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".

Four bags of three marbles gives twelve marbles. There are also 3 sets consisting of 4 marbles of the same colour.

Multiplication can also be thought of as scaling. In the above animation, we see 2 being multiplied by 3, giving 6 as a result

4 × 5 = 20, the rectangle is composed of 20 squares, having dimensions of 4 by 5.

Area of a cloth 4.5m × 2.5m = 11.25m²; 4½ × 2½ = 11¼

There are differences amongst educationalists as to which number should normally be considered as the number of copies and whether multiplication should even be introduced as repeated addition.^[1]

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 is measured first which illustrates that the order in which numbers are multiplied together does not 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 in which the operands are multiplied sometimes does matter.

Notation and terminology

The multiplication sign ×
(HTML entity is ×)

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

(verbally, "two times three equals six")

There are several other common notations for multiplication. Many of these are intended to reduce confusion between the multiplication sign × and the commonly used variable x:

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

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 other countries that use a comma as a decimal point, either the period or a middle dot is used for multiplication.^{[citation needed]}

§ The asterisk (as in 5*2) is often used in programming languages because it appears on every keyboard. This usage originated in theFORTRAN

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 quantities 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 similar 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 if the other factor 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.

Computation

The common methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9), however one method, the peasant multiplication algorithm, does not.

Multiplying numbers to more than a couple of decimal places by hand is tedious and error prone. Common logarithms were invented to simplify such calculations. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early twentieth century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.

http://www.khanacademy.org/video/basic-multiplication?playlist=Arithmetic

http://www.khanacademy.org/video/multiplication-2--the-multiplication-tables?playlist=Arithmetic

http://www.khanacademy.org/video/multiplication-3--10-11-12-times-tables?playlist=Arithmetic

http://www.khanacademy.org/video/multiplication-4--2-digit-times-1-digit-number?playlist=Arithmetic

http://www.khanacademy.org/video/multiplication-5---2-digit-times-a-2-digit-number?playlist=Arithmetic

http://www.khanacademy.org/video/multiplication-6--multiple-digit-numbers?playlist=Arithmetic

http://www.khanacademy.org/video/multiplication-7--old-video-giving-more-examples?playlist=Arithmetic