Search found 555 matches

Tue Aug 11, 2015 2:45 pm UTC
Forum: Mathematics
Topic: Prove by induction 3^n < (n+2)!
Replies: 12
Views: 3293

Re: Prove by induction 3^n < (n+2)!

The key is that from a <= b and b < c, we can conclude a < c. We only need *one* of the inequalities to be strict.
Thu Jul 09, 2015 2:24 am UTC
Forum: Mathematics
Topic: Help with a lemma about rationals and coprime integers
Replies: 4
Views: 1963

Help with a lemma about rationals and coprime integers

For something I'm working on, I would like the following lemma to be true. I'm reasonably certain that it is true, but it's one of those things where at first glance, you think "That's intuitive", but when you try to write down a proof, you realize that there are messy details to wade thro...
Wed Jul 08, 2015 4:37 pm UTC
Forum: Mathematics
Topic: Question about calculation of the limit Indeterminate
Replies: 3
Views: 1703

Re: Question about calculation of the limit Indeterminate

0 divided by 0 is a famous indeterminate form, but 0 times 0 is not indeterminate.

If you have x times sin(x), that's of the form f(x) times g(x), where both f and g are approaching 0 as x approaches 0. Hence the product x*sin(x) also approaches 0.
Thu Mar 19, 2015 5:01 pm UTC
Forum: Language/Linguistics
Topic: Can you help me identify this language?
Replies: 5
Views: 3480

Can you help me identify this language?

My friend asked for help identifying the language in this picture.

http://i.imgur.com/X2MQ8VU.jpg
Mon Mar 16, 2015 4:32 pm UTC
Forum: Mathematics
Topic: Happy Super Pi Day, Everyone!
Replies: 37
Views: 7430

Re: Happy Super Pi Day, Everyone!

I can't see why anyone would ever want any date format other than year-month-day. That's the only way you can actually sort/order dates meaningfully. It's true that that format is best for sorting. However, if you have data where the year remains constant (e.g. data for a particular course that tak...
Thu Feb 26, 2015 3:04 am UTC
Forum: Mathematics
Topic: Mathematical Induction - Help for a Beginner?
Replies: 75
Views: 12588

Re: Mathematical Induction - Help for a Beginner?

At that early point, phlip was merely pointing out that proving something true for all natural numbers requires proving infinitely many statements. Nope proving an infinite collection of statements is based on infinitary logic. Traditional first order logic only includes finite length proofs. Furth...
Tue Feb 24, 2015 5:06 am UTC
Forum: Mathematics
Topic: Mathematical Induction - Help for a Beginner?
Replies: 75
Views: 12588

Re: Mathematical Induction - Help for a Beginner?

This is not the correct understanding of induction on natural numbers. It's not meant to be the rigorous proof of induction, it's meant to be an explanation of how/why it works. Moreover, it was just an early part of a long verbal exposition about induction, and hadn't even got into induction per s...
Thu Jan 15, 2015 8:17 pm UTC
Forum: Mathematics
Topic: Seeking reference or proof for an integral inequality
Replies: 4
Views: 2223

Seeking reference or proof for an integral inequality

I asked this last night on math.stackexchange, but people ask a lot of questions there of varying quality, and I think questions sometimes get lost in the shuffle. I thought posting the same question here might be useful. It seems like the type of thing people here would find interesting. It's an in...
Tue Nov 18, 2014 6:33 pm UTC
Forum: Mathematics
Topic: Help me with this equations please!
Replies: 13
Views: 4833

Re: Help me with this equations please!

I was playing around with this a bit more, and I think I see how to prove that there are solutions for all n. Furthermore, using ideas suggested by eta oin shrdlu, I think it's possible to prove that there are infinitely many solutions when n is a nonsquare (this relies on knowing certain things abo...
Sun Nov 16, 2014 5:52 pm UTC
Forum: Mathematics
Topic: Help me with this equations please!
Replies: 13
Views: 4833

Re: Help me with this equations please!

This is really interesting. Does the equation (a^2+b)/(a+b^2) = n have a name that anyone's aware of? Does the equation have infinitely many solutions for all n? Did the OP get the question from a textbook or paper or anything, or just from playing around?
Sun Oct 26, 2014 4:09 pm UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 485510

Re: Favorite math jokes

Yep. Those explanations are all correct. I fully admit that my jokes were kind of forced and contrived.

