quadratic surd
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s which is
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their
least common denominator In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. Description The l ...
, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the complex numbers, are
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s of degree 2, and can therefore be expressed as :, for integers ; with , and non-zero, and with square-free. When is positive, we get real quadratic irrational numbers, while a negative gives complex quadratic irrational numbers which are not real numbers. This defines an injection from the quadratic irrationals to quadruples of integers, so their
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
. Quadratic irrationals are used in field theory to construct
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s of the field of rational numbers . Given the square-free integer , the augmentation of by quadratic irrationals using produces a quadratic field ). For example, the inverses of elements of ) are of the same form as the above algebraic numbers: : = . Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that ''all'' real quadratic irrationals, and ''only'' real quadratic irrationals, have periodic continued fraction forms. For example :\sqrt = 1.732\ldots= ;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 orbits of the dyadic transformation (for the binary digits) and the Gauss map h(x)=1/x-\lfloor 1/x \rfloor for continued fractions.


Real quadratic irrational numbers and indefinite binary quadratic forms

We may rewrite a quadratic irrationality as follows: :\frac d = \frac d. It follows that every quadratic irrational number can be written in the form :\frac d. This expression is not unique. Fix a non-square, positive integer c congruent to 0 or 1 modulo 4, and define a set S_c as : S_c = \left\. Every quadratic irrationality is in some set S_c, since the congruence conditions can be met by scaling the numerator and denominator by an appropriate factor. A
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
:\begin \alpha & \beta\\ \gamma & \delta\end with integer entries and \alpha \delta-\beta \gamma=1 can be used to transform a number y in S_c. The transformed number is :z = \frac If y is in S_c, then z is too. The relation between y and z above is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
. (This follows, for instance, because the above transformation gives a group action of the group of integer matrices with determinant 1 on the set S_c.) Thus, S_c partitions into
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es. Each equivalence class comprises a collection of quadratic irrationalities with each pair equivalent through the action of some matrix. Serret's theorem implies that the regular continued fraction expansions of equivalent quadratic irrationalities are eventually the same, that is, their sequences of partial quotients have the same tail. Thus, all numbers in an equivalence class have continued fraction expansions that are eventually periodic with the same tail. There are finitely many equivalence classes of quadratic irrationalities in S_c. The standard
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...
of this involves considering the map \phi from binary quadratic forms of discriminant c to S_c given by : \phi (tx^2 + uxy + vy^2) = \frac A computation shows that \phi is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
that respects the matrix action on each set. The equivalence classes of quadratic irrationalities are then in bijection with the equivalence classes of binary quadratic forms, and Lagrange showed that there are finitely many equivalence classes of binary quadratic forms of given discriminant. Through the bijection \phi, expanding a number in S_c in a continued fraction corresponds to reducing the quadratic form. The eventually periodic nature of the continued fraction is then reflected in the eventually periodic nature of the orbit of a quadratic form under reduction, with reduced quadratic irrationalities (those with a purely periodic continued fraction) corresponding to reduced quadratic forms.


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 :\frac. Thus quadratic irrationals are precisely those real 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 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not 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. The fundamental theorem of arithmetic is not actually required to prove the result, however. There are self-contained proofs by
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
, among others. The following proof was adapted by Colin Richard Hughes from a proof of the irrationality of the square root of 2 found by
Theodor Estermann Theodor Estermann (5 February 1902 – 29 November 1991) was a German-born American mathematician, working in the field of analytic number theory. The Estermann measure, a measure of the central symmetry of a convex set in the Euclidean plane ...
in 1975. 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 < − ''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'' is also an integer. Then: :( − ''n'')''q'' = ''qD'' − ''nq'' is also an integer. But 0 < ( − ''n'') < 1 so ( − ''n'')''q'' < ''q''. Hence ( − ''n'')''q'' is an integer smaller than ''q''. This is a contradiction since ''q'' was defined to be the smallest number with this property; hence cannot be rational.


See also

*
Algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
* Apotome (mathematics) * Periodic continued fraction * Restricted partial quotients * Quadratic integer


References


External links

*
Continued fraction calculator for quadratic irrationals


{{Algebraic numbers Number theory