Multiplication of whole numbers is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

$4\; +\; 4\; +\; 4\; =\; 12.\backslash !\backslash ,$

Multiplication can also be viewed as counting objects arranged in a rectangle, or finding the area of rectangle whose sides have given lengths.

Multiplication is one of four main operations in elementary arithmetic, and most people learn basic multiplication algorithms in elementary school. The inverse of multiplication is division.

Multiplication is generalized to many kinds of numbers and to more abstract constructs such as matrices.

Notation and terminology

Multiplication is written using the multiplication sign "×" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,

$2\backslash times\; 3\; =\; 6$ (verbally, "two times three equals six")

$3\backslash times\; 4\; =\; 12$

$2\backslash times\; 3\backslash times\; 5\; =\; 6\backslash times\; 5\; =\; 30$

$2\backslash times\; 2\backslash times\; 2\backslash times\; 2\backslash times\; 2\; =\; 32$

There are several other common notations for multiplication:

• Multiplication is sometimes denoted by either a middle dot or a period:

  $5\; \backslash cdot\; 2\; \backslash quad\backslash text\{or\}\backslash quad\; 5\backslash ,.\backslash ,2$

  The middle dot is standard in the United States, the United Kingdom, and other countries where the period is used as a decimal point. In some countries that use a comma as a decimal point, the period is used for multiplication instead.

• The asterisk (as in 5*2) is often used in programming languages because it appears on every keyboard and is easier to see on older monitors. This usage originated in the FORTRAN programming language.
• In algebra, multiplication involving variables is often written as a juxtaposition (e.g. xy for x times y or 5x for five times x). This notation can also be used for numbers that are surrounded by parentheses (e.g. 5(2) or (5)(2) for five times two).

In matrix multiplication, there is actually a distinction between the cross and the dot symbols. The cross symbol generally denotes a vector multiplication, while the dot denotes a scalar multiplication. A like convention distinguishes between the cross product and the dot product of two vectors.

The numbers to be multiplied are generally called the "factors" or "multiplicands". When thinking of multiplication as repeated addition, the number to be multiplied is called the "multiplicand", while the number of multiples is called the "multiplier". In algebra, a number that is the multiplier of a variable or expression (e.g. the 3 in 3xy²) is called a coefficient.

The result of a multiplication is called a product, and is a multiple of each factor that is an integer. For example 15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5.