Multiplication ($\times$)

Multiplication combines two factors into their product.

For a nonnegative integer exponent, exponentiation iterates multiplication: $a^0=1$ and $a^{b+1}=a{a}^b$. This convention defines $0^0=1$ in primitive recursive arithmetic. Extensions to other exponents may use different domains and conventions.

The hyperoperation hierarchy successively iterates basic arithmetic operations. In the usual indexing, successor is followed by addition, multiplication, exponentiation and their higher iterates. A sequence beginning with addition shifts this index by one. Each fixed rank is a primitive recursive function; this does not imply that the evaluator taking the rank as an additional variable is primitive recursive.

A product over the empty set has value one, the identity for multiplication. This makes splitting a product into successive pieces valid even when one piece contains no factors. In particular, the factorial convention is $0!=1$.

A multiplicative inverse of $x$ is an element $x^{-1}$ satisfying $x{x}^{-1}=x^{-1}x=1$.