An iterated logarithm sometimes appears in analytic number theory or other branches of math that deal with asymptotics. And it kind of sounds like "glug glug".
Mon Oct 20, 2014 5:52 pm UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 485510

Re: Favorite math jokes

These are a little silly, but...

What does an analytic number theorist say when drowning?

Log log, log log...

What kind of baseball bat does an analytic number theorist use?

A Liouville Slugger.

What's an analytic number theorist's favorite Star Wars character?

Landau Calrissian.
Wed Oct 01, 2014 5:18 pm UTC
Forum: Mathematics
Topic: Question on Mandelbrot Set
Replies: 6
Views: 3449

Re: Question on Mandelbrot Set

This might not answer the question exactly, or succinctly, but it's an exploration by John Baez of "why" certain processes lead to fractal sets.

http://www.math.ucr.edu/home/baez/roots/
Sat Jul 12, 2014 3:42 pm UTC
Forum: Mathematics
Topic: Need help with understanding equations
Replies: 6
Views: 2301

Re: Need help with understanding equations

Numbers that are being added or subtracted are pretty far away from each other, so they aren't really that "attached", and can be moved first, unless they are being held in by parenthesis. After you moved all the stuff that's added or subtracted to the other side, you go on to the next st...
Mon Jun 23, 2014 4:11 pm UTC
Forum: Mathematics
Replies: 88
Views: 18322

Infinite series sums were a gotcha for me for a while until I asked an instructor why the Riemann-Zeta function came out to (pi^2)/6 when all of the terms being added were rational. It's not really that weird for a sum of rational numbers to be equal to an irrational number -- as long as it's an in...
Mon Jun 02, 2014 6:51 pm UTC
Forum: Mathematics
Topic: Triangle geometry problem
Replies: 2
Views: 2412

Re: Triangle geometry problem

It's also sometimes called the "adventitious angles" problem, and Googling those words gives you some results.
Wed Apr 16, 2014 10:04 pm UTC
Forum: Mathematics
Topic: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?
Replies: 10
Views: 5777

Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Thanks for that, Bloopy. That's a good example of the type of thing I was missing, and hoping for. I knew how to write gamma as a slowly converging series, but I didn't know about generalizing the harmonic numbers to non-integer arguments, nor did I know about that specific definite integral that ev...
Tue Apr 08, 2014 5:41 pm UTC
Forum: Mathematics
Topic: how do i find F(g(x)) given f(x), g(x), or compute it?
Replies: 5
Views: 3294

Re: how do i find F(g(x)) given f(x), g(x), or compute it?

Or how Sum(n,n=1..4) is 1+2+3+4=10 and is not a function of n.
Mon Mar 17, 2014 7:35 pm UTC
Forum: Mathematics
Topic: Formula on a T-Shirt
Replies: 2
Views: 1991

Re: Formula on a T-Shirt

My best guess is the following. I think \pi(x) refers to the usual prime-counting function: \pi(x) = the number of primes less than or equal to x. I think \mu(n) refers to the Möbius function. I think J refers to the "Riemann prime-counting function", which is not the same function as the ...
Fri Mar 14, 2014 9:35 pm UTC
Forum: Language/Linguistics
Topic: list of words like bath/staff/clasp/dance
Replies: 1
Views: 2685

list of words like bath/staff/clasp/dance

Does anybody know if anybody has already compiled a reasonably complete list of words like the following: bath, branch, clasp, dance, master, staff which (according to what I have read) tend to be pronounced with the palm/calm/bra/father vowel in southern England, but the trap/cat/ham vowel in north...
Tue Feb 11, 2014 5:03 pm UTC
Forum: Mathematics
Topic: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?
Replies: 10
Views: 5777

Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

That's very similar to what I thought at first. If the number sqrt(1/3) comes from the method of approximate integration, as opposed to the function being integrated, then it does indeed seem like a bit of a cheat. However, somewhat ironically, when I tried to work through the details of that Wikipe...
Wed Feb 05, 2014 10:22 pm UTC
Forum: Mathematics
Topic: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?
Replies: 10
Views: 5777

"Why" is the Euler-Mascheroni constant near sqrt(1/3)?

