bloodhound Posted July 7, 2004 Share Posted July 7, 2004 This thread should be used to post your favourite theorems, not only their names, but also what they state. Fundamental Theorem of Arithmetic Each integer greater than 1 can be expressed as a product of primes, and, except for the order in which these primes are written, this can be done in only one way. and Central Limit Theorem If [math]X_1,...,X_n[/math] are independent identically distributed, with mean [math]\mu[/math] and finite variance [math]\sigma^{2}[/math], then [math]\bar X[/math] is approximately [math] N(\mu ,{{\sigma ^2 } \mathord{\left/ {\vphantom {{\sigma ^2 } n}} \right. \kern-\nulldelimiterspace} n}) [/math] for large n, no matter what the distribution of [math]X[/math] Link to comment Share on other sites More sharing options...
Dave Posted July 7, 2004 Share Posted July 7, 2004 For me, it'd have to be something like the intermediate value theorem, because it seems so inanely pointless, but allows you to prove very useful things. Link to comment Share on other sites More sharing options...
bloodhound Posted July 7, 2004 Author Share Posted July 7, 2004 could u state it please. so that anyone who might be interested would know what your talking about Link to comment Share on other sites More sharing options...
Tesseract Posted July 7, 2004 Share Posted July 7, 2004 Euclid's Parallel Postulate Through a point, not on a line, there exists exactly 1 line parallel to the given line. sum (k=1..inf) 1/kn = ? Although others have found that this expression equals PI2 / 6 when n=2, PI4 / 90 when n = 4 and simular solutions for all possible even values of n, no one has discovered an exact value when n is an odd integer (3, 5, 7, ...) (note: when n=1, the sum does not converge, but it does has relations to the gamma constant). Sorry I copied and pasted: http://www.geocities.com/RainForest/Vines/2977/gauss/euclidpp.html Link to comment Share on other sites More sharing options...
Dave Posted July 7, 2004 Share Posted July 7, 2004 Certainly. IVT For a function [math]f : [a,b] \to \mathbb{R}[/math] which is continuous, then [math]\forall\, c \in [a,b] \, \exists \, v \in (f(a), f(b))[/math] such that [math]f© = v[/math]. Link to comment Share on other sites More sharing options...
bloodhound Posted July 7, 2004 Author Share Posted July 7, 2004 I also quite like the Monotonic Sequence Theorem. Let [math](x_n)[/math] be a sequence which is non-decreasing for [math]n \ge N[/math]. If [math](x_n)[/math] is bounded above, then [math](x_n)[/math] converges, and the limit is the supremum s of the set [math]{x_n:n \in \mathbb{Z}, n \ge N}[/math]. If [math](x_n)[/math] is not bounded above then [math](x_n)[/math] diverges to infinity. Link to comment Share on other sites More sharing options...
fourier jr Posted July 7, 2004 Share Posted July 7, 2004 - Heine-Borel Theorem: a subspace of R^n is (with the usual topology) is compact iff it is closed and bounded. - Bolzano-Weierstrass Theorem: Every bounded infinite set in R^n has an accumulation point - Sylow's Theorems: http://mathworld.wolfram.com/SylowTheorems.html - Mean Value Theorem: http://mathworld.wolfram.com/Mean-ValueTheorem.html Link to comment Share on other sites More sharing options...
MandrakeRoot Posted July 7, 2004 Share Posted July 7, 2004 Euclids parallel postulate is an axiom and not a theorem. I would say Riesz's Representation theorem : The dual of every hilbert space can be isometrically isomophically embedded into itself. Mandrake Link to comment Share on other sites More sharing options...
JaKiri Posted July 7, 2004 Share Posted July 7, 2004 e^ipi = -1 It's fun and fruity! Link to comment Share on other sites More sharing options...
MandrakeRoot Posted July 7, 2004 Share Posted July 7, 2004 That is not a theorem either, but more a direct result from an identity exp(i phi) =cos(phi) + i*sin(phi) Mandrake Link to comment Share on other sites More sharing options...
JaKiri Posted July 7, 2004 Share Posted July 7, 2004 That is not a theorem either' date=' but more a direct result from an identityexp(i phi) =cos(phi) + i*sin(phi) Mandrake[/quote'] I know it is, but e^i theta isn't as fun and fruity as e^i pi. Link to comment Share on other sites More sharing options...
NSX Posted July 8, 2004 Share Posted July 8, 2004 I like De Moivre's Theorem. Other than finding complex roots, I like it as a easy(?) way to remember expansions of the sin & cos funx. De Moivre's Theorem If z = r cos(θ) + i r sin(θ), then zn = r cos(nθ) + i r sin(nθ) for all n ∈ N Source: Anton, Rogers. Elementary Linear Algebra Applications Version Link to comment Share on other sites More sharing options...
