Editing Split-complex number (section)

==Algebraic properties==
In [[abstract algebra]] terms, the split-complex numbers can be described as the [[quotient ring|quotient]] of the [[polynomial ring]] {{tmath|\R[x]}} by the [[ideal (ring theory)|ideal]] generated by the [[polynomial]] <math>x^2-1,</math>

<math display=block>\R[x]/(x^2-1 ).</math>

The image of {{mvar|x}} in the quotient is the "imaginary" unit {{mvar|j}}. With this description, it is clear that the split-complex numbers form a [[commutative algebra (structure)|commutative algebra]] over the real numbers. The algebra is ''not'' a [[field (mathematics)|field]] since the null elements are not invertible. All of the nonzero null elements are [[zero divisor]]s.

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

<math display=block>\lVert zw \rVert = \lVert z \rVert \lVert w \rVert ~</math>

for any numbers {{mvar|z}} and {{mvar|w}}.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the [[group ring]] {{tmath|\R[C_2]}} of the [[cyclic group]] {{math|C{{sub|2}}}} over the real numbers {{tmath|\R.}}

Elements of the [[identity component]] in the [[group of units]] in '''D''' have four square roots.: say <math>p = \exp (q), \ \ q \in D. \text{then} \pm \exp(\frac{q}{2}) </math> are square roots of ''p''. Further, <math>\pm j \exp(\frac{q}{2})</math> are also square roots of ''p''.

The [[idempotent element (ring theory)|idempotents]] <math>\frac{1 \pm j}{2}</math> are their own square roots, and the square root of <math>s \frac{1 \pm j}{2}, \ \ s > 0, \ \text{is} \ \sqrt{s} \frac{1 \pm j}{2}</math>