# Multiplication Distributive Law

An Multiplication Distributive Law is a distributive operation relation between multiplication ($\displaystyle{ \times }$) addition ($\displaystyle{ + }$) or subtraction ($\displaystyle{ - }$) binary operations.

## References

### 2017

∗ is left-distributive over + if, given any elements x, y, and z of S,
$\displaystyle{ x * (y + z) = (x * y) + (x * z), }$
∗ is right-distributive over + if, given any elements x, y, and z of S,
$\displaystyle{ (y + z) * x = (y * x) + (z * x), }$ and
∗ is distributive over + if it is left- and right-distributive.[1]
Notice that when ∗ is commutative, the three conditions above are logically equivalent.
(...) In either case, the distributive property can be described in words as: To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).
If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa.

### 2017

$\displaystyle{ (x+y)z=xz+yz }$
for every $\displaystyle{ x }$, $\displaystyle{ y }$, and $\displaystyle{ z }$. Similarly, it is said to be left distributive if
$\displaystyle{ z(x+y)=zx+zy }$
for every $\displaystyle{ x }$, $\displaystyle{ y }$, and $\displaystyle{ z }$.
If a multiplication is both right- and left-distributive, it is simply said to be distributive. For example, the real numbers R are distributive.