JaKiri Posted July 8, 2004 Share Posted July 8, 2004 I like De Moivre's Theorem. Other than finding complex roots' date=' I like it as a easy(?) way to remember expansions of the sin & cos funx. [b']De Moivre's Theorem[/b] If z = r cos(?) + i r sin(?), then zn = r cos(n?) + i r sin(n?) for all n ? N Source: Anton, Rogers. Elementary Linear Algebra Applications Version De Moivre's is winking at me. It's never done that before Link to comment Share on other sites More sharing options...
MandrakeRoot Posted July 8, 2004 Share Posted July 8, 2004 I think the statement of the moivre's theorem is : If z = rcos(x) + irsin(x), then z^n = r^n \cos(n x) + i r^n \sin(n x) for all n \in \mathbb{N} Mandrake Link to comment Share on other sites More sharing options...
MandrakeRoot Posted July 8, 2004 Share Posted July 8, 2004 Hey how do you guys make the math signs ? Mandrake Link to comment Share on other sites More sharing options...
Sayonara Posted July 8, 2004 Share Posted July 8, 2004 There's a thread on using the LaTex maths renderer in the General Maths forum. LaTex is like bb code only more... mathsy. Link to comment Share on other sites More sharing options...
YT2095 Posted July 8, 2004 Share Posted July 8, 2004 Murphys Law, relating to the maximum cussedness of matter, Stating that: "If anything CAN go wrong, it Will!" Link to comment Share on other sites More sharing options...
bloodhound Posted July 8, 2004 Author Share Posted July 8, 2004 I think the statement of the moivre's theorem is :If z = rcos(x) + irsin(x)' date=' then z^n = r^n \cos(n x) + i r^n \sin(n x) for all n \in \mathbb{N} Mandrake[/quote'] think it works for all rational numbers as well Link to comment Share on other sites More sharing options...
MandrakeRoot Posted July 8, 2004 Share Posted July 8, 2004 If [math]z = rcos(x) + irsin(x)[/math], then [math]z^n = r^n \cos(n x) + i r^n \sin(n x) \; \forall \; n \in \mathbb{N} [/math] Mandrake PS : why my math formula doesn't show up ? The only thing to do to have full LaTeX use is to write [ math ], [/math ] around the LaTeX code right ? Link to comment Share on other sites More sharing options...
MandrakeRoot Posted July 8, 2004 Share Posted July 8, 2004 Yes bloodhound, it does indeed Mandrake Link to comment Share on other sites More sharing options...
dryan Posted July 8, 2004 Share Posted July 8, 2004 My favorites: The Fundamental Theorem of Algebra: An n-th order polynomial has exactly n roots in the complex complex. For example, [math]x^4-4[/math] has 4 complex roots (Two real, two imaginary). And of course, The Fundamental Theorem of Calculus http://archives.math.utk.edu/visual.calculus/4/ftc.9/ Link to comment Share on other sites More sharing options...
Sayonara Posted July 8, 2004 Share Posted July 8, 2004 [math]If z = rcos(x) + irsin(x)' date=' thenz^n = r^n \cos(n x) + i r^n \sin(n x) \; \forall \; n \in \mathbb{N} [/math'] PS : why my math formula doesn't show up ? The only thing to do to have full LaTeX use is to write [ math ], [/math ] around the LaTeX code right ? You need to use valid syntax. That includes not using invalid syntax, such as trying to write your formula in a sentence. If [math]z = rcos(x) + irsin(x)[/math], then [math]z^n = r^n \cos(n x) + i r^n \sin(n x) \; \forall \; n \in \mathbb{N}[/math] Link to comment Share on other sites More sharing options...
MandrakeRoot Posted July 9, 2004 Share Posted July 9, 2004 Hey thanks. Thought it could take the text also. Mandrake Link to comment Share on other sites More sharing options...
NSX Posted July 11, 2004 Share Posted July 11, 2004 ah...drats I tried C&P the theta sign into the quick-reply box. I guess it doesn't work. Thanks to Sayo for the non-winking version of De Moivre's Th'm. I found the Binomial Theorem neat too: The theorem that' date=' for positive integers n, [img']http://mathworld.wolfram.com/bimg2594.gif[/img] the so-called binomial series, where are binomial coefficients. It was very neat when we studied it with Pascal's triangle, combinatorics, and the like. Link to comment Share on other sites More sharing options...
zaphod Posted November 29, 2004 Share Posted November 29, 2004 without a doubt Cantor's theorem on the nondenumberability of the continuum. Link to comment Share on other sites More sharing options...
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