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In algebra, a split complex number has two real number components x and y, and is written z = x + y j, where j = 1. The conjugate of z is z = x − y j. Since j = 1, the product of a number z with its conjugate is zz = x − y, an isotropic quadratic form, =N = x − y.
The collection D of all split complex numbers z = x + y j for x, y ∈ R forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies N = NN. This composition of N over the algebra product makes a composition algebra.
A similar algebra based on R and component-wise operations of addition and multiplication, , where xy is the quadratic form on R, also forms a quadratic space. The ring isomorphism
Split-complex numbers have many other names; see § Synonyms below. See the article Motor variable for functions of a split-complex number.