## ::Multiplication

### ::concepts

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**Multiplication** (often denoted by the cross symbol "**×**", by a point "**·**" or by the absence of symbol) is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

The multiplication of two whole numbers, when thinking of multiplication as repeated addition, is equivalent to adding as many copies of one of them (multiplicand, written second) as the value of the other one (multiplier, written first):<ref name="Devlin2011">"With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first). " See {{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

- <math>a\times b = \underbrace{b + \cdots + b}_a</math>

For example, 4 multiplied by 3 (often written as <math>3 \times 4 </math><ref name="Devlin2011" /> and said as "3 times 4") can be calculated by adding 3 copies of 4 together:

- <math>3 \times 4 = 4 + 4 + 4 = 12</math>

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

One of the main properties of multiplication is the commutative property, adding 3 copies of 4 gives the same result as adding 4 copies of 3:

- <math>4 \times 3 = 3 + 3 + 3 + 3 = 12</math>

The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition.

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. The area of a rectangle does not depend on which side is measured first, which illustrates the commutative property.

The inverse operation of the multiplication is the division. For example, since 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 (since the division of a number other than 0 by itself equals 1).

Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter. A listing of the many different kinds of products that are used in mathematics is given in the product (mathematics) page.

**Multiplication sections**

Intro [[Multiplication?section={{safesubst:#invoke:anchor|main}}Notation_and_terminology|{{safesubst:#invoke:anchor|main}}Notation and terminology]] Computation Products of measurements Properties Axioms Multiplication with set theory Exponentiation See also Notes References External links

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Times::numbers Number::product ''n''::right Example::notation Other::division Group::integers

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**Multiplication** (often denoted by the cross symbol "**×**", by a point "**·**" or by the absence of symbol) is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

The multiplication of two whole numbers, when thinking of multiplication as repeated addition, is equivalent to adding as many copies of one of them (multiplicand, written second) as the value of the other one (multiplier, written first):<ref name="Devlin2011">"With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first). " See {{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

- <math>a\times b = \underbrace{b + \cdots + b}_a</math>

For example, 4 multiplied by 3 (often written as <math>3 \times 4 </math><ref name="Devlin2011" /> and said as "3 times 4") can be calculated by adding 3 copies of 4 together:

- <math>3 \times 4 = 4 + 4 + 4 = 12</math>

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

One of the main properties of multiplication is the commutative property, adding 3 copies of 4 gives the same result as adding 4 copies of 3:

- <math>4 \times 3 = 3 + 3 + 3 + 3 = 12</math>

The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition.

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. The area of a rectangle does not depend on which side is measured first, which illustrates the commutative property.

The inverse operation of the multiplication is the division. For example, since 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 (since the division of a number other than 0 by itself equals 1).

Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter. A listing of the many different kinds of products that are used in mathematics is given in the product (mathematics) page.

**Multiplication sections**

Intro [[Multiplication?section={{safesubst:#invoke:anchor|main}}Notation_and_terminology|{{safesubst:#invoke:anchor|main}}Notation and terminology]] Computation Products of measurements Properties Axioms Multiplication with set theory Exponentiation See also Notes References External links

PREVIOUS: Intro | NEXT: [[Multiplication?section={{safesubst:#invoke:anchor|main}}Notation_and_terminology|{{safesubst:#invoke:anchor|main}}Notation and terminology]] |

<< | [[Multiplication?section={{safesubst:#invoke:anchor|main}}Notation_and_terminology|>>]] |