Multiplication

Multiplication 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. Multiplication is often denoted by the cross symbol, ×, by the mid-line dot operator, ⋅, by juxtaposition, or, in programming languages, by an asterisk, *. 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. This is to be distinguished from terms, which are added. a × b = b + ⋯ + b ⏟ a times . {\displaystyle a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}.} Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 × 4 {\displaystyle 3\times 4} can be phrased as "3 times 4", or "Three groups of four", notating the multiplication symbol with the word of, and evaluated as 4 + 4 + 4 {\displaystyle 4+4+4} , where 3 is the multiplier, but also as "3 multiplied by 4", in which case 3 becomes the multiplicand. 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. 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 calculating the area of a...