In algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying A split-complex number has two real number components and , and is written The conjugate of is Since the product of a number with its conjugate is an isotropic quadratic form. The collection of all split complex numbers for forms an algebra over the field of real numbers. Two split-complex numbers and have a product that satisfies This composition of over the algebra product makes a composition algebra. A similar algebra based on and component-wise operations of addition and multiplication, where is the quadratic form on also forms a quadratic space. The ring isomorphism relates proportional quadratic forms, but the mapping is an isometry since the multiplicative identity of is at a distance from 0, which is normalized in . Split-complex numbers have many other names; see below. See the article Motor variable for functions of a split-complex number. Definition A split-complex number is an ordered pair of real numbers, written in the form where and are real numbers and the hyperbolic unit satisfies In the field of complex numbers the imaginary unit i satisfies The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit is not a real number but an independent quantity. The collection of all such is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by 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 then the conjugate of is defined as The conjugate satisfies similar properties to usual complex conjugate. Namely, These three properties imply that the split-complex conjugate is an automorphism of order 2. The squared modulus of a split-complex number is given by the isotropic quadratic form It has the composition algebra property: 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 where and Another expression for the squared modulus is then 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