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1. What is division?

This doesn't work quite smoothly for division by real numbers or fractions in general. It's still a good perspective for natural numbers nevertheless.
2. What is division?

If you are happy with your "meaning" of multiplication but not with the one for division, why don't you view division from a multiplication perspective? For example: x/y = k is the same as saying x = ky. That is x/y is that number by which you multiply y in order to get x.
3. Is empty set part of every set?

{} is a subset of every set, so it is a subset of s. Therefore, {} is an element of p(s). However, it is clear that {} is not in set d. Therefore, {} is not an element of the intersection of p(s) and d. Hope that answers your question.
4. Methods of Proving Irrational

I suppose so. Proof: Suppose p2 is irrational. Suppose p is rational. Then, p = a/b for some integers a and b. Therefore, we can conclude that p2= a2/b2, which is clearly rational. But that clearly contradicts our initial assumption that p2 is irrational. Thus, we conclude p is...
5. Methods of Proving Irrational

Does the following work? Suppose √2 + √5 = p/q for some integers p and q. Then 2 + 2√10 + 5 = p2/q2. It follows that 2√10 = p2/q2 - 7 = r/s for some integers r and s. Thus, √10 = r/2s = m/n for some integers m and n. But it is easy to show that √10 is irrational. Thus, our...
6. After calculus first pure math course, having a hard time.

I would advise you to be careful and not to adopt such an attitude. Think about it for a second: you posted some of your homework problems, and evidently, all of them involved proofs. Given that you don't have much experience with proof-writing, I believe that it would be wiser to carefully...
7. Some information on this topic?

I don't know how this concept is called; it sounds like something in economics or financial mathematics to me. Having said that, I believe your problem is a purely probabilistic one. Here is an explanation: The chances of winning or losing is 50%-50%. If you win, your new balance is 110%...
8. After calculus first pure math course, having a hard time.

Hi jgens, I am pretty sure that your sketch of the proof is very elegant and efficient, but I feel "mod" is an unnecessary artifice for somebody trying to understand (and like) this type of pure math course. I have something which is a little bit simpler to handle for a novice: Suppose k...
9. Calculating number of character combinations in password

yeah, the calculation is more complicated than that. Because your calculation says that 12345678 ,for example, is a possible password. But there must be at least a number, at least an alphabet and at least one of the 3 symbols you mentioned above.
10. How do we know Pi is Constant?

I know what you mean but he offered a simple and effective graphical representation from which it can perfectly be grasped that C/d for any circle is just as good as scaling d/C by making the same circle smaller or bigger by any factor. So it gives a good idea, though it isn't rigorous at all...
11. How do we know Pi is Constant?

That was brilliant! A very simple graphical proof. Good job dodo.
12. Can any one give me the derivation of this series

If I'm not mistaken, Dexterdev, you are trying to find a formula for the finite sum of the above-mentioned series? Sure this could be difficult. But just out of curiousity, why would you want to find the formula for this sum?
13. How do we know Pi is Constant?

This is just a very basic proof where you take a general circle of radius R. You'll find that the area will always converge to a value, Pi*(R^2). Similarly, you can do the same for the circumference and find a clever way to show how the 2 are related. All of this can be done without calculus...
14. How do we know Pi is Constant?

Actually Pi is obtained by taking a limit so that it converges to 3.14.... It's like e (base of natural log) which converges to a given value by taking a limit. And I'm not really sure what you mean by "pi is a constant". Anyway all I know is how to compute the value of pi (proof).
15. Finite sum formula for tangent (trigonometry)

Wow. I had no idea it was that complicated. I'm in high school right now. These functions in real/complex analysis is way beyond me. Anyway, thank you everyone though. At least now I know which direction I must be heading to learn more about it.
16. Finite sum formula for tangent (trigonometry)

:cry: ok, thx for your consideration though. I'll wait a bit more; maybe someone's got a way to do it.
17. Finite sum formula for tangent (trigonometry)

Hi everyone, I've been looking for the finite sum formulae of trig functions. I've found the easiest ones (sine and cosine). But the one for the tangent seems to be very hard. No mathematical tricks work. Plus I've looked it up on the internet. Nothing. I will greatly appreciate your help...
18. Doing theoretical/pure mathematics

Sorry but I seriously think that even number theory originated from observing the physical world. For instance, it's completely logical to deduce that numbers were invented in the first place only to quantify things that we can see. All the other theories simply added up to that main basic idea.
19. Trying to solve any equation ever with Recurrences

I've learnt this before what you found is actually called iteration... the idea is that as you keep on feeding in the values generated by f(x) in x=f(x), you will converge to the solution. However the sequence won't always converge (i.e it will diverge from the solution). As you might have...