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I didn't feel like reading all the replies. Anyway here is mine: MASTER algebra and trig.

sqrt(ab) does equal sqrt(a) * sqrt(b), equivalently (ab)^(1/2) = a^(1/2)*b^(1/2) Of course when a,b < 0 you run into problems. sqrt(-25) = sqrt(25*-1) = sqrt(25)*sqrt(-1) = 5i, where i is sqrt(-1) But when one factor is negative and one is positive you're fine. For instance: sqrt(-5) * sqrt(-5) = -5, because by definition (sqrt(x))^2 = x. However if you group (for lack of a better term) the power 1/2 over the factors which are negative you run into problems.... sqrt(-5) * sqrt(-5) as sqrt(-5*-5) as sqrt(25) = 5. Anyway, in short, the rules of radicals for real numbers are not explicitly the same for complex. If that didn't make sense, I'll let someone else try to explain. For the most part, I think it is a matter of definitions of radicals which give conflicts with negative numbers. --------------- 0.99999... does not terminate. Do not look at it as a finite decimal. 0.99999... is a converging geometric series. It's in the form of S_n = a_1/(1-r), where S_n is the sum of the nth terms in the series, and a_1 is the first term and r is the ratio between each terms a_(n+1)/a_n 0.99999... converges (approaches a limit) at 1, hence if the number of terms in this particular series goes into infinity (indicated by the "..." notation) the series can never be greater than 1 so we say that 0.999... = 1 Anyway, if I missed something again, let me know.

I forgot to comment on sin^2(x) + cos^2(x) = 1. It is merely the Pythagorean Theorem on a unit circle. I'll explain why this is so: The sine function, sin(x) (read sine of x) is defined to give the ratio of the side opposite of an angle on a right triangle to its hypotenuse. The cosine function, cos(x) is defined to give the ratio of the side adjacent (the side that is next to the angle but is not the hypotenuse) of an angle on a right triangle to its hypotenuse. sine = side opposite / hypotenuse cosine = side adjacent / hypotenuse You need to remember that a ratio can be given in many ways: One half can be written as 5/10, 30/60, 3/6, or 0.5/1 among an infinite amount of other ways. Now, recall sin(30 degrees) = 1/2. Which is saying side opposite / hypotenuse is 1/2, which therefore means: side opposite / hypotenuse = 0.5/1 Since all ratios can be written in the form of x/1 (meaning x can be reduced to satisify this property), the hypotenuse can always "be 1". Hence when using the pythagorean theorem you're always going to get 1 when evaluating sin^2(x) + cos^2(x). Hopefully, that made some sense In any case, I felt like trying to help you all understand this identity to see it is not magic.

Unforunately, you're probably learning the material too fast to retain any of it for long-term memory. All I can recommend in that case is that you practice a lot. Btw, I can't wait for this forum to grow

(Great) this only allows us to use fossil fuels even longer... hint: ozone problems... But then again, this is also a good thing, for the reasons already stated. Anyone else have any thoughts?

Very roughly, Calculus is a branch of mathematics that deals with the measure of change. Here are some links to get you started: http://mathforum.org/library/drmath/view/51436.html I'm assuming you're perfectly caught up on your precalculus concepts... Learning from the web is the worst way to learn calculus. Buy a good textbook. I heard Anton is great: http://www.amazon.com/exec/obidos/tg/stores/detail/-/books/0471153060/contents/ref=pm_dp_ln_b_2/104-7890783-6061567 Some people in #math (on efnet) said they have used Stewart http://www.amazon.com/exec/obidos/ASIN/0534362982/qid=1027212824/sr=1-1/ref=sr_1_1/104-7890783-6061567 This book from Spivak is an excellent calculus book and can help you prepare for Analysis (a more generalized term for the branches of math that involve calculus) http://www.amazon.com/exec/obidos/ASIN/0914098896/qid=1027213056/sr=2-2/ref=sr_2_2/104-7890783-6061567 I'm planning on buying Spivak soon to see how it is. I can make a review of the book later on if you want. I'd give an intro, but I'm terrible when it comes to teaching (besides I'm still learning this myself!) Anyway, good luck!

