Different authors have used a great variety of names for the split-complex numbers.

See the article Motor variable for functions of a split-complex number.

Split-complex numbers have many other names; see the synonyms section below.

It is this sign change which distinguishes the split-complex numbers from the ordinary complex ones.

In the split-complex number plane there is an alternative "diagonal basis".

Consider the group of units U in the ring of split-complex numbers.

A second instance involves functions of a motor variable where arguments are split-complex numbers.

This leads to the split-complex numbers which have normalized basis with .

Split-complex numbers which are not invertible are called null elements.

The complexification of the split-complex numbers is the tessarines.

The split-complex number multiplication is used for spacetime transformation.

He used split-complex numbers for scalars in his split-biquaternions.

The construction uses homogeneous coordinates with split-complex number components.

The case (1,1) corresponds to the split-complex numbers.

This parameter is part of the polar decomposition of a split-complex number.

In the plane, split-complex numbers and dual numbers form alternative topological rings.

See also split-complex numbers in general.

Such elements arise in algebras of mixed signature, for example split-complex numbers or split-quaternions.

When , then z is a split-complex number and conventionally j replaces epsilon.

Using stereographic projection the plane of split-complex numbers is closed up with these lines to a hyperboloid of one sheet.

Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication.

The basis after Musès is identical to j from the split-complex numbers, and is a non-real root of .

A split-complex number is invertible if and only if its modulus is nonzero ().

The case where the determinant is negative only arises in a plane with , that is a split-complex number plane.

This insight follows from a study of split-complex number multiplications and the "diagonal basis" which corresponds to the pair of light lines.