[JonChesterfield]

It's a slight tangent, but in the general realm of division being complicated, divide by zero is a longstanding annoyance. Does anyone know (or can refer to) a reasonable definition of +,-,*,/,% (in their usual meanings) on rational numbers such that A op B always evaluates to a rational?

A/1 divide 0/1 being stored as 1/0 and then propagated around seems reasonable but I keep putting off the bookwork of finding out what arithmetic relations still hold at that point. I think one loses multiply by zero folding to zero, for example A * 0 is now only 0 for A known to have non-zero denominator.

Context is a language with arbitrary precision rationals as the number type, and all I'm really looking for is a way to declare that the type of division is a rational for any rational arguments. Searching for this mostly turns up results on the naturals or integers which I'm not particularly interested in.

Long shot but worth asking. Thanks

[jasomill]

Does anyone know (or can refer to) a reasonable definition of +,-,*,/,% (in their usual meanings) on rational numbers such that A op B always evaluates to a rational?

Given the usual definition of multiplication (and division as its inverse), this is not possible:

Assume some multiplicative inverse 0⁻¹ of 0 exists. Then, writing · for multiplication, since

a·0 = b·0

for all rational a and b, we have

a·0·0⁻¹ = b·0·0⁻¹

and therefore

a = b

for all rational a and b, which is absurd.

[jasomill]

As for extending the rationals by 0⁻¹ and giving up properties, a·0 = 0 can still hold for all a ≠ 0⁻¹.

To get an idea of what operations do and don't make sense when extending the rationals in the way you propose, define ∞ as our 0⁻¹ and refer to

https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

substituting ℚ and ℚ̂ for ℝ and ℝ̂.

For a similar construction that maintains ordering properties, see

https://en.wikipedia.org/wiki/Extended_real_number_line#Arit...

Finally, note that the complex number equivalent to your construction is really interesting and useful:

https://en.wikipedia.org/wiki/Riemann_sphere

https://www-users.cse.umn.edu/~arnold/moebius/

[JonChesterfield]

The Riemann sphere has always existed beyond my grasp of mathematics. I remember the maths undergraduates being excited about it. Thank you for the references, it looks like I need to learn more before proceeding with certainty.

[DougMerritt]

In ordinary arithmetic and algebra, there isn't any reasonable definition of n/0 because if it is allowed, then one can prove contradictory things.

Outside of proof systems/pure math, like in software you write, you can define n/0 to be anything you like, you just can't depend on being able to derive mathematically consistent results.

There are also more exotic math systems. Infinity is not literally a number (it would lead to contradictions), but mathematicians can happily deal with numbers systems augmented with infinity as an extra member -- one just has to step carefully, since it still isn't a number. The "number system" has one non-number member requiring special handling.

Similarly there are modern approaches to infinitesimals where there's a thing "e" which is not zero, but e^2 is zero. That is not normal arithmetic but when handled carefully it can be useful.

[discarded1023]

The standard arithmetic classes of Isabelle/HOL show that you can extend division with the equation `x / 0 = 0` and things (seemingly) work out.

https://lawrencecpaulson.github.io/2021/12/01/Undefined.html

https://www.hillelwayne.com/post/divide-by-zero/

https://xenaproject.wordpress.com/2020/07/05/division-by-zer...

You'll need to do some thinking/proof to find out if it works for you.

[JonChesterfield]

Huh. That's surprising but quite persuasive, thank you for the references. I like the general view that a non-axiomatic divide can't introduce unsoundness so define it however is useful.

It has somewhat kicked the can down the road to defining the multiplicative inverse, but that's still a good step forward. Thank you

[amenhotep]

Not a very satisfying answer (feels like if a satisfying answer to this question was possible, maths would look very different) but I think the trick to not getting annoyed by it is to shift your perspective rather than try and redefine maths. It's not that division is a badly behaved operator; it's that division isn't an operator. It's a shorthand for multiplication by the reciprocal. Since the reciprocal of 0 doesn't exist, we can no more multiply by it than we can compute 5 + the sound of one hand clapping. And we don't fret over how the addition operator's inadequacy there is unsatisfying!