quadratic scale - ορισμός. Τι είναι το quadratic scale
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MATHEMATICAL CONCEPT
Quadratic surd; Quadratic irrationality; Quadratic Irrational Number; Quadratic irrationalities; Quadratic irrational; Quadratic irrational numbers

Quadratic irrational number         
In 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 coefficients which is irreducible over the rational numbers.Jörn Steuding, Diophantine Analysis, (2005), Chapman & Hall, p.
Quadratic reciprocity         
THEOREM
Law of quadratic reciprocity; Quadratic reciprocity rule; Aureum Theorema; Law of Quadratic Reciprocity; Quadratic reciprocity law; Quadratic reciprocity theorem; Quadratic Reciprocity; Qr theorem
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:
Linear–quadratic regulator         
LINEAR OPTIMAL CONTROL TECHNIQUE
Linear-quadratic control; Dynamic Riccati equation; Linear-quadratic regulator; Quadratic quadratic regulator; Quadratic–quadratic regulator; Quadratic-quadratic regulator; Polynomial quadratic regulator; Polynomial–quadratic regulator; Polynomial-quadratic regulator; Linear quadratic regulator
The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem.

Βικιπαίδεια

Quadratic irrational number

In 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 coefficients which is irreducible 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, 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

a + b c d , {\displaystyle {a+b{\sqrt {c}} \over d},}

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:

d a + b c = a d b d c a 2 b 2 c . {\displaystyle {d \over a+b{\sqrt {c}}}={ad-bd{\sqrt {c}} \over a^{2}-b^{2}c}.}

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

3 = 1.732 = [ 1 ; 1 , 2 , 1 , 2 , 1 , 2 , ] {\displaystyle {\sqrt {3}}=1.732\ldots =[1;1,2,1,2,1,2,\ldots ]}

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 1 / x {\displaystyle h(x)=1/x-\lfloor 1/x\rfloor } for continued fractions.