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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). The result of this operation is called **product** and the multiplied numbers are called **factors**, or the **multiplier** and the **multiplicand**.

The product *mn* of a positive whole number *m* times another quantity *n* agrees with the result of successively adding *n* to itself a total of *m* times. For example, 2 multiplied by 3 (often said as "3 **times** 2") gives the same result as 2 taken 3 times, i.e., of addding 3 copies of 2 : 2 × 3 = 2 + 2 + 2. Similarly, multiplied by *2* is the same value obtained by adding the number pi to itself. Multiplication does not give the result of a repeated addition when the multiplier is not a whole number. We can calculate the product

using the rules for manipulating radicals and products, but we cannot view this computation as a repeatedly adding . Instead of being viewed as a special type of repeated addition, multiplication should be viewed as a basic numerical operation, separate from and as important as addition.

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 can be defined outside of the context of real numbers. There are natural multiplication operations on the complex numbers, matrices, tensors, sequences, and functions. The common thread that allows all of these operations to be called "multiplication" is the existence of an additional "addition" operation and a collection of common axioms satisfied by the addition and multiplication operations in each context. The most general context in which a multiplication operation exist, encompassing all of the above examples, is that of the abstract ring encountered in abstract algebra.

## Properties

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

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

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:

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.