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# 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). A result of this operation is called *product* and the multiplied numbers are called *factors*. 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

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

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

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

## Pruducts 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:

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

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.