Multiplication

Multiplication is the binary mathematical operation of scaling one number or quantity by another (multiplying). It is one of the four basic operations in elementary arithmetic (with addition, subtraction and division). The result of this operation is called product and the multiplied numbers are called factors, or the multiplier and the multiplicand. Multiplication is defined in terms of repeated addition: for example, 2 multiplied by 3 (often said as "2 times 3") is the same as adding 3 copies of 2: 2 × 3 = 2 + 2 + 2.

Multiplication can be visualised as counting objects arranged in a rectangle (for natural numbers) or as finding the area of a rectangle whose sides have given lengths (for numbers generally). The inverse of multiplication is division: as 2 times 3 equals to 6, so 6 divided by 3 equals to 2.

Multiplication is generalized further to other types of numbers (such as complex numbers) and to more abstract constructs such as matrices, groups, sets and tensors.

Properties

Commutativity

Multiplication is commutative, meaning a × b = b × a.

Associativity

Multiplication is associative, meaning a × (b × c) = (a × b) × c.

Distributivity

Multiplication is distributive over addition, meaning a × (x + y) = a × x + a × y.

Products of sequences

Capital pi notation

The product of a sequence can be written using capital Greek letter Π (Pi). Unicode position U+220F (∏) contains a symbol for the product of a sequence, distinct from U+03A0 (Π), the letter. The meaning of this notation is given by:

$\prod _{i=m}^{n}x_{i}=x_{m}\cdot x_{m+1}\cdot x_{m+2}\cdot \,\,\cdots \,\,\cdot x_{n-1}\cdot x_{n},$

where i is an index of multiplication, m is its lower bound and n is its upper bound. Example:

$\prod _{i=2}^{4}2^{i}=2^{2}\cdot 2^{3}\cdot 2^{4}=4\cdot 8\cdot 16=512.$

If m = n, the value of the product just equals to x_m. If m > n, the product is the empty product, with the value 1.