Editing Quadratic irrational number (section)

{{Short description|Mathematical concept}}
In [[mathematics]], a '''quadratic irrational number''' (also known as a '''quadratic irrational''' or '''quadratic surd''') is an [[irrational number]] that is the solution to some [[quadratic equation]] with [[rational number|rational]] [[coefficient]]s which is [[Irreducible polynomial|irreducible]] over the [[rational number]]s.<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 [[least common denominator]], a quadratic irrational is an irrational root of some quadratic equation with [[integer]] coefficients. The quadratic irrational numbers, a [[subset]] of the [[complex number]]s, are [[algebraic number]]s of [[Algebraic number#Properties|degree 2]], and can therefore be expressed as

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

for integers {{math|''a'', ''b'', ''c'', ''d''}}; with {{math|''b''}}, {{math|''c''}} and {{math|''d''}} non-zero, and with {{math|''c''}} [[Square-free integer|square-free]]. When {{math|''c''}} is positive, we get '''real quadratic irrational numbers''', while a negative {{math|''c''}} gives '''complex quadratic irrational numbers''' which are not [[real number]]s. This defines an [[injective function|injection]] from the quadratic irrationals to quadruples of integers, so their [[cardinality]] is at most [[countable]]; since on the other hand every square root of a [[prime number]] is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a [[countable set]]. [[Abu Kamil]] was the first mathematician to introduce irrational numbers as valid solutions to quadratic equations.<ref>{{Cite book |last1=Gowers |first1=Timothy |url=https://books.google.com/books?id=GLumDwAAQBAJ&q=%22irrational+solutions+to+quadratics%22+&pg=PA1010 |title=The Princeton Companion to Mathematics |last2=Barrow-Green |first2=June |last3=Leader |first3=Imre |date=2008-09-28 |publisher=Princeton University Press |isbn=978-0-691-11880-2 |language=en}}</ref><ref>{{Cite book |last=Meisner |first=Gary B. |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 |date=2018-10-23 |publisher=Race Point Publishing |isbn=978-1-63106-486-9 |language=en}}</ref>

Quadratic irrationals are used in [[field theory (mathematics)|field theory]] to construct [[field extension]]s of the [[Field (mathematics)|field]] of rational numbers {{math|'''Q'''}}. Given the square-free integer {{math|''c''}}, the augmentation of {{math|'''Q'''}} by quadratic irrationals using {{math|{{sqrt|''c''}}}} produces a [[quadratic field]] {{math|'''Q'''({{sqrt|''c''}}}}). For example, the [[Multiplicative inverse|inverses]] of elements of {{math|'''Q'''({{sqrt|''c''}}}}) are of the same form as the above algebraic numbers:

:<math>{d \over a+b\sqrt{c}} = {ad - bd\sqrt{c} \over a^2-b^2c}. </math>

Quadratic irrationals have useful properties, especially in relation to [[continued fraction]]s, where we have the result that ''all'' real quadratic irrationals, and ''only'' real quadratic irrationals, have [[periodic continued fraction]] forms. For example

:<math>\sqrt{3} = 1.732\ldots=[1;1,2,1,2,1,2,\ldots]</math>

The periodic continued fractions can be placed in one-to-one correspondence with the rational numbers. The correspondence is explicitly provided by [[Minkowski's question mark function]], and an explicit construction is given in that article. It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark function. Such repeating sequences correspond to [[periodic orbit]]s of the [[dyadic transformation]] (for the binary digits) and the [[Gauss–Kuzmin–Wirsing operator|Gauss map]] <math>h(x)=1/x-\lfloor 1/x \rfloor</math> for continued fractions.