The Algebra of Wheels
Wheels discard the usual notion of division being a binary operator, replacing it with multiplication by a unary operator similar (but not identical) to the reciprocal, such that 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 is not the same as the multiplicative inverse of .
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 with, then we may define negation by and .
Other identities that may be derived are
However, if and 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 is an invertible element of the commutative ring, then . Thus, whenever makes sense, it is equal to, but the latter is always defined, also when .
Read more about this topic: Wheel Theory
Famous quotes containing the words algebra and/or wheels:
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)
“The curfew tolls the knell of parting day,
The lowing herd wind slowly o’er the lea,
The ploughman homeward plods his weary way,
And leaves the world to darkness and to me.
Now fades the glimmering landscape on the sight,
And all the air a solemn stillness holds,
Save where the beetle wheels his droning flight,
And drowsy tinklings lull the distant folds.”
—Thomas Gray (1716–1771)