A
split-complex number is an ordered pair of real numbers, written in the form z = x + jy where and are
real numbers and the
hyperbolic unit , which is not a real number but an independent quantity, satisfies j^2 = +1 In the field of
complex numbers the
imaginary unit i satisfies i^2 = -1 . The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The collection of all such is called the
split-complex plane.
Addition and
multiplication of split-complex numbers are defined by \begin{align} (x + jy) + (u + jv) &= (x + u) + j(y + v) \\ (x + jy)(u + jv) &= (xu + yv) + j(xv + yu). \end{align} This multiplication is
commutative,
associative and
distributes over addition.
Conjugate, modulus, and bilinear form Just as for complex numbers, one can define the notion of a
split-complex conjugate. If z = x + jy ~, then the conjugate of is defined as z^* = x - jy ~. The conjugate is an
involution which satisfies similar properties to the
complex conjugate. Namely, \begin{align} (z + w)^* &= z^* + w^* \\ (zw)^* &= z^* w^* \\ \left(z^*\right)^* &= z. \end{align} The squared
modulus of a split-complex number z=x+jy is given by the
isotropic quadratic form \lVert z \rVert^2 = z z^* = z^* z = x^2 - y^2 ~. It has the
composition algebra property: \lVert z w \rVert = \lVert z \rVert \lVert w \rVert ~. However, this quadratic form is not
positive-definite but rather has
signature , so the modulus is
not a
norm. The associated
bilinear form is given by \langle z, w \rangle = \operatorname\mathrm{Re}\left(z^* w\right) = \operatorname\mathrm{Re} \left(z w^*\right) = xu - yv ~, where z=x+jy and w=u+jv. Here, the
real part is defined by \operatorname\mathrm{Re}(z) = \tfrac{1}{2}(z + z^*) = x. Another expression for the squared modulus is then \lVert z \rVert^2 = \langle z, z \rangle ~. Since it is not positive-definite, this bilinear form is not an
inner product; nevertheless the bilinear form is frequently referred to as an
indefinite inner product. A similar abuse of language refers to the modulus as a norm. A split-complex number is invertible
if and only if its modulus is nonzero Numbers of the form have no inverse and are called
null vectors. The
multiplicative inverse of an invertible element is given by z^{-1} = \frac{z^*}{ {\lVert z \rVert}^2} ~.
The diagonal basis There are two nontrivial
idempotent elements given by e=\tfrac{1}{2}(1-j) and e^* = \tfrac{1}{2}(1+j). Idempotency means that ee=e and e^*e^*=e^*. Both of these elements are null: \lVert e \rVert = \lVert e^* \rVert = e^* e = 0 ~. 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 z = x + jy = (x - y)e + (x + y)e^* ~. If we denote the number z=ae+be^* for real numbers and by , then zero is , one is , split-complex addition is given by \left( a_1, b_1 \right) + \left( a_2, b_2 \right) = \left( a_1 + a_2, b_1 + b_2 \right) ~, and split-complex multiplication is given by \left( a_1, b_1 \right) \left( a_2, b_2 \right) = \left( a_1 a_2, b_1 b_2 \right) ~. The split-complex conjugate in the diagonal basis is given by (a, b)^* = (b, a) and the squared modulus by \lVert (a, b) \rVert^2 = ab.
Isomorphism relates the action of the hyperbolic versor on to squeeze mapping applied to 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 z = x + jy and making the mapping (u, v) = (x, y) \begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix} = (x, y) S ~. Now the quadratic form is uv = (x + y)(x - y) = x^2 - y^2 ~. Furthermore, (\cosh a, \sinh a) \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} = \left(e^a, e^{-a}\right) so the two
parametrized hyperbolas are brought into correspondence with . The
action of
hyperbolic versor e^{bj} \! then corresponds under this linear transformation to a
squeeze mapping \sigma: (u, v) \mapsto \left(ru, \frac{v}{r}\right),\quad r = e^b ~. 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 \{(a,b) \in \R \oplus \R : ab=1\}. The contracted
unit hyperbola \{\cosh a+j\sinh a : a \in \R\} 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==