Editing Split-complex number (section)

===The diagonal basis===
There are two nontrivial [[idempotent element (ring theory)|idempotent element]]s given by <math>e=\tfrac{1}{2}(1-j)</math> and <math>e^* = \tfrac{1}{2}(1+j).</math> Idempotency means that <math>ee=e</math> and <math>e^*e^*=e^*.</math> Both of these elements are null:

<math display=block>\lVert e \rVert = \lVert e^* \rVert = e^* e = 0 ~.</math>

It is often convenient to use {{mvar|e}} and {{mvar|e}}<sup>∗</sup> as an alternate [[basis (linear algebra)|basis]] for the split-complex plane. This basis is called the '''diagonal basis''' or '''null basis'''. The split-complex number {{mvar|z}} can be written in the null basis as

<math display=block> z = x + jy = (x - y)e + (x + y)e^* ~.</math>

If we denote the number <math>z=ae+be^*</math> for real numbers {{mvar|a}} and {{mvar|b}} by {{math|(''a'', ''b'')}}, then split-complex multiplication is given by

<math display=block>\left( a_1, b_1 \right) \left( a_2, b_2 \right) = \left( a_1 a_2, b_1 b_2 \right) ~.</math>

The split-complex conjugate in the diagonal basis is given by
<math display=block>(a, b)^* = (b, a)</math>
and the squared modulus by

<math display=block> \lVert (a, b) \rVert^2  = ab.</math>