Product (mathematics)

In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and ${\displaystyle x\cdot (2+x)}$ is the product of ${\displaystyle x}$ and ${\displaystyle (2+x)}$ (indicating that the two factors should be multiplied together). When one factor is an integer, the product is called a multiple.

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, is non-commutative, and so is multiplication in other algebras in general as well.

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.

Originally, a product was and is still the result of the multiplication of two or more numbers. For example, 15 is the product of 3 and 5. The fundamental theorem of arithmetic states that every composite number is a product of prime numbers, that is unique up to the order of the factors.

With the introduction of mathematical notation and variables at the end of the 15th century, it became common to consider the multiplication of numbers that are either unspecified (coefficients and parameters), or to be found (unknowns). These multiplications that cannot be effectively performed are called products. For example, in the linear equation ${\displaystyle ax+b=0,}$ the term ${\displaystyle ax}$ denotes the product of the coefficient ${\displaystyle a}$ and the unknown ${\displaystyle x.}$

Later and essentially from the 19th century on, new binary operations have been introduced, which do not involve numbers at all, and have been called products; for example, the dot product. Most of this article is devoted to such non-numerical products.

The product operator for the product of a sequence is denoted by the capital Greek letter pi Π (in analogy to the use of the capital Sigma Σ as summation symbol). For example, the expression ${\displaystyle \textstyle \prod _{i=1}^{6}i^{2}}$ is another way of writing ⁠${\displaystyle 1\cdot 4\cdot 9\cdot 16\cdot 25\cdot 36}$⁠.

The product of a sequence consisting of only one number is just that number itself; the product of no factors at all is known as the empty product, and is equal to 1.