# Distributive property

Distribution is a concept from algebra: It tells how binary operations are to be handled. The most simple case is that of addition and multiplication of numbers. For example, in arithmetic:

2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3).

In the left-hand side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards.Because these give the same final answer (8), it is said that multiplication by 2 distributes over addition of 1 and 3.Since one could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

## Definition

Given a set S and two binary operators ∗ and + on S, we say that the operation:

∗ 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.

Other Languages
العربية: توزيعية
asturianu: Distributividá
башҡортса: Дистрибутивлыҡ
беларуская: Дыстрыбутыўнасць
čeština: Distributivita
Esperanto: Distribueco
euskara: Banakortasun
français: Distributivité
한국어: 분배법칙
Bahasa Indonesia: Distributif
íslenska: Dreifiregla
italiano: Distributività
македонски: Дистрибутивност
Bahasa Melayu: Kalis agihan
Nederlands: Distributiviteit

norsk nynorsk: Distributivitet