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. 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. The Riemann sphere can also be extended to a wheel by adjoining an element ⊥ {displaystyle ot } , where 0 / 0 = ⊥ {displaystyle 0/0=ot } . The Riemann sphere is an extension of the complex plane by an element ∞ {displaystyle infty } , where z / 0 = ∞ {displaystyle z/0=infty } for any complex z ≠ 0 {displaystyle z eq 0} . However, 0 / 0 {displaystyle 0/0} is still undefined on the Riemann sphere, but is defined in its extension to a wheel. The term wheel is inspired by the topological picture ⊙ {displaystyle odot } of the projective line together with an extra point ⊥ = 0 / 0 {displaystyle ot =0/0} . A wheel is an algebraic structure ( W , 0 , 1 , + , ⋅ , / ) {displaystyle (W,0,1,+,cdot ,/)} , satisfying: Wheels replace the usual division as a binary operator with multiplication, with a unary operator applied to one argument / x {displaystyle /x} similar (but not identical) to the multiplicative inverse x − 1 {displaystyle x^{-1}} , such that a / b {displaystyle a/b} becomes shorthand for a ⋅ / b = / b ⋅ a {displaystyle acdot /b=/bcdot a} , and modifies the rules of algebra such that If there is an element a {displaystyle a} such that 1 + a = 0 {displaystyle 1+a=0} , then we may define negation by − x = a x {displaystyle -x=ax} and x − y = x + ( − y ) {displaystyle x-y=x+(-y)} .