Content deleted Content added
Citation bot (talk | contribs) Add: s2cid, page. | Use this bot. Report bugs. | Suggested by Whoop whoop pull up | #UCB_webform 292/1877 |
|||
| (6 intermediate revisions by 6 users not shown) | |||
Line 1:
{{Short description|Real number uniquely specified by description}}
[[File:Square root of 2 triangle.svg|thumb|200px|The [[square root of 2]] is equal to the length of the [[hypotenuse]] of a [[right triangle]] with legs of length 1 and is therefore a '''constructible number''']]
Line 14 ⟶ 15:
[[File:Algebraicszoom.png|thumb|Algebraic numbers on the [[complex plane]] colored by degree (red=1, green=2, blue=3, yellow=4)]]
A real number <math>r</math> is called a real [[algebraic number]] if there is a [[polynomial]] <math>p(x)</math>, with only integer coefficients, so that <math>r</math> is a root of <math>p</math>, that is, <math>p(r)=0</math>.
Each real algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial <math>q(x)</math> has 5 real roots, the third one can be defined as the unique <math>r</math> such that <math>q(r)=0</math> and such that there are two distinct numbers less than <math>r</math> at which <math>q</math> is zero.
All rational numbers are
The real algebraic numbers form a [[field extension|subfield]] of the real numbers. This means that 0 and 1 are algebraic numbers and, moreover, if <math>a</math> and <math>b</math> are algebraic numbers, then so are <math>a+b</math>, <math>a-b</math>, <math>ab</math> and, if <math>b</math> is nonzero, <math>a/b</math>.
Line 62 ⟶ 63:
* ''[[Entscheidungsproblem]]''
* [[Ordinal definable set]]
* [[Richard's paradox]]
* [[Tarski's undefinability theorem]]
Line 71 ⟶ 73:
<ref name=kunen>{{Citation | last1=Kunen | first1=Kenneth | author1-link=Kenneth Kunen | year=1980 | title=[[Set Theory: An Introduction to Independence Proofs]] | publisher=North-Holland | ___location=Amsterdam | isbn=978-0-444-85401-8 | page=153}}</ref>
<ref name=tsirelson>{{Citation | last1=Tsirelson | first1=Boris | author1-link=Boris Tsirelson | year=2020 | title=Can each number be specified by a finite text? | periodical=WikiJournal of Science | volume=3 | issue=1 | page=8 | doi=10.15347/WJS/2020.008 | doi-access=free | arxiv=1909.11149 | s2cid=202749952 }}</ref>
<ref name=turing>{{Citation | last1=Turing | first1=A. M. | author1-link=Alan Turing | year=1937 | title=On Computable Numbers, with an Application to the Entscheidungsproblem | journal=[[Proceedings of the London Mathematical Society]] | series=2 | volume=42 | issue=1 | pages=230–65 | doi=10.1112/plms/s2-42.1.230 | s2cid=73712 | url=http://www.abelard.org/turpap2/tp2-ie.asp }}</ref>
| |||