quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation...

root of 2 was likely the first number proved irrational. The golden ratio is another famous quadratic irrational number. 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 quadratic irrational number, which is an algebraic...

all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational. The algebraic numbers are dense in the reals. This follows...

In algebraic number theory, a quadratic field is an algebraic number field of degree two over Q {\displaystyle \mathbf {Q} } , the rational numbers. Every...

In mathematics, an irrationality measure of a real number x {\displaystyle x} is a measure of how "closely" it can be approximated by rationals. If a function...

Quadratic field, an algebraic number field of degree two over the field of rational numbers Quadratic irrational or "quadratic surd", an irrational number...

quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics...

represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers...

{\displaystyle SL(2,\mathbb {Z} ).} A quadratic irrational number is an irrational real root of the quadratic equation a x 2 + b x + c = 0 {\displaystyle...

subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals. Applying any non-constant single-variable algebraic...

square is irrational. For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite...

is a quadratic irrational number in the upper half plane then j(τ) is an algebraic integer. In addition he proved that if τ is an algebraic number but...

theorem Irrational number Square root of two Quadratic irrational Integer square root Algebraic number Pisot–Vijayaraghavan number Salem number Transcendental...

{\displaystyle M(\alpha )<\infty } if α {\displaystyle \alpha } is a quadratic irrational number. In fact, the lower bound for M ( α ) {\displaystyle M(\alpha...

number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational...

real-valued function of an integer or natural number variable). Examples of quadratic growth include: Any quadratic polynomial. Certain integer sequences such...

apotome can be interpreted as a quadratic irrational number formed by subtracting one square root of a rational number from another. This concept of the...

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...

other integer arguments, Pythagorean addition can produce a quadratic irrational number as its result. The operation ⊕ {\displaystyle \oplus } is associative...

In mathematics, a quadratic function of a single variable is a function of the form f ( x ) = a x 2 + b x + c , a ≠ 0 , {\displaystyle f(x)=ax^{2}+bx+c...

it solves the quadratic equation ⁠ x 2 − 5 = 0 {\displaystyle x^{2}-5=0} ⁠, making it a quadratic integer, a type of algebraic number. ⁠ 5 {\displaystyle...

to various issues. The number of votes to add is determined by a quadratic cost function, which simply means that the number of votes an individual casts...

In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if...

quadratic irrationals that are PV numbers are: Pisot–Vijayaraghavan numbers can be used to generate almost integers: the nth power of a Pisot number approaches...

imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property...

identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents...

} ⁠ satisfies the quadratic equation ⁠ φ 2 = φ + 1 {\displaystyle \textstyle \varphi ^{2}=\varphi +1} ⁠ and is an irrational number with a value of φ...

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...

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...