Search found 555 matches

Thu Jan 20, 2011 11:09 pm UTC
Forum: Mathematics
Topic: Does this series converge!?
Replies: 32
Views: 4368

Re: Does this series converge!?

It's probably quite true that many of the standard tests don't apply, since many of those tests are stated for positive-term series. Likely the easiest way to prove convergence is directly from the definition. Note that every partial sum is 0, so the sequence of partial sums converges to 0. I guess ...
Thu Jan 20, 2011 8:56 pm UTC
Forum: Mathematics
Topic: ln(x) or log(x)?
Replies: 63
Views: 9391

Re: ln(x) or log(x)?

But don't you ever say or write ln in a lecture? I do, all the time. But I write the lower case L cursive style, with a loop. Edit: And to clarify, I don't mind reading something that's ambiguous between a lower case L and a capital i -- i.e. a vertical line. It's the capital i's with serifs that b...
Thu Jan 20, 2011 4:45 pm UTC
Forum: Mathematics
Topic: ln(x) or log(x)?
Replies: 63
Views: 9391

Re: ln(x) or log(x)?

Yeah, I meant that some of my students write it as a capital i with serifs, not just a vertical line.
Thu Jan 20, 2011 1:35 am UTC
Forum: Mathematics
Topic: ln(x) or log(x)?
Replies: 63
Views: 9391

Re: ln(x) or log(x)?

I always write lower case 'l's in cursive to avoid confusion. This, this, a thousand times this. If any of you teach, please, please, please do NOT write "ln" using just a vertical line for the lower case L! I cannot stand it when my students write it like "in" (rhymes with &quo...
Sun Jan 16, 2011 8:15 pm UTC
Forum: Mathematics
Replies: 25
Views: 2987

Even though the result is a unwieldy and probably not terribly useful, good job. One of the best ways to get better at math is exposing yourself to it, and it sounds like this is a really good way to give some great practice with rearranging equations. So don't worry about justifying yourself. Embr...
Sun Jan 16, 2011 6:05 pm UTC
Forum: Mathematics
Replies: 25
Views: 2987

Wouldn't the formula for c be

c = - ax^2 - bx

?
Mon Jan 10, 2011 8:46 pm UTC
Forum: Mathematics
Replies: 10
Views: 2292

This is just my two cents, and thus may not be worth more than anyone else's two cents, but: I have a PhD in math. I didn't do an enormous amount of computing as a student (I took intro to programming and numerical analysis as an undergrad, and did a fair bit of experimentation using Maple as a grad...
Fri Jan 07, 2011 3:52 pm UTC
Forum: Mathematics
Topic: Pascals Triangle, Why don't we learn it sooner
Replies: 17
Views: 2281

Re: Pascals pyramid, Why don't we learn it sooner

But I haven't been able, in my idle moments, to come up with a good demonstration of why the r+1th element of the nth row gives you the value for nCr. When you expand (x+y)^n, which terms contain x^r (and therefore y^{n-r})? To form a typical term in the expansion, you choose one term from each of ...
Mon Jan 03, 2011 4:18 pm UTC
Forum: Mathematics
Topic: Probability Query
Replies: 10
Views: 1513

Re: Probability Query

I know this topic has been discussed thoroughly elsewhere, but I also think it may be helpful to point out that probability questions, especially in scenarios like these, are "really" just about proportions. Questions such as this essentially have the form, "In what proportion of situ...
Tue Dec 28, 2010 10:02 pm UTC
Forum: Mathematics
Topic: Principle of Induction for Reals?
Replies: 8
Views: 2522

Re: Principle of Induction for Reals?

