In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality $${\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}$$ is always true in elementary algebra. For example, in elementary arithmetic, one has $${\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).}$$ Therefore, one would say that multiplication distributes over addition.

This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted ${\displaystyle \,\land \,}$) and the logical or (denoted ${\displaystyle \,\lor \,}$) distributes over the other.

Given a set ${\displaystyle S}$ and two binary operators ${\displaystyle \,*\,}$ and ${\displaystyle \,+\,}$ on ${\displaystyle S,}$

$${\displaystyle x*(y+z)=(x*y)+(x*z);}$$

$${\displaystyle (y+z)*x=(y*x)+(z*x);}$$

When ${\displaystyle \,*\,}$ is commutative, the three conditions above are logically equivalent.

The operators used for examples in this section are those of the usual addition ${\displaystyle \,+\,}$ and multiplication ${\displaystyle \,\cdot .\,}$

If the operation denoted ${\displaystyle \cdot }$ is not commutative, there is a distinction between left-distributivity and right-distributivity: