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  • Natural numbers are closed under addition and multiplication, but not subtraction. Fixed by...

  • Integers are closed under subtraction, but not division. Fixed by...

  • Rational numbers are closed under division, but not root. Fixed by...

  • Real numbers are closed under roots, but not negative roots. Fixed by...

  • Complex numbers are closed under negative roots.

But wait, rational numbers are not closed under division, because division-by-zero is not defined.

My question is: Given the above strategy of defining new number systems to cope with non-closure properties, has anything been done or attempted to fix divide-by-zero?

Obviously, we can have a "value" like NaN, but it's not very algebraically useful, except as a kind of error.

BTW: It seems to me that we can't do better than this, it's in the nature of even a semi-ring like the natural numbers, that multiply-by-zero is an annihilator, which is what causes the problem. (e.g. regular expressions also have this property). But what do I know?

The suggested "duplicate" is about extending the natural numbers to allow division-by-zero. The question here is more general: ensuring closure. This allows other approaches e.g. not having zero at all (see my answer).

hyperpallium
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    Yes, you cannot define an inverse of something noninjective (or at least not a two-sided inverse, i..e, we cannot keep the $\frac ab\cdot b=a$ cake). – Hagen von Eitzen Oct 18 '18 at 06:45
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    https://en.wikipedia.org/wiki/Riemann_sphere Note that if you allow division by zero, certain other things will break like associativity and you are left with other undefined operations. – JMoravitz Oct 18 '18 at 06:45
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    Why is "divide-by-zero" something that needs to be "fix"ed? – Angina Seng Oct 18 '18 at 06:47
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    See also: https://en.wikipedia.org/wiki/Wheel_theory – Hans Lundmark Oct 18 '18 at 08:29
  • @LordSharktheUnknown I amswer with a question: why were all the others "fix"ed? Closure seems to be an important property. I like it because it enables arbitrary algebraic manipukation to always work (i.e. without checkmg for specific values). In coding terms, compile-time checks then obviate all runtime checks. Of course, it needn't be fixed - e.g. people subtract natural numbers, despite non-closure - it just seems logical to me to try it, and I'm wondering what people have tried. – hyperpallium Oct 18 '18 at 11:09
  • Thanks @HagenvonEitzen, nice exposition. But why should the additive identity be a multiplicative annihilator? Addition and multiplication are only related via distributivity, so it must be something to do with that...? – hyperpallium Oct 19 '18 at 04:58
  • @HagenvonEitzen I think it's $a(b+0)=ab+a0$. Because $b+0=b$ and so $a(b+0)=ab$, $a0$ must equal zero. – hyperpallium Oct 19 '18 at 23:16

1 Answers1

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positive rational numbers: add division (multiplicative inverse) to the natural numbers instead of subtraction (additive inverse). No subtraction means no need for zero, and no division-by-zero.

hyperpallium
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