# Is IEEE 754 float arithmetic associative, commutative, distributive, etc? Why?

Does the associative/commutative/distributive/etc property hold for arithmetic performed with IEEE 754 floats?

Obviously the answer is no to most of those questions, but do any of the properties of standard arithmetic hold? How can one formulate proofs (or at least proof-like statements) explaining why any particular property does or does not hold?

• A counterexample suffices to prove that a property doesn't hold (e.g., floats $x$, $y$ and $z$ such that $x+(y+z)\neq (x+y)+z$). As for proving that properties do hold, what are you looking for? A proof of $X$ is a convincing argument that $X$ is true. How one formulates a convincing argument that something is true depends very much on what that thing is. If you're asking for a proof or counterexample for each of the three-plus-etc properties, that's too much for a single question. – David Richerby Apr 18 '18 at 17:48

• @tel, associative laws don't hold. Others do, like $x - x = 0$, $x + 0 = x$, $1 \cdot x = x$, $x + y = y + x$. – vonbrand May 2 '18 at 14:25