What exactly is "exact" (was Clean Singleton Docstrings)

[Random832]

On Mon, Jul 18, 2016, at 01:37, Rustom Mody wrote:
> The INTENTION is that in addition to capturing the usual number tower
> ℕ ⊂ ℤ ⊂ ℝ (or parts therefore)

Well, ℚ. You hardly ever see representations of numbers that are in ℝ-ℚ.

> scheme also captures ORTHOGONALLY (word in the docs) the (in)exactness
> attribute
>
> In the realization, the designers need to (over)load these different
> attributes onto numeric constants. So the ruse chosen is that fraction-
> constants by default are exact Decimal constants are inexact And one
> can override any of these by suitable a function [At least that's my
> understanding]

The other thing you're missing is that implementations are free to
choose any representation they want (such as an IEEE binary float, or a
rational of bounded denominator) for inexact numbers, no matter how they
were written, because they're not required to preserve the exact value.
The fact that an implementation *could* also use an IEEE binary float to
represent an exact rational number whose denominator happens to be power
of two is immaterial to this discussion.

It's not required to actually be "orthogonal" in the sense that you're
imagining - i.e. being able to have inexact numbers whose machine
representations are big integers, big rationals, etc. An implementation
could choose to represent *all* inexact numbers as floats.