• 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).


marked as duplicate by Hans Lundmark, Michael Hoppe, Rahul, GNUSupporter 8964民主女神 地下教會, Blue Oct 18 '18 at 11:14

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    $\begingroup$ 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). $\endgroup$ – Hagen von Eitzen Oct 18 '18 at 6:45
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    $\begingroup$ 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. $\endgroup$ – JMoravitz Oct 18 '18 at 6:45
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    $\begingroup$ Why is "divide-by-zero" something that needs to be "fix"ed? $\endgroup$ – Lord Shark the Unknown Oct 18 '18 at 6:47
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    $\begingroup$ See also: en.wikipedia.org/wiki/Wheel_theory $\endgroup$ – Hans Lundmark Oct 18 '18 at 8:29
  • $\begingroup$ @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. $\endgroup$ – hyperpallium Oct 18 '18 at 11:09

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.


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