Some proofs of some classic results in analysis "resemble" induction over the reals, speaking somewhat vaguely. (Loosely speaking, induction is about "going from one integer to the next" -- something similar for real numbers might be "you can always go a little bit further t...
Mon Dec 27, 2010 6:18 am UTC
Forum: Mathematics
Topic: Linear system problem
Replies: 4
Views: 859

Re: Linear system problem

You are correct; the number of linearly independent rows and linearly independent columns are the same. These are sometimes referred to as the dimension of the row space/dimension of the column space. I don't remember the proof off of the top of my head... Neither do I (in any detail) at 10:17 pm l...
Wed Dec 22, 2010 10:17 pm UTC
Forum: Mathematics
Topic: A Probability Question
Replies: 10
Views: 1647

Re: A Probability Question

In the way I usually interpret problems like this (I'm not positive this problem was meant this way), the players don't "take turns" -- instead, there is ONE die that's rolled many times, generating one sequence consisting of numbers from 1 to 6. There can be no ties, because a single roll...
Sat Dec 18, 2010 2:32 am UTC
Forum: Mathematics
Topic: ln(x) or log(x)?
Replies: 63
Views: 9391

Re: ln(x) or log(x)?

My personal preference is for log(x) to mean the natural log. But when teaching calculus, I defer to all the textbooks, and use ln. In a more specialized academic context -- writing a research article, or giving a presentation to fellow PhDs -- I would use "log" to mean the natural log. (P...
Fri Dec 17, 2010 2:53 am UTC
Forum: Mathematics
Topic: Useless Math
Replies: 60
Views: 10365

Re: Useless Math

I remember when I was learning more about mathematics, being almost disappointed at how mundane (in a sense) higher dimensions really are. To most mathematicians, having more than three dimensions just means you have more than three coordinates. That's pretty much it. (We can't necessarily visualize...
Thu Dec 16, 2010 6:25 pm UTC
Forum: Mathematics
Topic: Geo. and Arith. Seq. (and some induction!)
Replies: 23
Views: 2594

Re: Geo. and Arith. Seq. (and some induction!)

I wonder if the question was supposed to say x^n < 1. The other questions that start with P(n) contain the variable n. (Incidentally, the ones that start with P(n) aren't quite questions as written. Probably the question is "Use induction to prove that the statement P(n) is true for all n.")
Thu Dec 16, 2010 4:17 pm UTC
Forum: Mathematics
Topic: Work Done by a Force Field
Replies: 2
Views: 507

Re: Work Done by a Force Field

Also, remember that there's a "nonclever mechanical way" of checking whether a given vector field is the gradient of something. You don't have to just creatively "notice" it out of the blue.
Wed Dec 15, 2010 3:01 am UTC
Forum: Mathematics
Topic: Topology Resources
Replies: 30
Views: 3678

Re: Topology Resources

Gallian might not be the most encyclopedic algebra text, but it's very user-friendly and a joy to learn from.
Wed Dec 15, 2010 1:29 am UTC
Forum: Mathematics
Topic: Cantor Diagonalization Formula
Replies: 11
Views: 2906

Re: Cantor Diagonalization Formula

If you're willing to invoke Schroeder-Bernstein, it's enough to show a bijection between N and part of NxN, and another bijection between NxN and part of N. The first is trivial. For the second, say each of your pairs of natural numbers is written with a comma separating the two numbers. Interpret t...
Wed Dec 15, 2010 1:23 am UTC
Forum: Mathematics
Topic: Topology Resources
Replies: 30
Views: 3678

Re: Topology Resources

If we're talking about introductions to abstract algebra for undergraduates, my vote is for Gallian.
Tue Dec 14, 2010 9:28 pm UTC
Forum: Mathematics
Topic: Useless Math
Replies: 60
Views: 10365

Re: Useless Math

On simpler level, do the complex numbers exist in reality? The mathematical concept is very useful, and lets us prove things about real numbers, but did you ever see 2-3i cows in a field? You also never see exactly two-sevenths of a cow. But people don't tend to think of fractions as being as "...
Tue Dec 14, 2010 4:17 pm UTC
Forum: Mathematics
Topic: I have a measure theory final on tuesday
Replies: 24
Views: 2428

Re: I have a measure theory final on tuesday

Marbas wrote:
Now if you'll excuse me, I'll be over here trying to convince myself that this is not the end of my chances at becoming a decent mathematician.