Most of this is just a test of your algebra (simplifications) and the very basics of your knowledge of trig identities.... int(sqrt[(t*cos(t) + sin(t))^2 + (cos(t) - t*sin(t))^2]) dt We first expand the binomials, therefore we have (a+b)^2 = (a^2 + 2ab + b^2); (a-b)^2 = (a^2 - 2ab + b^2) int(sqrt[[t^2*cos^2(t) + 2t*sin(t)cos(t) + sin^2(t)] + [cos^2(t) - 2t*sin(t)cos(t) + t^2*sin^2(t)]]) dt 2t*sin(t)cos(t) - 2t*sin(t)cos(t) = 0, therefore we have: int(sqrt[t^2*cos^2(t) + [sin^2(t) + cos^2(t)] + t^2*sin^2(t)]) dt Indeed, since since sin^2(t) + cos^2(t) = 1, we have: int(sqrt[1 + t^2*cos^2(t) + t^2*sin^2(t)]) dt Factoring inside the radical, we obtain: int(sqrt[1 + t^2(cos^2(t) + sin^2(t))]) dt Once again, since sin^2(t) + cos^2(t) = 1, we have: int(sqrt[1 + t^2]) dt Anyway, I get sqrt(t + (t^3)/3) + C (hmm, i think?) for my integral in case you want to check your answer. Note: Scratch that, this involves hyperbolic functions which I have no clue about. Sorry. Hope that helps!

New Scientist.com reports of a possible amino acid found in space. This will certainly add to the debate of potential life outside our planet. The article can be accessed here.

I'm thinking of buying a number theory book for self-study. I was wondering.. what are the prerequisites in math subjects for successfully learning number theory? (besides arithmetic ) Also, any good book recommendations would help. Thanks

yeah, now that you mention it, you're right... like you said, this forum is too new, you should add it in its own section if there is enough interest later on

haha, memorize your basic trig identities! i knew a guy who had tables of them in his ti-89, needless to say, his exams were easy heh but i think it helps to memorize your basic identites, like: tan(x) = sin(x)/cos(x) sin^2(x) + cos^2(x) = 1 sin(x) = cos(pi/2 - x) (same with any trig function and its cofunction) sin(a +- b), cos(a +-b) and some others i didn't list... and use what you know to derive something else, e.g. using sin(a +- b)/cos(a +-b) = tan(a +- b) to derive an identity for the sum/difference tangent formula... at least, this helped me out. if i knew d/dx sin(x) = cos(x), then i can find out d/dx cos(x) by using the fact that cos(x) = sin(pi/2 - x), therefore i find d/dx sin(pi/2 - x) which is cos(pi/2-x) * (-1), and since cos(pi/2-x) = sin(x), we have d/dx cos(x) = -sin(x) i can go on but you get the idea

Right, I know that one can compute the area under a curve using FTC. However I don't see the connection by just subtracting the two antiderivatives. It seems like you should just make a different notation for antidifferentation and a different notation for computing definite integrals ... Sorry about that. I should have said the limit of a riemann sum as the number of intervals approach infinity. (Which would yield an exact answer assuming you can find its limit [whether or not the sums are left, right, or mid-point sums])

Hi all. Just a small suggestion... maybe you can add Linear Algebra to the list of math topics in the math forum.

Okay, I am in the middle of learning single variable calculus and I can understand most of the topics I've learned so far. There are some things that are not obvious or easy understand to me, e.g. the Fundamental Theorem of Calculus: int(a,b, f(x) dx) = F(b) - F(a) , where F is an antiderivative of f. I can understand that in order to find the area between two intervals one would subtract the smaller area from the larger. So I guess this would imply that taking the antiderivative of a function and evaluating it at some point (we'll call it b) would give you the area from 0 to b. (Someone verify this) I see computing the area under a curve from the development of the Riemann Sum, but how is that the same as evaluating the difference between two antiderivatives ( F(b), and F(a) ) ? If anyone can give a nice explanation or reference on FTC I would appreciate it. Sorry for any typos. Thanks.