Thu Apr 07, 2011 9:51 pm UTC
Forum: Logic Puzzles
Topic: Troll math: x^x^x^...=2=4
Replies: 10
Views: 4563

Re: Troll math: x^x^x^...=2=4

Well, (different starting value) and (similar recurrence) = different sequence.

Certainly, different sequences can have different limits.

If you specify one particular sequence of real numbers, it has at most one limit.
by skullturf
Thu Apr 07, 2011 9:00 pm UTC
Forum: Logic Puzzles
Topic: Troll math: x^x^x^...=2=4
Replies: 10
Views: 4563

Re: Troll math: x^x^x^...=2=4

That isn't (x^a)^b, it is x^(a^b). Correct. That's the standard way to interpret a "tower" x^a^b. Furthermore, the standard way to interpret an infinite such tower would be as the limit of the sequence x, x^x, x^x^x, x^x^x^x, ... if that limit exists. You'd expect that if x is "large...
by skullturf
Thu Apr 07, 2011 6:50 pm UTC
Forum: Mathematics
Topic: A series whose convergence or divergence is unknown
Replies: 3
Views: 1037

A series whose convergence or divergence is unknown

I was surprised recently to learn that one can give a specific, explicit example of an infinite series ("built" out of rational functions, trig functions, etc) such that nobody currently knows whether the series converges or diverges. Namely, the sum of the reciprocals of n^3*sin(n)^2. (Th...
by skullturf
Thu Apr 07, 2011 4:17 pm UTC
Forum: Mathematics
Topic: Would this ever end? (infinities)
Replies: 6
Views: 1185

Re: Would this ever end? (infinities)

Finding the exact probability that it terminates (assuming we're at "step 0") is an interesting exercise. Certainly, there is a nonzero probability that it terminates, because it's possible to start with a run of 10 tails. The probability of that is small, but is definitely NOT zero -- in ...
by skullturf
Thu Apr 07, 2011 3:02 pm UTC
Forum: Mathematics
Topic: a million positive divisors
Replies: 22
Views: 3790

Re: a million positive divisors

Interestingly, jaap's reply to Sizik shows that there's something subtle at play here. To create an integer with exactly 1000000 divisors, we first factor 1000000. The "obvious" way to factor it is 5 5 5 5 5 5 2 2 2 2 2 2. So certainly Sizik's number is an example of a number with exactly ...
by skullturf
Thu Apr 07, 2011 3:52 am UTC
Forum: Mathematics
Topic: What will I be expected to know?
Replies: 3
Views: 990

Re: What will I be expected to know?

Yesila wrote:Also know your trig functions and some of their identities.


ESPECIALLY the definitions of all trig functions, and the Pythagorean identities.

Other identities can be useful too, but the above are the really truly important ones.
by skullturf
Wed Apr 06, 2011 3:43 am UTC
Forum: Logic Puzzles
Topic: My write-up of the "Blue Eyes" solution (SPOILER A
Replies: 1368
Views: 424629

Re: My write-up of the "Blue Eyes" solution (SPOILER A

I hesitated before adding to this thread. On the one hand, it's 23 pages long and the answer is pretty well established, but on the other hand, the very fact that it's 23 pages long and people continue to ask questions is evidence that more input could be useful -- sometimes paraphrasing things in a...
by skullturf
Wed Apr 06, 2011 2:43 am UTC
Forum: Logic Puzzles
Topic: Box Paradox
Replies: 13
Views: 2821

Re: Box Paradox

So, I WAS convinced, but then I thought of it this way, and now I am not so sure again. If I reach into a box, and there are 3 gold coins in there, and 1 silver coin, and I pull out a gold coin, the odds that the next coin I draw is gold, is 2/3. Are you talking about a situation where these 4 coin...
by skullturf
Tue Apr 05, 2011 1:49 am UTC
Forum: Logic Puzzles
Topic: Box Paradox
Replies: 13
Views: 2821

Re: Box Paradox

...but AFTER you ahve already drawn the coin, isnt the chance just 50/50? It matters how you got there. Loosely speaking, you don't get to ignore what happened in the past, just because it happened in the past. In a sense, every (discrete) probability problem is really a proportions problem. If you...
by skullturf
Mon Apr 04, 2011 9:38 pm UTC
Forum: Logic Puzzles
Topic: Partitioning A Formula
Replies: 10
Views: 2708

Re: Partitioning A Formula

I was sufficiently interested in this combinatorial problem to start writing up some general notes on it and to explore connections to other studied problems. I may post later summarizing my findings. Edit: I didn't bother writing up my own notes, but here are the connections I noticed. A "sche...
by skullturf
Mon Apr 04, 2011 8:02 pm UTC
Forum: Mathematics
Topic: Infinite Series Fun, Who's Excited?!
Replies: 3
Views: 872

Re: Infinite Series Fun, Who's Excited?!

Say we have convergent series an which converges to A and bn which converges to B. Is there such a case that the series an*bn would not converge to A*B? I teach calculus, so I may be overly sensitive on this point, but I strongly recommend against writing things like "the series a_n". The...
by skullturf
Sun Apr 03, 2011 10:14 pm UTC
Forum: Logic Puzzles
Topic: Sharing a pie between three persons?
Replies: 4
Views: 2145

Re: Sharing a pie between three persons?

Here's something I read a long time ago; I don't remember where. (Possibly a puzzle column at the back of Discover magazine.) If you don't mind your pie looking unpretty: Gradually slice off very small amounts of pie, placing them on a plate. As the amount of pie on the plate gradually increases, an...
by skullturf
Sun Apr 03, 2011 3:25 am UTC
Forum: Logic Puzzles
Topic: Partitioning A Formula
Replies: 10
Views: 2708

Re: Partitioning A Formula

I like the purely combinatorial problem implicit in the original post, and it might be worth exploring ways to formulate the "right" general question. Let the "pieces of the formula" be the integers 1,2,...,n. Suppose the number of sons is k. Definition: By a "scheme", ...

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