# Algebraic structure for the floating-point arithmetic.

Is it possible to consider floating-point arithmetic as some common algebraic structure? For example, consider something like simplified IEEE754 single precision binary floating-point subset of numbers that consist of 1 sign bit, 23-bit mantissa and 8-bit exponent

Let $\mathcal{F}$ be a floating-point set which consists of:

• finite floating-point numbers, such that each number is described by three integers: $(-1)^{s}\cdot M\cdot 2^{q}$. $s$ is a sign($0$ or $1$), $\ M$ and $q$ must be integers in the ranges $0$ through $2^{23} -1$ and $-128$ to $127$ respectively.
• two infinities: $+\infty$ and $-\infty$
• NaN (not a number)

Let's consider four operations defined on the $\mathcal{F}$ : summation, subtraction, multiplication and division(as multiplication by a reciprocal).
The result is NaN in each of the following cases: - On of the operands is NaN - The divisions 0/0 and ±∞/±∞ - The multiplications 0×±∞ and ±∞×0 - The additions ∞ + (−∞), (−∞) + ∞ and equivalent subtractions

If sum or product of finite numbers cannot be represented as finite floating-point number, the result is infinity.

Finally, let $\mathfrak F$ be an algebraic structure with the set and operations we have defined.

So, is it possible to characterize this structure somehow? Excuse me if my question is too absurd :)

• Well, it is an algebraic structure, that's for sure. But it's not a group, not a ring, not a semiring, or any of the most common object studied in algebra. What do you mean by characterize? This question doesn't seem absurd, but it does seem very vague. Voting to close as too vague for now. – tomasz Jul 18 '13 at 11:49
• I think it's specific enough to keep, and that Hagen von Eitzen's answer is already pretty good. I think by "characterize" Shem might have been asking something more like "categorize," as in "is it some type of already-studied algebraic structure?" – rschwieb Jul 18 '13 at 13:06
• I agree with rschwieb. I think it is quite clear what Shemhamforasch is asking, and not vague at all. If the answer was "Ring" then there would be no close votes. Just because the answer is "Sorry, we do not know of any formal name for a structure which has these properties", does not mean we should close the question! – user1729 Jul 18 '13 at 13:43

## 2 Answers

This is a finite set with a couple of binary operations on it that are not even associative (though commutative as far as they should) or distributive.

• Well, nothing could be added to this?.. – Max Malysh Jul 18 '13 at 0:29
• @Shemhamforasch There sure can, but anything lackinmg associativity is beyond my liking. Probably the best thing to do is compare it with what it is intended to model: $\mathbb R$. – Hagen von Eitzen Jul 18 '13 at 12:48
• It should be pointed out that an operation being non-associative is not grounds for not studying it. For example, the Lie Bracket is non-associative, and Lie Algebras are much-studied (and for good reason). – user1729 Jul 18 '13 at 13:41

Look into D.E. Knuth's The Art of Computer Programming, Vol. 2, Section 4.2.2 A, p. 214-223, he gives an axiomatic approach of floating point arithmetic which seems to be what you are looking for.