a right and left branch passing through and . The case is called the unit hyperbola. The conjugate hyperbola is given by with an upper and lower branch passing through and . The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements: These two lines (sometimes called the null cone) are perpendicular in and have slopes ±1. Split-complex numbers and are said to be hyperbolic-orthogonal if . While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the simultaneous hyperplane concept in spacetime. The analogue of Euler's formula for the split-complex numbers is This formula can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers. For all real values of the hyperbolic angle the split-complex number has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as have been called hyperbolic versors. Since has modulus 1, multiplying any split-complex number by preserves the modulus of and represents a hyperbolic rotation (also called a Lorentz boost or a squeeze mapping). Multiplying by preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself. The set of all transformations of the split-complex plane which preserve the modulus (or equivalently, the inner product) forms a group called the generalized orthogonal group . This group consists of the hyperbolic rotations, which form a subgroup denoted , combined with four discrete reflections given by and The exponential map sending to rotation by is a group isomorphism since the usual exponential formula applies: If a split-complex number does not lie on one of the diagonals, then has a polar decomposition. Algebraic properties In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring by the ideal generated by the polynomial The image of in the quotient is the "imaginary" unit . With this description, it is clear that the split-complex numbers form a commutative algebra over the real numbers. The algebra is not a field since the null elements are not invertible. All of the nonzero null elements are zero divisors. 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