TITLE: Terminology: algebraic structure for "floating point" arithmetic
QUESTION [12 upvotes]: "floating point arithmetic" is a terminology that refer to the arithmetic perform over (finite) representation of real number. See the wikipedia article for more details. 
In the formal specification  of floating point arithmetic (that should be used by all the major programming languages), it is specified that a "Not A number" (NaN) value should be a number.
If we abstract the fact that floating point arithmetic care only of finitely many values and abstract a bit, we get a sort of arithmetic arithmetic over $\mathbb{R}\cup\{\textrm{NaN}\}$ with $\textrm{NaN}$ being a zero for any arithmetic operation. Formally, for all $x$,
$$x\times \textrm{NaN} = x + \textrm{NaN} = \frac{x}{0} = \frac{0}{0}= \cdots =\textrm{NaN}$$
Is there any algebraic structures $(E,+,\times)$ having axioms allowing this or is it just to arbitrary to have been introduced even in weird part of algebra?

REPLY [12 votes]: The abstract picture here is that of adjoining an undefined or bottom element to a (partial) algebra. Doing this to a total algebra is boring, but it is a useful trick to turn partial algebras into total ones.
One just picks an element $\bot$ not in the carrier set, and extends the operations as follows: Any operation applied to a tuple outside its original domain yields $\bot$, including any tuple containing a $\bot$ somewhere.
The structure you are describing is then just the $1$-point totalirization of the partial algebra $(\mathbb{R},+,\times,/)$. The algebras arising in such a manner will hardly ever have very familiar properties. In particular, I don't think people name them directly, and would rather speak about the underlying partial algebras.