## Search found 145 matches

- Wed Jul 21, 2010 11:01 pm UTC
- Forum: Mathematics
- Topic: How do we know i is well defined?
- Replies:
**16** - Views:
**2730**

### Re: How do we know i is well defined?

The k you chose is not the square root of -1. If you let k=i/\pi , then k^2=-1/\pi . There are actually two square roots of -1 , i and -i . To see this, suppose i and k were both square roots of -1. i^2=1=k^2 , so i=\pm k . Edit: To be fair, you could use any imaginary number you wanted as a "b...

- Wed Jan 20, 2010 6:02 am UTC
- Forum: Mathematics
- Topic: Some Basic Analysis Questions
- Replies:
**13** - Views:
**2542**

### Re: Some Basic Analysis Questions

That looks good to me. The only issue I noticed is that you should call your new subsequence \{ a_{n_i} \} . (Or else define \{ b_{n_i} \} a little more explicitly, but it seems easier to use the same name as in the problem statement. This has the added benefit of looking exactly like the definition...

- Tue Jan 19, 2010 11:49 pm UTC
- Forum: Mathematics
- Topic: Some Basic Analysis Questions
- Replies:
**13** - Views:
**2542**

### Re: Some Basic Analysis Questions

Weird. For some reason, I actually have the fifth and third editions, so now my problems and suchlike should be numbered correctly. You haven't covered the theorem jestingrabbit mentioned yet, but problem 26 actually gives you the special case you need (that if \{ a_n \} converges to a, then \{ \sqr...

- Tue Jan 19, 2010 12:32 pm UTC
- Forum: Mathematics
- Topic: Some Basic Analysis Questions
- Replies:
**13** - Views:
**2542**

### Re: Some Basic Analysis Questions

I seem to have a different version than you do, so my section numbers may be a little off, but you should take a closer look at the theorems in section 1.3 (arithmetic operations on sequences). These theorems are exactly what skeptical scientist was referring to. The gist is that you can figure out ...

- Thu Jan 14, 2010 4:28 am UTC
- Forum: Mathematics
- Topic: Mathematics and intellect
- Replies:
**85** - Views:
**9379**

### Re: Mathematics and intellect

I had a somewhat related discussion with one of my professors a couple years ago. He used to run some sort of biotech business, and he told me that he generally preferred to hire math majors rather than biology majors. While the biology majors may have had previous knowledge about the material they ...

- Thu Jan 07, 2010 10:55 pm UTC
- Forum: Mathematics
- Topic: Shuffling Cards
- Replies:
**17** - Views:
**2596**

### Re: Shuffling Cards

Just riffle shuffle eleven or twelve times. Riffle shuffling is a lot faster than pile shuffling, and eleven shuffles should be close enough to random for a 60 card M:tG deck. This. The only real uses for pile shuffling are counting your deck and cheating. If you're just playing casually, the count...

- Thu Jan 07, 2010 6:21 am UTC
- Forum: Mathematics
- Topic: Shuffling Cards
- Replies:
**17** - Views:
**2596**

### Re: Shuffling Cards

I don't know anything about the composition of a Pokemon deck, but I've shuffled a lot of magic decks before. In practice, I've found that seven riffle shuffles really isn't enough. This is mostly because, assuming you care at all about the condition of your cards, your riffle shuffles will be far f...

- Wed Dec 16, 2009 5:40 am UTC
- Forum: Mathematics
- Topic: Closure of sets in a continuous function
- Replies:
**8** - Views:
**2383**

### Re: Closure of sets in a continuous function

Do you know any other definitions of continuous? In particular, have you learned that a function [imath]f: X\rightarrow Y[/imath] is continuous iff [imath]\forall V \in Y[/imath] such that V is open, [imath]f^{-1}(V)[/imath] is open.

- Wed Dec 16, 2009 3:51 am UTC
- Forum: Mathematics
- Topic: Integration of e^( x^2)
- Replies:
**24** - Views:
**4130**

### Re: Integration of e^( x^2)

If you rename your constants, it's not bad at all. Let b = \frac{-\beta\epsilon}{\sigma\lambda-1} and a = -b\lambda\sigma . It should be: e^{a+br}(\frac{r^2}{b}-\frac{2r}{b^2}+\frac{2}{b^3}) It's at most twice that long when you plug in your limits, although it looks like it might factor pre...

