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From wikipedia on algebraic numbers: > "In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational c
by NAFV_P 12y ago
From wikipedia on algebraic numbers:
> "In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently—by clearing denominators—with integer coefficients)."
I almost forgot, the set of algebraic numbers also includes complex numbers.
- xyzzyz 12y agoIf you mean that some complex numbers are algebraic, then yes -- the imaginary unit i itself is algebraic, as it satisfies an polynomial equation x^2 + 1 = 0 with rational (in fact integer) coefficients. The set of algebraic numbers does not include all complex numbers, though.
- deleted 12y ago[deleted]
- auvrw 12y agomore generally, from Lang: > Let F be a subfield of a field E. An element \alpha of E is said to be {algebraic} over F if there exist elements a_0, ..., a_n (n >= 1) of F, not all equal to 0, such that a_0 + a_1\alpha^n + ... + a_n\alpha^n = 0. point being, even a sub-par undergrad knows how to generalize algebraic numbers using some machinery that was just beginning to be built in Ramanujan's time (and of course was completely unavailable to the man himself), and now these guys have plugged fields into something that i'd only heard of in the context of group theory and proved something useful. way to go. but at the same time, if Ramanujan saw this, i have to imagine he'd be thinking something along the lines of, "the game ain't the same." (not that it's a game.)
- surement 12y agoFun fact: the set of algebraic numbers is countable. If you recall that the set of rational numbers is also countable, then you get that the reals are uncountable only "because" of transcendental numbers (pi, e, Chapernowne's number, etc.).
- pash 12y agoAs Alonzo Church and Alan Turing showed, the computable numbers [0] are countable too. The computable numbers include all the algebraic numbers and some transcendental numbers (including π and e), so the reals are uncountable "because" of those other transcendentals, the uncomputable numbers. Put differently, almost all reals are uncomputable. 0. https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number
- NAFV_P 12y agoLike Omega: http://en.wikipedia.org/wiki/Chaitin's_constant http://en.wikipedia.org/wiki/Chaitin's_constant
- GregBuchholz 12y agoCan't mention Omega without linking to: Meta Math! http://arxiv.org/abs/math/0404335 http://arxiv.org/abs/math/0404335
- NAFV_P 12y agoGood book, I read it a few years ago. His own proof that there are infinitely many primes is a good head spinner.
- Ind007 12y agoSuch a good read.Thanks for sharing this.
- JonnieCache 12y agoCheck him out on youtube too, he's a fun speaker.
- surement 12y agoI've only ever heard of one uncomputable number (Chaitin's constant), so this is rather mind-blowing.