# Are there number systems that fix divide-by-zero? [duplicate]

• 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, BlueOct 18 '18 at 11:14

• 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 6:45