- Tue Dec 15, 2009 11:51 pm UTC
- Forum: Mathematics
- Topic: Readable Encryption
- Replies:
**15** - Views:
**1914**

### Re: Readable Encryption

What you're referring to is a specific case of steganography. The most basic way to do this would be through some sort of code word system. I'm not sure if that's what you mean when you say "encryption method" though.

- Mon Dec 14, 2009 8:30 am UTC
- Forum: Mathematics
- Topic: How To Learn (Advanced) Math
- Replies:
**12** - Views:
**3770**

### Re: How To Learn (Advanced) Math

Oh no, leap right into differential geometry! If you've done diff eq's you should be all set for a curves and surfaces course. Classical differential geometry is awesome. I really liked do Carmo's book, but there are others. I think the style of differential geometry that comes with the full force ...

- Mon Dec 14, 2009 6:21 am UTC
- Forum: Mathematics
- Topic: How To Learn (Advanced) Math
- Replies:
**12** - Views:
**3770**

### Re: How To Learn (Advanced) Math

All you should need for topology, linear algebra and number theory are good books. I don't know any particularly good linear algebra or number theory books, but Topology by James Munkres is an excellent place to start that subject. Real analysis is another area you might want to consider. (A good gr...

- Fri Dec 11, 2009 1:50 am UTC
- Forum: Mathematics
- Topic: So, what can I do to keep myself entertained and interested?
- Replies:
**14** - Views:
**1942**

### Re: So, what can I do to keep myself entertained and interested?

You might want to look for a book about number theory. It really only requires algebra and some elbow grease (basically just the willingness to keep playing with problems until you figure them out). I wouldn't recommend anything that you're going to cover in the next year or two, since you'll still ...

- Tue Oct 27, 2009 4:46 am UTC
- Forum: Mathematics
- Topic: What kind of function is this?
- Replies:
**9** - Views:
**1415**

### Re: What kind of function is this?

As far as I know, equations of that form don't have a special name (besides exponential, but that's pretty broad). I'm sure that if you refer to it as "the equation for an underdamped harmonic oscillator", people will understand. If you need to talk to people who haven't studied differenti...

- Wed Oct 21, 2009 11:25 pm UTC
- Forum: Mathematics
- Topic: Euler's Elements of Algebra
- Replies:
**5** - Views:
**919**

### Re: Euler's Elements of Algebra

I haven't read it either, but a quick google search suggests that you'll be able to understand it just fine. Apparently Euler sometimes chose some strange subjects to focus on, but it's pretty similar to the algebra you've already done. Besides, you might as well take a look at any free book that st...

- Sat Oct 17, 2009 9:17 pm UTC
- Forum: Mathematics
- Topic: Need help developing an algorithm!
- Replies:
**7** - Views:
**1015**

### Re: Need help developing an algorithm!

What about [imath]v=50\sqrt{s}[/imath] or [imath]v=\frac{50}{log(2)}log(s+1)[/imath] (where v is the value and s is the skill level)? These won't usually give you whole numbers, but you can just round down.

- Sat Oct 17, 2009 9:01 pm UTC
- Forum: Mathematics
- Topic: Calculus Midterm Question
- Replies:
**3** - Views:
**832**

### Re: Calculus Midterm Question

Have you learned about tangent lines? A tangent line of a function f at a point a (also called a linear approximation) is approximately equal to f as long as you are close to a. Does that sound like part ii?

- Sat Oct 17, 2009 8:15 pm UTC
- Forum: Mathematics
- Topic: Got a new calculator for Math 10, woot
- Replies:
**29** - Views:
**2765**

### Re: Got a new calculator for Math 10, woot

My high school was similar to RogerMurdock's. Everyone was supposed to have a graphing calculator, but if you couldn't afford one the school had spares that you could use whenever you needed to. They seemed very useful at the time, but I've found that calculators usually instill a lot of bad habits....

- Sat Oct 17, 2009 9:33 am UTC
- Forum: Mathematics
- Topic: Definition of dy and dx?
- Replies:
**16** - Views:
**1864**

### Re: Definition of dy and dx?

If you're really curious, dy and dx are differential forms . (Specifically, one forms ). Don't worry if this goes way over your head. It's way over mine too, and I've been studying it. The basic idea is that calculus is a way to talk about infinitely small changes. One of the best ways to describe a...

