and only if its modulus is nonzero thus numbers of the form have no inverse. The multiplicative inverse of an invertible element is given by Split-complex numbers which are not invertible are called null vectors. These are all of the form for some real number . The diagonal basis There are two nontrivial idempotent elements given by and Recall that idempotent means that and Both of these elements are null: It is often convenient to use and ∗ as an alternate basis for the split-complex plane. This basis is called the diagonal basis or null basis. The split-complex number can be written in the null basis as If we denote the number for real numbers and by , then split-complex multiplication is given by The split-complex conjugate in the diagonal basis is given by and the modulus by Isomorphism On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum with addition and multiplication defined pairwise. The diagonal basis for the split-complex number plane can be invoked by using an ordered pair for and making the mapping Now the quadratic form is Furthermore, so the two parametrized hyperbolas are brought into correspondence with . The action of hyperbolic versor then corresponds under this linear transformation to a squeeze mapping Though lying in the same isomorphism class in the category of rings, the split-complex plane and the direct sum of two real lines differ in their layout in the Cartesian plane. The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a dilation by . The dilation in particular has sometimes caused confusion in connection with areas of a hyperbolic sector. Indeed, hyperbolic angle corresponds to area of a sector in the plane with its "unit circle" given by The contracted unit hyperbola of the split-complex plane has only half the area in the span of a corresponding hyperbolic sector. Such confusion may be perpetuated when the geometry of the split-complex plane is not distinguished from that of . Geometry A two-dimensional real vector space with the Minkowski inner product is called -dimensional Minkowski space, often denoted Just as much of the geometry of the Euclidean plane can be described with complex numbers, the geometry of the Minkowski plane can be described with split-complex numbers. The set of points is a hyperbola for every nonzero in The hyperbola consists of