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[Jim Groovester]

Don't you need the integral from -1 to 1? That's the x values where y = 1 and y = x² intersect.

That would double your result, just like you need.

[Darvi]

Ya, what he said.

[Vector]

Isn't it pi times the integral of (x^2-1)^2 from 0 to 1 that's asked for?

... Er... I think so?  I mean, the problem is written out verbatim, and my interpretation is written out, but I vaguely suspect that something is going criminally wrong, given that I'm getting exactly half of what I should.

Don't you need the integral from -1 to 1? That's the x values where y = 1 and y = x² intersect.

That would double your result, just like you need.

Oh my god, I suck.  I actually kept seeing pictures of that in my head, but I had no idea why.

Thank you.

[MaximumZero]

Man, it's a good thing I created this thread. I have no freaking clue what's going on in here, but I like to read the big words and fancy ways you folks arrange letters to mean numbers of some sort. It's almost like watching a stage show. I know that there are specific techniques involved in what you guys are doing, but from where I'm sitting, it may as well be magic.

[Dutchling]

I wish my math lessons were given in English, because most of the stuff they talk about is like wuh? to me

[ed boy]

Once again, this is bordering on being more logic than maths, but here it is anyway.

I've been doing a lot of pondering about the various fundamental definitions and properties of mathematical structures, and of course paradoxes have come up. Take, for example, russel's paradox. For those of you who are unfamiliar with russel's paradox, it is a such:

Let us call Russel's set the set of all sets that are not members of themselves. That is, if any set is not a member of itself, it is a member of Russel's set. The paradox arises when you consider whether Russel's set is a member of itself.

Now, this contradiction fouls up naive set theory, and we therefore have lots of different and complicated forms of set theory and logic, like morse-kelley and ZMF, which are been constructed so paradoxes like the above do not occur. However, this seems to be a bit of an unnescessary complication. The problem with naive set theory is that its domain of discourse (that is, the collection of all objects which we are considering and applying logic to) is all sets, which includes Russel's set. Why can we not just define an improved domain of discourse to be all sets which do not reult in a paradox?

Of course, we will have to carefully define what a paradox is. So, for the purposes of this post, we will say that:
-A logical system is called "paradoxical" if, by working within that logical system, it is possible to deduce that some statement within the system is not true xor false (Where xor is the exclusive or. That is, it is possible to deduce that some statement is both true and false, or neither true nor false).
-Similarly, we say a logical system is "not paradoxical" if it is not a paradoxical logical system.
-Within a paradoxical logical system, we say that a subset of that logical system is a paradox if, when removed from that logical system, the paradoxical logical system becomes not paradoxical.
-Similarly, within the context of a not paradoxical logical system, we call any logical object a paradox if including it in the logical system results in a paradoxical logical system.

This concept, although it may seem like cheating somewhat, is not entirely unfamiliar. Indeed, proofs by contradiction are a common thing in maths, and they work of the assumption that the logical system does not contain any paradoxes. In the majority of maths, the lack of paradoxes is not questioned, but at the fundamental level it is seemingly abandoned.

However, this arises another question: can you prove that something does not result in a paradox? Within a finite logical system, a proof by exhaustion would be possible, but how can one extend this to an infinite logical system, such as the natural numbers?

[Vector]

No, you can't.  Any model which permits Peano's Axioms (i.e. the natural numbers) disallows it.

Look up Godel.

[Virex]

Or to put it in other words: You're tossing out so much that you can't call the theory close to an overarching theory for sets any more. It may describe a subset of all sets, and that subset may or may not be trivial, but there are already a lot of theories that apply to a subset of all sets, so I don't see what this is going to add, especially if it doesn't permit Peano's Axioms (not just natural numbers. All vector spaces are build around Peano's Axioms for example)

[Another]

"All sets that do not result in a paradox" sounds similar to both the original "all sets that are not members of themselves" and to "maximum natural number that can be expressed in under fifteen English words". Statements like these have much in common with "this statement is false" (or "a lie").

If you didn't mean really all sets but a sufficiently large for practical needs set of sets - than see previous posters ^ .

I think that it would be not so hard to formally reduce "all/maximum [anything] that do not result in a paradox" to Godel's proof that either there will be a paradox, not "all/maximum" is possible or some really useful stuff (arithmetics that can be based on Peano's Axioms) will never make it into your axiom system.

[lordnincompoop]

'Ello. I'm currently working my way through a trigonometry section, and they're mentioning the u,v coordinate system. Now, I can't seem to find a satisfactory explanation, so if anyone can link me to a concise and understandable explanation or can write one, that'd be just awesome.

Also, is A = cos^(-1)((-a^2+b^2+c^2)/(2 b c)) really the formula to use when finding the angle A for triangle ΔABC? I've tried it for a few triangles (such as a = 2, b = 3, c = 4) and I'm not getting the right answer (~28.96 when it should be 28.57), and I can't figure out why.

[Darvi]

Lemme put that in a way that I can read.

A=Arccos((-a²+b²+c²)/(2bc)) ?

Also, that result might be due to rounding errors. Try something easier, like a=5, b=4, c=3 (Pythagoras, ya know)

A=arccos((-25+16+9)/(24))=arccos(0)=pi/2

Seems right.

[lordnincompoop]

Lemme put that in a way that I can read.

A=Arccos((-a²+b²+c²)/(2bc)) ?

Also, that result might be due to rounding errors. Try something easier, like a=5, b=4, c=3 (Pythagoras, ya know)

A=arccos((-25+26+9)/(24))=arccos(0)=pi/2

Seems right.

Ah, thanks. Hmm.

[Another]

28.57 in the previous example must have been 28°57'00'' which is equal to 28.95000...°. Some calculators just give angles with angle minutes and angle seconds which is IMHO a good reason to just keep all angles in radians.

[Christes]

'Ello. I'm currently working my way through a trigonometry section, and they're mentioning the u,v coordinate system. Now, I can't seem to find a satisfactory explanation, so if anyone can link me to a concise and understandable explanation or can write one, that'd be just awesome.

Also, is A = cos^(-1)((-a^2+b^2+c^2)/(2 b c)) really the formula to use when finding the angle A for triangle ΔABC? I've tried it for a few triangles (such as a = 2, b = 3, c = 4) and I'm not getting the right answer (~28.96 when it should be 28.57), and I can't figure out why.

A more standard way of writing the same thing is the Law of Cosines, which is typically stated as:

a²=b²+c²-2bc*cos(A)

Stated this way, it can also easily find the length of a, given A, b, and c.  Not sure if you've done that or not, but it's good to have.

Also, I'm not sure what you mean by u,v coordinate system.  Are you rotating the x and y axes?

[Miggy]

So I'm prepping for my linear algebra exam, and I'm preparing for all of the various questions I might end up facing.

Reading through all of my materials, one potential question entered my mind: Schur's decomposition works for complex numbers and unitary matrices. Does it also apply to Real numbers and orthogonal matrices? Looking through the proof and turning the general concepts in my head, I can't see why it shouldn't, but there is no mention of it anywhere, which leads me to believe that it isn't the case.