# Multiplication

Multiplication (often denoted by the cross symbol "×") is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division).

Because the result of scaling by whole numbers can be thought of as consisting of some number of copies of the original, whole-number products greater than 1 can be computed by repeated addition; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 4 copies of 3 together:

Here 3 and 4 are the "factors" and 12 is the "product".

Educators differ as to which number should normally be considered as the number of copies, and whether multiplication should even be introduced as repeated addition. For example 3 multiplied by 4 can also be calculated by adding 3 copies of 4 together:

Multiplication of rational numbers (fractions) and real numbers is defined by systematic generalization of this basic idea.

Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have given lengths (for numbers generally). The area of a rectangle does not depend on which side you measure first, which illustrates that the order numbers are multiplied together in doesn't matter.

In general the result of multiplying two measurements gives a result of a new type depending on the measurements. For instance:

The inverse operation of multiplication is division. For example, 4 multiplied by 3 equals 12. Then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number.

Multiplication is also defined for other types of numbers (such as complex numbers), and for more abstract constructs such as matrices. For these more abstract constructs, the order that the operands are multiplied in sometimes does matter.

### Other articles related to "multiplication":

Multiplication - Exponentiation
... When multiplication is repeated, the resulting operation is known as exponentiation ... For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 23, a two with a superscript three ...
Classical Hamiltonian Quaternions - Four Operations - Cardinal Operations - Multiplication
... Classical quaternion notation had only one concept of multiplication ... Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation ... Multiplication of a scalar and a vector was accomplished with the same single multiplication operator multiplication of two vectors of quaternions used this same operation as did ...
Logrithm - History - Predecessors
... Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition ... Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use ... and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division ...
Church Of The Multiplication
... The Church of the Multiplication of the Loaves and Fishes, shortened to (The Church of the Multiplication), is a Roman Catholic church located in Tabgha, on the northwest ...
Semifield
... In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to the field, but with ... More precisely, it is a Z-algebra whose nonzero elements form a loop under multiplication ... S with two operations + (addition) and · (multiplication), such that (S,+) is an abelian group, multiplication is distributive on both the left and ...