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en>Gilliam: Reverted edits by 108.3.211.60 (talk) to last version by ClueBot NG

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Reverted edits by 108.3.211.60 (talk) to last version by ClueBot NG

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{{Refimprove|date=May 2008}}
In [[mathematics]], a '''quadratic irrational''' (also known as
a '''quadratic irrationality''' or '''quadratic surd''') is an [[irrational number]] that is the solution to some [[quadratic equation]] with rational coefficients.<ref>Jörn Steuding, ''Diophantine Analysis'', (2005), Chapman & Hall, p.72.</ref> Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their [[common denominator]], a quadratic irrational is an irrational root of some quadratic equation whose coefficients are [[integer]]s. The quadratic irrationals form the [[Real number|real]] [[algebraic number]]s of degree 2 and can, therefore, be expressed in this form:

:<math>{a+b\sqrt{c} \over d}</math>

for [[integer]]s ''a'', ''b'', ''c'', ''d''; with ''b'' and ''d'' non-zero, and with ''c'' > 1 and [[Square-free integer|square-free]]. This implies that the quadratic irrationals have the same [[cardinality]] as ordered quadruples of integers, and are therefore [[countable]].

The [[rational number]]s together with all quadratic irrationals with a given ''c'' form a [[Field (mathematics)|field]], called a [[Quadratic field|real quadratic field]]. In particular, their inverses are of the same form, since
:<math>{d \over a+b\sqrt{c}} = {ad - bd\sqrt{c} \over a^2-b^2c}. \, </math>
This field is often called the field obtained by adjoining √{{Overline|''c''}} to the rational numbers, and denoted '''Q'''(√{{Overline|''c''}}).

Quadratic irrationals have useful properties, especially in relation to [[continued fraction]]s, where we have the result that ''all'' quadratic irrationals, and ''only'' quadratic irrationals, have [[periodic continued fraction]] forms. For example
:<math>\sqrt{3}=1.732\ldots=[1;1,2,1,2,1,2,\ldots]</math>

==Square root of non-square is irrational==
The definition of quadratic irrationals requires them to satisfy two conditions: they must satisfy a quadratic equation and they must be irrational. The solutions to the quadratic equation ''ax''2 + ''bx'' + ''c'' = 0 are
:<math>\frac{-b\pm\sqrt{b^2-4ac}}{2a}.</math>
Thus quadratic irrationals are precisely those numbers in this form that are not rational. Since ''b'' and 2''a'' are both integers, asking when the above quantity is irrational is the same as asking when the square root of an integer is irrational. The answer to this is that the square root of any natural number that is not a [[ square number]] is irrational.

The [[square root of 2]] was the first such number to be proved irrational. [[Theodorus of Cyrene]] proved the irrationality of the square roots of whole numbers up to 17, but stopped there, probably because the algebra he used couldn't be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes. The original proof of the irrationality of the non-square natural numbers depends on [[Euclid's lemma]].

Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the [[fundamental theorem of arithmetic]], which was first proven by [[Carl Friedrich Gauss]] in his [[Disquisitiones Arithmeticae]]. This asserts that every integer has a unique factorization into primes. For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator. When the numerator is squared that prime will still not divide into it because of the unique factorization. Therefore the square of a rational non-integer is always a non-integer; by [[contrapositive]], the square root of an integer is always either another integer, or irrational.

[[Euclid]] used a restricted version of the fundamental theorem and some careful argument to prove the theorem. His proof is in [[Euclid's Elements]] Book X Proposition 9.<ref>{{cite web | url=http://aleph0.clarku.edu/~djoyce/java/elements/bookX/propX9.html |title=Euclid's Elements Book X Proposition 9 |accessdate=2008-10-29 |author=Euclid | publisher=D.E.Joyce,
Clark University }}</ref>

The fundamental theorem of arithmetic is not actually required to prove the result though. There are self-contained proofs by [[Richard Dedekind]],<ref>[http://www.cut-the-knot.org/proofs/sq_root.shtml Cut the knot: Square root of 2 is irrational]</ref> among others. The following proof was adapted by Colin Richard Hughes from a proof of the irrationality of the square root of two found by [[Theodor Estermann]] in 1975.<ref>Colin Richard Hughes, "Irrational roots", ''[[Mathematical Gazette]]'', Vol. 83, No. 498 (1999), pp. 502–503.</ref><ref>Theodor Estermann, "The irrationality of √2", ''Mathematical Gazette'', Vol. 59, No. 408 (1975), p. 110.</ref>

Assume ''D'' is a non-square natural number, then there is a number ''n'' such that:

:''n''2 < ''D'' < (''n'' + 1)2,

so in particular

:0 < √''D'' − ''n'' < 1.

Assume the square root of ''D'' is a rational number ''p''/''q'', assume the ''q'' here is the smallest for which this is true, hence the smallest number for which ''q''√''D'' is also an integer. Then:

:(√''D'' − ''n'')''q''√''D'' = ''qD'' − ''nq''√''D''

is also an integer. But 0 < (√''D'' − ''n'') < 1 so (√''D'' − ''n'')''q'' < ''q''. Hence (√''D'' − ''n'')''q'' is an integer smaller than ''q'' such that (√''D'' − ''n'')''q''√''D'' is also an integer. This is a contradiction since ''q'' was defined to be the smallest number with this property; hence √''D'' cannot be rational.

==See also==
* [[Algebraic number field]]
* [[Apotome (mathematics)]]
* [[Periodic continued fraction]]
* [[Restricted partial quotients]]
* [[Quadratic integer]]

==References==
{{reflist}}

==External links==
*{{mathworld|QuadraticSurd}}
*[http://www.numbertheory.org/php/surd.html Continued fraction calculator for quadratic irrationals]
*[http://planetmath.org/encyclopedia/EIsIrrational.html Proof that e is not a quadratic irrational]

[[Category:Number theory]]

Legume - Revision history

en>Rgdboer: /* Nutritional facts */ rm POV, see Talk#Facts?

en>Northamerica1000: + ==See also== * List of legume dishes

en>Gilliam: Reverted edits by 108.3.211.60 (talk) to last version by ClueBot NG