If it's any consolation, I'm a PhD with a university job who bombed a couple of grad courses when I was a student.
Tue Dec 14, 2010 4:10 pm UTC
Forum: Mathematics
Topic: I don't get quaternions
Replies: 22
Views: 2573

Re: I don't get quaternions

Why are we limited to only 4 real division algebras? Couldn't we have a "trinion" where we define i*i, i*j, and j*j to all equal -1? If I remember the history correctly, that's exactly what William Rowan Hamilton tried to do around the 1840s or so. In fact, I remember reading that his son...
Tue Dec 14, 2010 1:54 pm UTC
Forum: Mathematics
Topic: Geo. and Arith. Seq. (and some induction!)
Replies: 23
Views: 2594

Re: Geo. and Arith. Seq. (and some induction!)

Not sure if this will help:

If you had to describe a geometric sequence in words, what would you say?
Tue Dec 14, 2010 12:06 am UTC
Forum: Mathematics
Topic: Primes vs. Natural Numbers
Replies: 9
Views: 1805

Re: Primes vs. Natural Numbers

One possible way to try to make more precise the idea of the ratio (Amount of primes)/(Amount of natural numbers) is to look at what proportion of the first n natural numbers are prime, and then take the limit of that proportion as n approaches infinity. http://en.wikipedia.org/wiki/Natural_density ...
Mon Dec 13, 2010 7:31 pm UTC
Forum: Mathematics
Topic: I don't get quaternions
Replies: 22
Views: 2573

Re: I don't get quaternions

It's also possible to represent them by a certain family of 4x4 real matrices, if you like matrices.

http://en.wikipedia.org/wiki/Quaternion ... sentations
Sun Dec 12, 2010 9:33 pm UTC
Forum: Mathematics
Topic: Inflection point
Replies: 12
Views: 1276

Re: Inflection point

Well you could do \frac{x^2}{(x^2-3)(x^2-1)^2} But does that make things easier? I get the feeling it's really something simple, but I'm just not seeing it. I don't understand what you did there. Did x^2-3 move from the top to the bottom? Anyway, the trick I'm alluding to isn't real...
Sun Dec 12, 2010 9:05 pm UTC
Forum: Mathematics
Topic: Inflection point
Replies: 12
Views: 1276

Re: Inflection point

pietertje wrote:
Do you mean

$f'(x) = \frac{x^2(x^2-3)}{x^2-1)^2}$?

Yes, I believe that's what I got.

Can you introduce a new variable in a clever shortcutty way? (Granted, I guess you can argue it still ends up being a little tedious.)
Sun Dec 12, 2010 7:23 pm UTC
Forum: Mathematics
Topic: Inflection point
Replies: 12
Views: 1276

Re: Inflection point

Which give a second derivative of: f''(x) = \frac{6x}{x^2-1} + \frac{8x^5}{(x^2-1)^4} - \frac{14x^3}{(x^2-1)^2} This disagrees slightly with what Maple gives me. You're right that this is a nuisance to do by hand. But I'm pretty sure I eventually get an expression fo...
Sun Dec 12, 2010 6:49 pm UTC
Forum: Mathematics
Topic: Inflection point
Replies: 12
Views: 1276

Re: Inflection point

This may help:

When you compute the first derivative of f, what's the tidiest form you can put it in? Does anything jump out at you? Is there a possible substitution that would simplify things?
Sun Dec 12, 2010 12:17 am UTC
Forum: Mathematics
Topic: Cantor Diagonalization Formula
Replies: 11
Views: 2906

Re: Cantor Diagonalization Formula

The expression "Cantor's diagonalization" almost always refers to the proof that the real numbers are uncountably infinite, a.k.a. nondenumerably infinite. There's a different argument belonging to the same general topic that's superficially similar because it also involves "diagonals...
Fri Dec 10, 2010 2:57 pm UTC
Forum: Mathematics
Topic: Intersection between a plane and a straight line
Replies: 5
Views: 942

