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In chemistry, a radical, also known as a free radical, is an atom, ... Their chemical properties are distinct from the properties of their singlet analogues.
In general, the radical of is , where is the product of all distinct prime factors of , the largest square-free factor of (see Radical of an integer). In fact, this generalizes to an arbitrary ideal (see the Properties section).
Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.
The hydroxyl radical, • HO, is the neutral form of the hydroxide ion (HO –). Hydroxyl radicals are highly reactive and consequently short-lived; however, they form an important part of radical chemistry .
For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...
An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd [2] or a radical. [3] Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression , and if it contains no transcendental functions or transcendental numbers it is called an algebraic ...
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
In chemistry, the amino radical, ·NH 2, also known as the aminyl or azanyl, is the neutral form of the amide ion (NH − 2). Aminyl radicals are highly reactive and consequently short-lived, like most radicals; however, they form an important part of nitrogen chemistry. In sufficiently high concentration, amino radicals dimerise to form hydrazine.