Whole numbers multiplication models and properties

Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12).

Multiplication can also be thought of as scaling. Here we see 2 being multiplied by 3 using scaling, giving 6 as a result.

Animation for the multiplication 2 × 3 = 6.

4 × 5 = 20. The large rectangle is made up of 20 squares, each 1 unit by 1 unit.

Area of a cloth 4.5m × 2.5m = 11.25m²; $4{\tfrac {1}{2}}\times 2{\tfrac {1}{2}}=11{\tfrac {1}{4}}$

Multiplication (often denoted by the cross symbol ×, by the mid-line dot operator ⋅, by juxtaposition, or, on computers, by an asterisk *) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.

The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier. Both numbers can be referred to as factors.

$a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}$

For example, 4 multiplied by 3, often written as $3\times 4$ and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:

$3\times 4=4+4+4=12$

Here, 3 (the multiplier) and 4 (the multiplicand) are the factors, and 12 is the product.

One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:

$4\times 3=3+3+3+3=12$

Thus the designation of multiplier and multiplicand does not affect the result of the multiplication.

Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers.

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 some given lengths. The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property.

The product of two measurements is a new type of measurement. For example, multiplying the lengths of the two sides of a rectangle gives its area. Such a product is the subject of dimensional analysis.

The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the original number. 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 for more abstract constructs, like matrices. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products used in mathematics is given in Product (mathematics).

Properties

For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:

Commutative property

The order in which two numbers are multiplied does not matter:

$x\cdot y=y\cdot x.$

Associative property

Expressions solely involving multiplication or addition are invariant with respect to the order of operations:

$(x\cdot y)\cdot z=x\cdot (y\cdot z)$

Distributive property

Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:

$x\cdot (y+z)=x\cdot y+x\cdot z$

Identity element

The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property:

$x\cdot 1=x$

Property of 0

Any number multiplied by 0 is 0. This is known as the zero property of multiplication:

$x\cdot 0=0$

Negation

−1 times any number is equal to the additive inverse of that number.

$(-1)\cdot x=(-x)$ where $(-x)+x=0$

–1 times –1 is 1.

$(-1)\cdot (-1)=1$

Inverse element

Every number x, except 0, has a multiplicative inverse, ${\frac {1}{x}}$, such that $x\cdot \left({\frac {1}{x}}\right)=1$.

Order preservation

Multiplication by a positive number preserves the order:

For a > 0, if b > c then ab > ac.

Multiplication by a negative number reverses the order:

For a < 0, if b > c then ab < ac.

The complex numbers do not have an ordering that is compatible with both addition and multiplication.

Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices and quaternions.

Licensing

Content obtained and/or adapted from:

• Multiplication, Wikipedia under a CC BY-SA license