Wheel theory
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Wheels are a kind of algebra where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring.
Also the Riemann sphere can be extended to a wheel by adjoining an element 0 / 0. The Riemann sphere is an extension of the complex plane by an element , where for any complex . However, 0 / 0 is still undefined on the Riemann sphere, but defined in wheels.
[edit] The algebra of wheels
Wheels discard the usual notion of division being a binary operator, replacing it with a unary operator / x similar (but not identical) to the reciprocal x − 1 such that a / b becomes short-hand for , and modifies the rules of algebra such that
- in the general case.
- in the general case.
- in the general case, as / x is not the same as the multiplicative inverse of x.
Precisely, a wheel is an algebraic structure with operations binary addition + , multiplication , constants 0, 1 and unary / , satisfying:
- Addition and multiplication are commutative and associative, with 0 and 1 as identities respectively
- and
If there is an element a with 1 + a = 0, then we may define negation by − x = ax and x − y = x + ( − y).
Other identities that may be derived are
However, if 0x = 0 and 0 / x = 0 we get the usual
The subset is always a commutative ring if negation can be defined as above, and every commutative ring is such a subset of a wheel. If x is an invertible element of the commutative ring, then x − 1 = / x. Thus, whenever x − 1 makes sense, it is equal to / x, but the latter is always defined, also when x = 0.
[edit] References
- Carlström, Jesper: Wheels — on division by zero . Mathematical Structures in Computer Science, 14(2004): no. 1, 143-184 (also available online here).