The Euler-Mascheroni constant (also called "Euler's constant", or lower case gamma) is approximately equal to 0.57722. The square root of 1/3 is approximately equal to 0.57735. I remember noticing this a while ago, and wondering if there was any informal or intuitive "reason" for...
Mon Jan 27, 2014 4:07 pm UTC
Forum: Mathematics
Topic: Strange constant. Apparently useless.
Replies: 33
Views: 9117

Re: Strange constant. Apparently useless.

This is a little subjective, but personally, I think it could be argued that it's *more* surprising when the value of an infinite sum turns out to be, say, sqrt(2*e) or log(pi) or something similarly related to "known" constants.
Mon Dec 02, 2013 8:30 pm UTC
Forum: Mathematics
Topic: Limit exponential/power function
Replies: 3
Views: 2794

Re: Limit exponential/power function

Yes. So, for example, if ln(y) approached 15 (it doesn't, I just made that up) then y would approach e^15.
Sat Nov 30, 2013 7:03 pm UTC
Forum: Mathematics
Topic: Limit exponential/power function
Replies: 3
Views: 2794

Re: Limit exponential/power function

Another thing I tried was writing it as an exponential function (e ln(x/3)... ), but that wasn't useful either That should work. Personally, the way that I prefer to write these types of limits is as follows: Let y = (x/3) 1/(x-3) (which of course is of the form a^b where a and b bo...
Tue Nov 26, 2013 11:53 pm UTC
Forum: Mathematics
Topic: Different Numbers of Coins, Different Totals
Replies: 3
Views: 2393

Re: Different Numbers of Coins, Different Totals

I like this problem, even if we're the only two here who do. Now that some time has passed, I'll post spoilers freely. Hope that's OK. For the three coin case, the condition is that the three values not be in arithmetic progression. Here's another way of saying that. If the values of the three types...
Sat Nov 23, 2013 9:08 pm UTC
Forum: Mathematics
Topic: Different Numbers of Coins, Different Totals
Replies: 3
Views: 2393

Re: Different Numbers of Coins, Different Totals

This is really interesting. I know you said you solved the 3-coin case, but let's look at it in a little more detail. Let's say the values of the three types of coin are v1 < v2 < v3. Let's also define f(a,b,c) = a*v3 + b*v2 + c*v1 = the total value of a of the most valuable coin, b of the next most...
Sun Nov 10, 2013 12:54 am UTC
Forum: Mathematics
Topic: Uniform Probability Distribution Over All Natural Numbers
Replies: 13
Views: 5198

Re: Uniform Probability Distribution Over All Natural Number

It's fair to ask for something "close" to a uniform probability distribution on the natural numbers. We can't have uniformity together with countable additivity. However, can we get something "close" that still satisfies some of the intuitive properties of "probability"...
Sun Oct 27, 2013 5:00 pm UTC
Forum: Mathematics
Topic: n sided polygon, n < 3
Replies: 35
Views: 6671

Re: n sided polygon, n < 3

Yep. There are some contexts where we wouldn't want the prefix "poly-" to include "only one". Some such contexts are in everyday language, such as the words "polyglot" or "polymath" or "polyamory". There are some contexts where we do want the prefix ...
Tue Oct 01, 2013 7:23 pm UTC
Forum: Mathematics
Topic: Math Books
Replies: 379
Views: 274043

Re: Math Books

As an undergraduate math major, I very much enjoyed "Graph Theory" by Bollobas. I know I'm just one person, but take that as you will.
Wed Sep 25, 2013 9:59 pm UTC
Forum: Mathematics
Topic: Cardinality of the complex numbers
Replies: 41
Views: 11776

Re: Cardinality of the complex numbers

Just some admittedly very handwavy remarks about the "intuition" behind this stuff: Intuitively, one may think of R^2 as "bigger" than R. But that's only true if you're including all the geometric and algebraic structure of those sets. If we care only about cardinality, then we'r...
Thu Sep 19, 2013 3:31 pm UTC
Forum: Mathematics
Topic: Justifying L'Hopital to an ambitious undergrad
Replies: 8
Views: 4239

