In mathematics, a binary operation on a set is a calculation involving two elements of the set (called operands) and producing another element of the set (more formally, an operation whose arity is two, and whose two domains and one codomain are (subsets of) the same set). Examples include the familiar elementary arithmetic operations of addition, subtraction, multiplication and division. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups.
Molar solubility is the number of moles of a substance (the solute) that can be dissolved per liter of solution before the solution becomes saturated. It can be calculated from a substance's solubility product constant (Ksp) and stoichiometry. The units are mol/L, sometimes written as M.
Given excess of a simple salt AxBy in an aqueous solution of no common ions (A or B already present in the solution), the amount of it which enters solution can be calculated as follows:
In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven dimensional Euclidean space. It assigns to any two vectors a, b in ℝ7 a vector a × b also in ℝ7. Like the cross product in three dimensions the seven-dimensional product is anticommutative and a × b is orthogonal to both a and b. Unlike in three dimensions, it does not satisfy the Jacobi identity. And while the three-dimensional cross product is unique up to a change in sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to octonions as the three-dimensional product does to quaternions.
The seven-dimensional cross product is one way of generalising the cross product to other than three dimensions, and it turns out to be the only other non-trivial bilinear product of two vectors that is vector valued, anticommutative and orthogonal. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.
Mathematical analysis is a branch of mathematics that includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry. However, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In India, the 12th century mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem.