Product (mathematics)

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication), and x ⋅ ( 2 + x ) {displaystyle xcdot (2+x)} is the product of x {displaystyle x} and ( 2 + x ) {displaystyle (2+x)} (indicating that the two factors should be multiplied together). In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication), and x ⋅ ( 2 + x ) {displaystyle xcdot (2+x)} is the product of x {displaystyle x} and ( 2 + x ) {displaystyle (2+x)} (indicating that the two factors should be multiplied together). The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, and multiplication in other algebras is in general non-commutative. There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different algebraic structures. An overview of these different kinds of products is given here. Placing several stones into a rectangular pattern with r {displaystyle r} rows and s {displaystyle s} columns gives stones. Another approach to multiplication that applies also to real numbers is continuously stretching the number line from 0, so that the 1 is stretched to the one factor, and looking up the product, where the other factor is stretched to. Integers allow positive and negative numbers. Their product is determined by the product of their positive amounts, combined with the sign derived from the following rule, which is a necessary consequence of demanding distributivity of the multiplication over addition, but is no additional rule.