Re: Intersection between a plane and a straight line

The problem is: if 0λ=0; λ=?? Well, if this were a stand-alone problem, the answer is that the equation is always true for all values of lambda. I think what it means in the context is that the entire line lives in the plane -- but be careful with your logic here. In general, be careful of reasonin...
Fri Dec 10, 2010 1:52 pm UTC
Forum: Mathematics
Topic: relation which is symmetric, reflexive but not transitive?
Replies: 20
Views: 24216

Re: relation which is symmetric, reflexive but not transitiv

Since a relation is nothing but a set of ordered pairs, notice that only so many relations are even possible on a two-element set. You could list them all. This may seem a bit inelegant (and isn't the only approach to these problems), but is still very possible. Say your set is {a,b}. Then the only ...
Thu Dec 09, 2010 10:02 pm UTC
Forum: Mathematics
Topic: Homework: Eigenvalues
Replies: 18
Views: 3063

Re: Homework: Eigenvalues

Depending on who the grader is, this might be too nitpicky to worry much about, but note that somewhere in there, you are using the fact that

matrix times scalar times vector = scalar times matrix times vector.
Wed Dec 08, 2010 8:16 pm UTC
Forum: Mathematics
Topic: My problems with complex analysis
Replies: 6
Views: 1166

Re: My problems with complex analysis

Both to the original poster, and to anyone else who might be interested, I highly recommend two things: --the article "The Fundamental Theorem of Algebra: A Visual Approach" by Daniel J. Velleman, which you can find online --the book "Visual Complex Analysis" by Tristan Needham, ...
Wed Dec 08, 2010 5:36 pm UTC
Forum: Mathematics
Topic: My problems with complex analysis
Replies: 6
Views: 1166

Re: My problems with complex analysis

In response to your first question, people sometimes think of a complex-valued function of a complex variable as a "transformation" of the complex plane. Instead of trying to visualize everything in one picture, another approach is to draw one copy of the complex plane, which you can think...
Wed Dec 08, 2010 3:55 pm UTC
Forum: Mathematics
Topic: How hard is college level mathematics?
Replies: 36
Views: 9788

Re: How hard is college level mathematics?

wAriot wrote:(btw, as I said, I'm Spanish, so please, please correct me if i did any grammatical error, so I can correct it)

I would say "made any error" rather than "did any error".
Wed Dec 08, 2010 3:50 pm UTC
Forum: Mathematics
Topic: Double integrals in polar
Replies: 12
Views: 1368

Re: Double integrals in polar

Also, it may help to backtrack a little and think about rectangular coordinates again. When finding an area in rectangular coordinates, you basically just "integrate the function", to put it briefly. But why do you "integrate the function"? Typically, you're looking at the integr...
Mon Dec 06, 2010 4:23 am UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 494591

Re: Favorite math jokes

letterX wrote:
Yakk wrote:Did you intend to make a roman-numeral Christ pun? Because you did.

... I hesitate to ask, but... I did?

XX days before Xmas.
Mon Dec 06, 2010 2:04 am UTC
Forum: Mathematics
Topic: Polar Coordinates Intuition
Replies: 7
Views: 1664

Re: Polar Coordinates Intuition

This is partly a subjective matter and thus opinions may differ, but I tend to think that plotting in any coordinate system whatsoever is, fundamentally, "just" plugging points in. Of course, there are various useful shortcuts: look for horizontal and vertical tangents, consider concavity,...
Fri Dec 03, 2010 3:59 am UTC
Forum: Mathematics
Topic: Forbidden subwords -- logic vs intuition
Replies: 13
Views: 1883

Re: Forbidden subwords -- logic vs intuition

I vote product rule. Drawing a rectangle might help. Say its sides are u and v. So its current area is uv. How does the area change if both u and v are changing by small amounts du and dv respectively? If du and dv are both positive, we've added three things: --a strip of dimensions u times dv --a ...