Split-complex Number - Algebraic Properties

Algebraic Properties

In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring R by the ideal generated by the polynomial x2 − 1,

R/(x2 − 1).

The image of x in the quotient is the "imaginary" unit j. With this description, it is clear that the split-complex numbers form a commutative ring with characteristic 0. Moreover if we define scalar multiplication in the obvious manner, the split-complex numbers actually form a commutative and associative algebra over the reals of dimension two. The algebra is not a division algebra or field since the null elements are not invertible. In fact, all of the nonzero null elements are zero divisors. Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a topological ring.

The algebra of split-complex numbers forms a composition algebra since

for any numbers z and w.

The class of composition algebras extends the normed algebras class which also has this composition property.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring R of the cyclic group C₂ over the real numbers R.

The split-complex numbers are a particular case of a Clifford algebra. Namely, they form a Clifford algebra over a one-dimensional vector space with a positive-definite quadratic form. Contrast this with the complex numbers which form a Clifford algebra over a one-dimensional vector space with a negative-definite quadratic form. (NB: some authors switch the signs in the definition of a Clifford algebra which will interchange the meaning of positive-definite and negative-definite). In mathematics, the split-complex numbers are members of the Clifford algebra Cℓ_1,0(R) = Cℓ0_1,1(R) (the superscript 0 indicating the even subalgebra). This is an extension of the real numbers defined analogously to the complex numbers C = Cℓ_0,1(R) = Cℓ0_2,0(R).