- Fri Oct 02, 2009 10:13 pm UTC
- Forum: Mathematics
- Topic: which asshat came up with infix?
- Replies:
**23** - Views:
**3309**

### Re: which asshat came up with infix?

If you dont like infix, there is a simple way to fix that.

a+b=f(a,b)

To be fair, that doesn't solve the problem at all, since that's exactly what prefix notation does. Everyone just calls the function + instead of f.

- Sat Sep 26, 2009 11:30 am UTC
- Forum: Mathematics
- Topic: What's the coolest thing you've ever learned in math class?
- Replies:
**58** - Views:
**6601**

### Re: What's the coolest thing you've ever learned in math class?

That's certainly the sticking point, since what I meant by "an explicit construction is possible" is that a mathematician could state it. I suppose I should just suck it up and suspend my disbelief when I start talking about infinitely many mathematicians.

- Sat Sep 26, 2009 12:44 am UTC
- Forum: Mathematics
- Topic: What's the coolest thing you've ever learned in math class?
- Replies:
**58** - Views:
**6601**

### Re: What's the coolest thing you've ever learned in math class?

Depending on what you mean by "infinity," I think countable prisoners and hats is a lot more unsettling. This paradox bugs me because of the line "Using the axiom of choice, they select and memorize a representative sequence from each equivalence class." The Axiom of choice guar...

- Tue Sep 22, 2009 1:28 am UTC
- Forum: Mathematics
- Topic: What's the coolest thing you've ever learned in math class?
- Replies:
**58** - Views:
**6601**

### Re: What is...

This. more generally, how bifurcations work.

- Sat Sep 19, 2009 10:06 am UTC
- Forum: Mathematics
- Topic: Teaching myself higher mathematics
- Replies:
**30** - Views:
**3507**

### Re: Teaching myself higher mathematics

This looks like a good resource too. I have physical copies of several analysis books, so I've never used this one myself, but it looks very similar to other books I've used. (And it's free. That's the big part.) It's fairly dense, so it might be a little intimidating, but if you really want to lea...

- Fri Sep 18, 2009 4:25 am UTC
- Forum: Mathematics
- Topic: Weird Ass-Function
- Replies:
**8** - Views:
**1442**

### Re: Weird Ass-Function

You should keep in mind that "the n th partial derivative" is only defined for n \in \mathbb{N} (and usually 0, since the 0th derivative of a function is generally defined as the function itself). This means that f(.5) doesn't really make sense. The formula wolfram gives you may wo...

- Fri Sep 18, 2009 2:15 am UTC
- Forum: Mathematics
- Topic: Where did I go wrong on this question?
- Replies:
**26** - Views:
**2511**

### Re: Where did I go wrong on this question?

Gowers's weblog talked about a similar issue a little while ago. He was discussing problems where you're asked to show that a function is well-defined. Obviously, it's kind of a silly question. If f is a function, then f is well-defined by definition. Likewise, if you call f^{-1} "the inverse ...

- Wed Sep 16, 2009 9:41 pm UTC
- Forum: Mathematics
- Topic: Teaching myself higher mathematics
- Replies:
**30** - Views:
**3507**

### Re: Teaching myself higher mathematics

If you want to learn something that you won't be covering in high school, I would definitely recommend topology or number theory. These subjects are very simple in the sense that they don't have very much prerequisite knowledge attached, so with some dedication you would be able to pick them up with...

- Fri Aug 28, 2009 12:09 am UTC
- Forum: Mathematics
- Topic: Struggling with the concept of compact space
- Replies:
**40** - Views:
**4049**

### Re: Struggling with the concept of compact space

The important part isn't that you can find a finite cover. You have to be able to start with any cover and prune it down to a finite subcover. Basically, whenever someone gives you a collection of sets that covers (0,1), you have to be able to take away all but a finite number of them and still cove...

- Tue Dec 16, 2008 10:06 pm UTC
- Forum: Mathematics
- Topic: D'oh
- Replies:
**15** - Views:
**1649**

### Re: D'oh

You can just take the discrete topology on any uncountable set, so this isn't a very interesting question as written. Good point. What I meant was totally disconnected. I was thinking more along the lines of ordinals. To be fair though, if someone just started analysis, the discrete topology in its...

- Tue Dec 16, 2008 12:50 pm UTC
- Forum: Mathematics
- Topic: Negative Numbers Raised to Irrational Exponents
- Replies:
**20** - Views:
**9361**

### Re: Negative Numbers Raised to Irrational Exponents

It's actually discontinuous along the entire branch cut of the complex log function, unless you allow multi-valued functions (and excepting where b is an integer). Depending on your convention, this is typically when a is a negative real, which makes it a particularly bad definition when trying to ...

