veelvouding

Veelvouding, or multiplication, is the mathematical operation that combines two numbers to produce their product. For numbers a and b, the result is a × b, often written as ab. With natural numbers, multiplication can be interpreted as repeated addition: adding a to itself b times. This interpretation extends to integers, rationals, real numbers, and complex numbers, and beyond, within the framework of algebraic structures.

Properties: Many multiplication operations are binary operations that satisfy key rules. They are commutative: a × b = b × a. They are associative: (a × b) × c = a × (b × c). They are distributive over addition: a × (b + c) = a × b + a × c. There is a multiplicative identity 1, since a × 1 = a, and a property with zero: a × 0 = 0. Inverse concepts lead to division, where nonzero b allows a ÷ b to be expressed as a × (1/b), the multiplicative inverse of b; zero has no inverse. Multiplication is defined and studied within various number systems and is closed within those systems.

Notation and generalization: Multiplication is commonly written as ab, a × b, or a · b. In higher mathematics, multiplication is extended to objects such as matrices, where the product is defined by row-by-column sums and requires compatible dimensions; in vector spaces, scalar multiplication combines a scalar and a vector; for polynomials, multiplication follows the distributive law and adds degrees. Exponentiation describes repeated multiplication of a number by itself.

Applications and significance: Multiplication is a fundamental arithmetic operation used in calculations of area and volume, scaling, algebraic manipulation, physics, engineering, computer science, and many other fields.