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...

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

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...

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...

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

{\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...

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

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...

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

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

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...

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...

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...

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

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

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...

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...

elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form ⁠ a x 2 + b x + c {\displaystyle \textstyle ax^{2}+bx+c}...

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...

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...

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...

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

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

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...

}{\frac {1}{F_{2k}}}=3.359885666243\dots } Moreover, this number has been proved irrational by Richard André-Jeannin. Millin's series gives the identity...