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...ield]], an algebraic number field of degree two over the field of rational numbers * [[Quadratic irrational]] or "quadratic surd", an irrational number that is a root of a quadratic polynomial ...

...[[Diophantine approximation|Rational approximants]] of [[irrational number|irrational]] values can be mapped to points lying close to lines having gradients corr In [[mathematics]], the {{mvar|n}}-dimensional '''integer lattice''' (or '''cubic lattice'''), denoted {{tmath|\mathbb{Z}^n}}, is the [[lattice (group)|latti ...

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

...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 ...

...ndices Felix Kling.svg|thumb|300px|Planes with different Miller indices in cubic crystals]] ...hat [hkℓ] is ''not'' generally normal to the (''hkℓ'') planes, except in a cubic lattice as described below. ...

...) is a quantity produced by the [[division (mathematics)|division]] of two numbers.<ref>{{Cite web|title=Quotient|website=Dictionary.com|url=http://dictionary For example, [[density]] (mass divided by volume, in units of [[kilogram per cubic metre|kg/m³]]) is said to be a "quotient", whereas [[mass fracti ...

...{{mvar|c}} is not the square of a rational number, there are two rational numbers {{mvar|x}} and {{mvar|y}} such that If the nested radical is real, {{mvar|x}} and {{mvar|y}} are the two numbers ...

===Cubic equation=== The general [[cubic equation]] ...

...s irrational magnitudes. He then explained Book X using those rational and irrational magnitudes instead of geometric magnitudes like in the original.{{sfn|Dold- ...

the numbers ''h'' and ''k'' may be interpreted as the [[Cartesian coordinates]] of the ===Irrational and complex roots=== ...

...d their study in the 1940s. [[Salem number]]s are a closely related set of numbers. A characteristic property of PV numbers is that their powers [[almost integer|approach integers]] at an exponential ...

** [[Cubic function]]: Third degree polynomial. * [[Euler's totient function]]: Number of numbers [[coprime]] to (and not bigger than) a given one. ...

Using [[Halley's method]], {{math|&Omega;}} can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iter {{Irrational number}} ...

...[[abundant number]],<ref name="abundant">{{Cite OEIS|A005101|name=Abundant numbers (sum of divisors of m exceeds 2m)}}</ref> since [[Aliquot sum|the sum of it ...divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).}}</ref> ...

...)|square]] of an integer;<ref>Some authors also call squares of [[rational numbers]] perfect squares.</ref> in other words, it is the [[multiplication|product ...s]] (other examples being [[Cube (algebra)|cube numbers]] and [[triangular numbers]]). ...

{{about|nth-roots of real and complex numbers|other uses|Root (disambiguation)#Mathematics}} ...''. Roots of higher degree are referred by using [[ordinal numeral|ordinal numbers]], as in ''fourth root'', ''twentieth root'', etc. The computation of an { ...

...e theorem depends on, and is equivalent to, the [[completeness of the real numbers]], although [[Weierstrass Nullstellensatz]] is a version of the intermediat Consider the closed interval <math>I = [a,b]</math> of real numbers <math>\R</math> and a continuous function <math>f \colon I \to \R</math>. T ...

...o if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, ...understand how to work with complex numbers in order to solve all forms of cubic equation. ...

*The discovery of [[irrational number]]s by the school of [[Pythagoras]] in ancient Greece; for example, t ...mber]]s in the 16th century in order to find the roots of [[Cubic function|cubic]] and [[Quartic function|quartic]] [[polynomial function]]s. ...

...n integer|Euler integer||List of topics named after Leonhard Euler#Euler's numbers}} ...gorithm is as follows. First perform the division in the field of complex numbers, and write the quotient in terms of {{math|''ω''}}: ...