In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

that is the solution to some

quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...

with

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abil ...

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 ...

s which is irreducible over the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s. 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 low ...

, a quadratic irrational is an irrational root of some quadratic equation with

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

coefficients. The quadratic irrational numbers, a

subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...

of the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, 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 p ...

s of degree 2, and can therefore be expressed as :

,

for integers ; with , and non-zero, and with

square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. ...

. When is positive, we get real quadratic irrational numbers, while a negative gives complex quadratic irrational numbers which are not

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s. 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 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 number ...

; since on the other hand every square root of a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

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 number ...

. 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

of rational numbers . Given the square-free integer , the augmentation of by quadratic irrationals using produces a

quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...

). 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 fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...

s, where we have the result that ''all'' real quadratic irrationals, and ''only'' real quadratic irrationals, have

periodic continued fraction In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form : x = a_0 + \cfrac where the initial block of ''k'' + 1 partial denominators is followed by a block 'a'k''+1, ''a'k''+2,. ...

forms. For example :

/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 In mathematics, the Minkowski question-mark function, denoted , is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expressi ...

, 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 In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given ...

s of the

dyadic transformation The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to

Gauss map In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ...

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

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 (mathematics)">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'' (franchi ...

\begin \alpha & \beta\\ \gamma & \delta\end

with integer entries and

\alpha \delta-\beta \gamma=1

can be used to transform a number

y

S_c

. The transformed number is :

z = \frac

y

is in

S_c

, then

z

is too. The relation between

y

and

z

above is an equivalence relation. (This follows, for instance, because the above transformation gives a Group action (mathematics), group action of the Group (mathematics), 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 ...

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 of this involves considering the map

\phi

from

binary quadratic form In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables : q(x,y)=ax^2+bxy+cy^2, \, where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results ar ...

s of discriminant

c

S_c

given by :

\phi (tx^2 + uxy + vy^2) = \frac

A computation shows that

\phi

is a bijection 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

s 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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

that is not a

square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...

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 princi ...

was the first such number to be proved irrational.

Theodorus of Cyrene Theodorus of Cyrene ( el, Θεόδωρος ὁ Κυρηναῖος) was an ancient Greek mathematician who lived during the 5th century BC. The only first-hand accounts of him that survive are in three of Plato's dialogues: the ''Theaetetus'', the ...

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 In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as we ...

. Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the

fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the o ...

, which was first proven by

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...

in his

Disquisitiones Arithmeticae The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...

. 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 In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statem ...

, the square root of an integer is always either another integer, or irrational.

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...

used a restricted version of the fundamental theorem and some careful argument to prove the theorem. His proof is in

Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...

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 pla ...

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.

References

External links

*
Continued fraction calculator for quadratic irrationals

{{Algebraic numbers Number theory

Real quadratic irrational numbers and indefinite binary quadratic forms

Square root of non-square is irrational

See also

References

External links