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 coefficients which is irreducible over the rational numbers.[1] 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 numbers, are algebraic numbers of degree 2, and can therefore be expressed as
for integers a, b, c, d; with b, c and d non-zero, and with c square-free. When c is positive, we get real quadratic irrational numbers, while a negative c 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 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.
Quadratic irrationals are used in field theory to construct field extensions of the field of rational numbers Q. Given the square-free integer c, the augmentation of Q by quadratic irrationals using √c produces a quadratic field Q(√c). For example, the inverses of elements of Q(√c) 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
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 for continued fractions.
quadraticirrationalnumber (also known as a quadraticirrational or quadratic surd) is an irrationalnumber that is the solution to some quadratic equation...
of a non-zero polynomial, namely bx − a. Quadraticirrational numbers, irrational solutions of a quadratic polynomial ax2 + bx + c with integer coefficients...
root of 2 was likely the first number proved irrational. The golden ratio is another famous quadraticirrationalnumber. The square roots of all natural...
gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadraticirrationalnumber, which is an algebraic...
Quadratic field, an algebraic number field of degree two over the field of rational numbers Quadraticirrational or "quadratic surd", an irrational number...
{\displaystyle SL(2,\mathbb {Z} ).} A quadraticirrationalnumber is an irrational real root of the quadratic equation a x 2 + b x + c = 0 {\displaystyle...
In algebraic number theory, a quadratic field is an algebraic number field of degree two over Q {\displaystyle \mathbf {Q} } , the rational numbers. Every...
non-square natural number is irrational, see Quadraticirrationalnumber or Infinite descent. One proof of the number'sirrationality is the following proof...
subset of the algebraic numbers, including the quadraticirrationals and other forms of algebraic irrationals. Applying any non-constant single-variable algebraic...
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as a x 2 + b x + c = 0 , {\displaystyle...
theorem Irrationalnumber Square root of two Quadraticirrational Integer square root Algebraic number Pisot–Vijayaraghavan number Salem number Transcendental...
case that every real number is rational. A real number that is not rational is called irrational. A famous irrational real number is the π, the ratio of...
Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in...
principle of the number field sieve (both special and general) can be understood as an improvement to the simpler rational sieve or quadratic sieve. When using...
{\displaystyle \varphi } satisfies the quadratic equation φ 2 = φ + 1 {\displaystyle \varphi ^{2}=\varphi +1} and is an irrationalnumber with a value of φ = 1 + 5...
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree...
more closely approximated by rational numbers than any algebraic irrationalnumber can be. In 1844, Joseph Liouville showed that all Liouville numbers...
real-valued function of an integer or natural number variable). Examples of quadratic growth include: Any quadratic polynomial. Certain integer sequences such...
that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in Disquisitiones arithmeticae...
number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrationalnumber, and quadratic irrational...
also proves that e is not a root of a quadratic polynomial with rational coefficients; in particular, e2 is irrational. The most well-known proof is Joseph...
elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form a x 2 + b x + c {\displaystyle ax^{2}+bx+c} to...
Quadratic voting is a collective decision-making procedure which involves individuals allocating votes to express the degree of their preferences, rather...
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field Q [ − d ]...
(linguistics). The irrational numbers are a set of numbers that includes all real numbers that are not rational numbers. The irrational numbers are categorised...
}{\frac {1}{F_{2k}}}=3.359885666243\dots } Moreover, this number has been proved irrational by Richard André-Jeannin. Millin's series gives the identity...
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