Product (mathematics)

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. 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, and the multiplications in the other algebras, are non-commutative.

The product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏ (in analogy to the use of the capital Sigma ∑ as summation symbol). 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.

Many different kinds of products are studied in mathematics:

• Products of the various classes of numbers
• The product of matrices and vectors:
  • scalar multiplication,
  • matrix multiplication,
  • dot product,
  • cross product,
  • Hadamard product,
  • Kronecker product.
• The product of tensors:
  • Wedge product or exterior product
  • Interior product
  • Outer product
  • Tensor product
• The pointwise product of two functions.
• A function's product integral (as a continuous equivalent to the product of a sequence or the multiplicative version of the (normal/standard/additive) integral. The product integral is also known as "continuous product" or "multiplical".
• It is often possible to form the product of two (or more) mathematical objects to form another object of the same kind. Such products are generically called internal products, as they can be described by the generic notion of a monoidal category. Examples include:
  • the Cartesian product of sets,
  • the product of groups, and also the semidirect product, knit product and wreath product,
  • the free product of groups
  • the product of rings,
  • the product of ideals,
  • the product of topological spaces,
  • the Wick product of random variables.
  • the cap, cup and slant product in algebraic topology.
  • the smash product and wedge sum (sometimes called the wedge product) in homotopy.
• For the general treatment of the concept of a product, see product (category theory), which describes how to combine two objects of some kind to create an object, possibly of a different kind. But also, in category theory, one has:
  • the fiber product or pullback,
  • the product category, a category that is the product of categories.
  • the ultraproduct, in model theory.
• Complex multiplication, a theory of elliptic curves.