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{{wiktionary|quadratic}} In mathematics, the term '''quadratic''' describes something that pertains to [[Square (algebra)|squares]], to th ...

...quadratic equation with [[integer]] coefficients. The quadratic irrational numbers, a [[subset]] of the [[complex number]]s, are [[algebraic number]]s of [[Al ...|url=https://books.google.com/books?id=9Sl1DwAAQBAJ&dq=abu+kamil+quadratic+irrational+number&pg=PA39 |title=The Golden Ratio: The Divine Beauty of Mathematics |d ...

...{cite journal |author=R. O. Kuzmin |title=On a new class of transcendental numbers |journal=Izvestiya Akademii Nauk SSSR, Ser. Matem. |volume=7 |year=1930 |pa ...>\sqrt{2}^\sqrt{2}</math> is a rational which proves the theorem, or it is irrational (as it turns out to be) and then ...

In [[algebraic number theory]], a '''quadratic field''' is an [[algebraic number field]] of [[Degree of a field extension| ...<0</math>, it is called an '''imaginary quadratic field''' or a '''complex quadratic field''', corresponding to whether or not it is a [[Field extension|subfiel ...

*[[Even and odd numbers]] *[[Quadratic residue]] ...

...values of the [[Dedekind eta function]] at imaginary quadratic irrational numbers. The result was essentially found by {{harvs|txt|author-link=Mathias Lerch| ...bol]] modulo ''D'', where ''−D'' is the [[discriminant]] of an imaginary [[quadratic field]]. The sum is taken over 0 < ''r'' < ''D'', with the usual convention ...

...isomorphisms between the [[Quadratic irrational number|quadratic algebraic numbers]] and the [[rational number]]s, and between the rationals and the [[dyadic ...

...aic because it is a root of the polynomial <math>x^4 + 4</math>. Algebraic numbers include all [[integer]]s, [[rational number]]s, and [[nth root|''n''-th roo ...tably infinite]] algebraic numbers, hence [[almost all]] real (or complex) numbers (in the sense of [[Lebesgue measure]]) are transcendental. ...

...<ref name="wolfram">{{cite web |last=Weisstein |first=Eric Wolfgang |title=Quadratic Equation |url=https://mathworld.wolfram.com/QuadraticEquation.html |access- ...''quadratic polynomial'' are nearly synonymous and often abbreviated as ''quadratic''. ...

{{about|the ring of complex numbers integral over <math>\mathbb{Z}</math>|the general notion of algebraic integ ...therefore is a [[commutative ring|commutative]] [[subring]] of the complex numbers. ...

** [[Quadratic integral]] ==Special functions and numbers== ...

{{short description|Method for solving quadratic equations}} ...y algebra]], '''completing the square''' is a technique for converting a [[quadratic polynomial]] of the form {{tmath|\textstyle ax^2 + bx + c}} to the form {{t ...

...that can be described with an algebraic expression are called [[Algebraic numbers]].{{contradict-inline|section=In roots of polynomials|date=October 2024}} ...tion]], can always be written as algebraic expressions if ''n'' < 5 (see [[quadratic formula]], [[cubic function]], and [[quartic equation]]). Such a solution o ...

*[[Proof that e is irrational]] *[[Proof that π is irrational]] ...

...last = Lehmer|first = D.H.|author2=Powers, R.E.|title = On Factoring Large Numbers|journal = Bulletin of the American Mathematical Society|volume = 37|year = Since this is a [[quadratic irrational]], the continued fraction must be [[periodic continued fraction|periodic]] ...

...on-integer positional numeral system]] that uses the [[golden ratio]] (the irrational number <math display=inline>\frac{1+\sqrt{5}}{2}</math>&nbsp;≈&nbsp;1.61803 ...ch possess a finite base-φ representation is the [[ring (algebra)|ring]] [[quadratic integer|'''Z'''[<math display=inline>\frac{1+\sqrt{5}}{2}</math>]]]; it pla ...

...lar [[Aleksandr Yakovlevich Khinchin]] proved that for [[almost all]] real numbers ''x'', the coefficients ''a''_''i'' of the continued fraction exp ...eal number ''not'' specifically constructed for the purpose. The following numbers whose continued fraction expansions apparently do have this property (based ...

{{DISPLAYTITLE:Proof that {{mvar|e}} is irrational}} ...hann Bernoulli|Johann]], proved that {{math|''e''}} is [[Irrational number|irrational]]; that is, that it cannot be expressed as the quotient of two integers. ...

#''The Theory of Algebraic Numbers'', second edition, by [[Harry Pollard (mathematician)|Harry Pollard]] and H #''The Arithmetic Theory of Quadratic Forms'', by [[Burton Wadsworth Jones|B. W. Jones]] (out of print) ...

...m the theorem that the square roots of the non-square numbers up to 17 are irrational: ..., Hardy and Wright<ref>{{cite book |title=An Introduction to the Theory of Numbers |last1=Hardy |first1=G. H. |author1-link=G. H. Hardy |last2=Wright |first2= ...