# Wheel theory

Wheels are a type 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 ${\displaystyle 0/0}$. The Riemann sphere is an extension of the complex plane by an element ${\displaystyle \infty }$, where ${\displaystyle z/0=\infty }$ for any complex ${\displaystyle z\neq 0}$. However, ${\displaystyle 0/0}$ 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 ${\displaystyle /x}$ similar (but not identical) to the multiplicative inverse ${\displaystyle x^{-1}}$, such that ${\displaystyle a/b}$ becomes shorthand for ${\displaystyle a\cdot /b=/b\cdot a}$, and modifies the rules of algebra such that

• ${\displaystyle 0x\neq 0\ }$ in the general case.
• ${\displaystyle x-x\neq 0\ }$ in the general case.
• ${\displaystyle x/x\neq 1\ }$ in the general case, as ${\displaystyle /x}$ is not the same as the multiplicative inverse of ${\displaystyle x}$.

Precisely, a wheel is an algebraic structure with operations binary addition ${\displaystyle +}$, multiplication ${\displaystyle \cdot }$, constants 0, 1 and unary ${\displaystyle /}$, satisfying:

• Addition and multiplication are commutative and associative, with 0 and 1 as their respective identities.
• ${\displaystyle /(xy)=/x/y\ }$ and ${\displaystyle //x=x\ }$
• ${\displaystyle xz+yz=(x+y)z+0z\ }$
• ${\displaystyle (x+yz)/y=x/y+z+0y\ }$
• ${\displaystyle 0\cdot 0=0\ }$
• ${\displaystyle (x+0y)z=xz+0y\ }$
• ${\displaystyle /(x+0y)=/x+0y\ }$
• ${\displaystyle 0/0+x=0/0\ }$

If there is an element ${\displaystyle a}$ with ${\displaystyle 1+a=0}$, then we may define negation by ${\displaystyle -x=ax}$ and ${\displaystyle x-y=x+(-y)}$.

Other identities that may be derived are

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

And, for ${\displaystyle x}$ with ${\displaystyle 0x=0}$ and ${\displaystyle 0/x=0}$, we get the usual

• ${\displaystyle x-x=0\ }$
• ${\displaystyle x/x=1\ }$

The subset ${\displaystyle \{x\mid 0x=0\}}$ is always a commutative ring if negation can be defined as above, and every commutative ring is such a subset of a wheel. If ${\displaystyle x}$ is an invertible element of the commutative ring, then ${\displaystyle x^{-1}=/x}$. Thus, whenever ${\displaystyle x^{-1}}$ makes sense, it is equal to ${\displaystyle /x}$, but the latter is always defined, even when ${\displaystyle x=0}$.