--- ## Definition > [!tldr] Definition > The **commutative**, **associative**, and **distributive** properties of arithmetic state the following: > > - *Commutative property*: Given any two integers $a$ and $b$, we have $a+b = b+a$ and $ab = ba$. That is, we get the same results if we add two integers in opposite order, or multiply them in opposite order. > - *Associative property*: Given any three integers $a$, $b$, and $c$, we have $a+(b+c) = (a+b)+c$ and $a(bc) = (ab)c$. That is, we can group together addition or multiplication operations on three integers two at a time, in whatever order we wish and get the same results. > - *Distributive property*: Given any three integers $a$, $b$, and $c$, we have $a(b+c) = ab + ac$. Notes: - Combining the distributive property with the commutative property gives an additional formulation of the distributive property: $(a+b)c = ac + bc$. - There are other fundamental properties of the integers such as the existence of additive inverses; this article focuses only on the three properties used most commonly in [[Proof|proofs]].