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Integer operations in computers (i.e. 32-bit integers) probably can be represented best by modular arithmetic (because of integer overflows/underflows).

What about IEEE 754 floating point arithmetic? You have rounding and negative zeros. What would be the a space(?) in mathematics that comes very near to numerical floating points?

user133801
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  • My beginner's intuition says that this might be close to a subgroup or subring (not sure if that's a thing) of the rational numbers. Yet, I'm not sure about the effect of the two $0$'s... – apnorton Mar 07 '14 at 16:34
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    You will have trouble when taking into account $\pm\infty$s, and of course, the good old, $\mathrm{NaN}$. – Lord Soth Mar 07 '14 at 16:37
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    Because of rounding and catastrophic losses of precision, the IEEE operations are not associative. This rules out most of the better-known algebraic structures. – Chris Culter Mar 07 '14 at 16:44
  • You may find it interesting that the C standard specifies two types of integers, "signed" and "unsigned". Unsigned integer arithmetic is defined to be the same as arithmetic mod $2^k$ where $k$ is the word size, and overflow conditions never occur. But signed integer arithmetic is defined differently, and overflow conditions do occur, leading to what is called "undefined behavior". Underflow in integer arithmetic is impossible by definition. – MJD Mar 07 '14 at 18:44

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