Wheel theory

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 $\infty$ , where $z/0=\infty$ for any complex $z\neq 0$ . However, $0/0$ is still undefined on the Riemann sphere, but defined in wheels.

The algebra of wheels

Wheels discard the usual notion of division being a binary operator, replacing it with multiplication by a unary operator $/x$ similar (but not identical) to the reciprocal $x^{-1}$ , such that $a/b$ becomes shorthand for $a\cdot /b=/b\cdot a$ , and modifies the rules of algebra such that

$0x\neq 0\$ in the general case.
$x-x\neq 0\$ in the general case.
$x/x\neq 1\$ 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 $\cdot$ , constants 0, 1 and unary $/$ , satisfying:

Addition and multiplication are commutative and associative, with 0 and 1 as identities respectively
$/(xy)=/x/y\$ and $//x=x\$
$xz+yz=(x+y)z+0z\$
$(x+yz)/y=x/y+z+0y\$
$0\cdot 0=0\$
$(x+0y)z=xz+0y\$
$/(x+0y)=/x+0y\$
$0/0+x=0/0\$

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

$0x+0y=0xy\$
$x-x=0x^{2}\$
$x/x=1+0x/x\$

However, if $0x=0$ and $0/x=0$ we get the usual

$x-x=0\$
$x/x=1\$

The subset $\{x\mid 0x=0\}$ 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, even when $x=0$ .