A wheel algebra, usually called a wheel, is an algebraic structure in which reciprocal and hence division are defined for every element, including zero. A wheel has commutative addition and multiplication, constants 0 and 1, and a unary reciprocal operation. Its axioms agree with ordinary fractional arithmetic where denominators are nonzero but weaken ring identities that would make total division inconsistent.
In the wheel obtained from a commutative ring, 
is an unsigned point at infinity and
is a separate absorbing error element,
often denoted 
.
In particular,
 |
and the familiar ring law 
does not hold for every wheel element. The subset of elements 
satisfying 
forms a commutative ring or semiring. Thus a wheel
does not define division by zero while leaving
all ordinary field laws unchanged; it supplies a different algebra whose exceptional
elements obey different identities.