- Tue Dec 16, 2008 12:45 pm UTC
- Forum: Mathematics
- Topic: D'oh
- Replies:
**15** - Views:
**1649**

### Re: D'oh

You're on the right track, but saying "list the neighborhoods" assumes that they are countable, which is what you're trying to prove. Have you tried using a contradiction? What would happen if they weren't countable? That's my standard operating procedure for these types of problems. Also,...

- Fri Nov 07, 2008 9:52 pm UTC
- Forum: Mathematics
- Topic: Help with an infinite series.
- Replies:
**9** - Views:
**1465**

### Re: Help with an infinite series.

This is one of the important distinctions between finite series and infinite series. When you add two numbers, it doesn't matter what order you do it in, you'll always get the same answer. 1 + 2 = 2 + 1 = 3. When you take an infinite series, though, changing the order changes the sum. You can take t...

- Tue Nov 04, 2008 5:43 am UTC
- Forum: Mathematics
- Topic: Maximum Principle for Parabolic PDEs
- Replies:
**0** - Views:
**651**

### Maximum Principle for Parabolic PDEs

I have a homework problem due tomorrow for my PDE class, and I've been beating my head against it for a while now. Assume that \Omega is bounded, \partial\Omega is of class C^2 and that u,v\in C^2(D) \cap C^1 (D) . (Note that D = \Omega \times (0,T] and \overline{D} = \overline{\...

- Wed Oct 22, 2008 5:58 am UTC
- Forum: Mathematics
- Topic: Help for trig.
- Replies:
**36** - Views:
**2506**

### Re: Help for trig.

My favorite mnemonic is still Some Old Hippy Caught Another Hippy Tripping On Acid. Probably not allowed in a lot of high schools though.

- Wed Oct 22, 2008 5:34 am UTC
- Forum: Mathematics
- Topic: A Simple question
- Replies:
**3** - Views:
**679**

### Re: A Simple question

Does your calculator have a button labeled "log"? If so, then yes. log(x)=y means that y is a number such that 10^x = y. In your case, you don't want to solve 10^a = 1234, but your calculator doesn't have a log-base-6-button, so you have to resort to a little mathematical trickery. One of ...

- Tue Oct 21, 2008 11:55 pm UTC
- Forum: Logic Puzzles
- Topic: Random Questions
- Replies:
**10** - Views:
**2841**

### Re: Random Questions

**Spoiler:**

- Fri Oct 03, 2008 9:04 pm UTC
- Forum: Mathematics
- Topic: Constraints on the derivative and the nature of a function
- Replies:
**12** - Views:
**1384**

### Re: Constraints on the derivative and the nature of a function

Well, the original post just said continuous and differentiable everywhere, so I was technically within the rules =). But you're right, that "for instance" is wrong. I'm too lazy to check right now, but I believe you need f to be continuously differentiable on a connected domain. I can't a...

- Fri Oct 03, 2008 8:04 pm UTC
- Forum: Mathematics
- Topic: Constraints on the derivative and the nature of a function
- Replies:
**12** - Views:
**1384**

### Re: Constraints on the derivative and the nature of a function

In case it's not obvious, you can do exactly the same thing if f' < e for some e<0. Sadly, it's not enough to say that the derivative is bounded away from zero (unless you specify that f is continuously differentiable). For instance, f: R\{0}-->R s.t. f(x)=|x| is no good. And I was going to say Lips...

- Fri Sep 26, 2008 1:20 am UTC
- Forum: Mathematics
- Topic: I might have an opportunity to help teach math next semester
- Replies:
**6** - Views:
**1255**

### Re: I might have an opportunity to help teach math next semester

As long as you know them both fairly well, I would recommend teaching calculus. I've found that it's far harder to teach more introductory classes than it is to teach advanced classes. When someone doesn't understand integration by parts, I can just think back to when I was learning the subject and,...

- Thu Sep 11, 2008 12:41 am UTC
- Forum: Mathematics
- Topic: Mathematics and Beauty
- Replies:
**42** - Views:
**5517**

### Re: Mathematics and Beauty

For the most part, I agree that surprising yet simple proofs are beautiful: the kind where you read it and suddenly the theorem fits perfectly and you can't remember why it seemed complicated before. I also find some theorems beautiful in themselves. For instance, the maximum principle (more of a co...