Re: Justifying L'Hopital to an ambitious undergrad

One thing I tell my students, which is definitely vague and hand-wavy, is along the following lines. Consider something like 5x/exp(x) as x approaches infinity. The numerator and denominator each approach infinity. However, the numerator grows only linearly, whereas the denominator grows faster than...
Wed Sep 18, 2013 3:18 pm UTC
Forum: Mathematics
Topic: Justifying L'Hopital to an ambitious undergrad
Replies: 8
Views: 4239

Justifying L'Hopital to an ambitious undergrad

I teach calculus, and I use the classic textbook by Stewart. One topic we cover is L'Hopital's Rule for limits of the form 0/0 or infinity/infinity. Many students are fine with treating L'Hopital's Rule as a "magic box": For some reason, for the above types of indeterminate form, f/g has t...
Wed Sep 18, 2013 3:06 pm UTC
Forum: Mathematics
Topic: mean of geometric distribution: *intuitive* reason
Replies: 14
Views: 4987

Re: mean of geometric distribution: *intuitive* reason

Exactly. You're more likely to join the table during a long run between zeroes than a short one, and that weighting makes the expected length of the run that you join at double the ordinary average run length. It's sort of related to the following. You don't find the average family size by asking p...
Sun Sep 15, 2013 9:08 pm UTC
Forum: Mathematics
Topic: mean of geometric distribution: *intuitive* reason
Replies: 14
Views: 4987

Re: mean of geometric distribution: *intuitive* reason

I like lightvector's approach too. Thanks, everybody. Somewhat in the spirit of dudiobugtron's remarks: note that the median of the geometric distribution, as opposed to the mean, is less intuitive. The formula for the median appears on the Wikipedia page I linked in my first post. It's the ceiling ...
Sat Sep 14, 2013 6:53 pm UTC
Forum: Mathematics
Topic: mean of geometric distribution: *intuitive* reason
Replies: 14
Views: 4987

Re: mean of geometric distribution: *intuitive* reason

That's good, I like that. The little intuitive leap I wasn't making last night: One can ask about the expected number of successes on a single roll, even though it's not an integer. And that expected number of successes is 1/n. As far as "intuitive" explanations go, maybe yours is pretty m...
Sat Sep 14, 2013 2:48 am UTC
Forum: Mathematics
Topic: mean of geometric distribution: *intuitive* reason
Replies: 14
Views: 4987

mean of geometric distribution: *intuitive* reason

Suppose you roll a fair six-sided die repeatedly, and you count the number of rolls it takes to see your favorite side (say your favorite number is 5) at least once. The number of required rolls follows the well-known geometric distribution , using the first of the two slightly different definitions...
Thu Sep 05, 2013 11:58 pm UTC
Forum: Mathematics
Topic: Can x+3 and x^2+3 both be perfect cubes?
Replies: 14
Views: 4759

Re: Can x+3 and x^2+3 both be perfect cubes?

Cool. That's largely the same as what I suspect the intended "trick" is, except I didn't phrase the last step in terms of Fermat's Last Theorem. My phrasing of the argument was: If x+3 and x^2+3 are both cubes, then so is their product x^3+3x^2+3x+9. But it's also true that x^3+3x^2+3x...
Thu Sep 05, 2013 10:15 pm UTC
Forum: Mathematics
Topic: Can x+3 and x^2+3 both be perfect cubes?
Replies: 14
Views: 4759

Re: Can x+3 and x^2+3 both be perfect cubes?

arbiteroftruth wrote:Rather amusingly, the first solution I found to the original problem uses Fermat's last theorem as the final step.

Heh, cool. Would you mind sharing your solution?
Thu Sep 05, 2013 8:50 pm UTC
Forum: Mathematics
Topic: Can x+3 and x^2+3 both be perfect cubes?
Replies: 14
Views: 4759

Can x+3 and x^2+3 both be perfect cubes?

I actually saw this question in a Buzzfeed article about trivia addicts. There was a photograph of a blackboard at an ice cream shop that offered a free scoop of ice cream to the first person to answer the question. The question is: Find a whole number x such that x+3 and x^2+3 are both perfect cube...