Lesson plan
Split-complex number
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1 , where j \neq \pm 1 . A split-complex number has two real number components and , and is written z=x+yj . The conjugate of is z^*=x-yj. Since j^2=1, the product of a number with its conjugate is N(z) := zz^* = x^2 - y^2, an isotropic quadratic form. The collection of all split-complex numbers z=x+yj for forms an algebra over the field of real numbers. Two split-complex numbers and have a product that satisfies N(wz)=N(w)N(z). 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 \begin{align} D &\to \mathbb{R}^2 \\ x + yj &\mapsto (x - y, x + y) \end{align} is an isometry of quadratic spaces. Split-complex numbers have many other names; see below. See the article Motor variable for functions of a split-complex number.
Article summary
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1 , where j \neq \pm 1 . A split-complex number has two real number components and , and is written z=x+yj . The conjugate of is z^*=x-yj. Since j^2=1, the product of a number with its conjugate is N(z) := zz^* = x^2 - y^2, an isotropic quadratic form. The collection of all split-complex numbers z=x+yj for forms an algebra over the field of real numbers. Two split-complex numbers and have a product that satisfies N(wz)=N(w)N(z). 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 \begin{align} D &\to \mathbb{R}^2 \\ x + yj &\mapsto (x - y, x + y) \end{align} is an isometry of quadratic spaces. Split-complex numbers have many other names; see below. See the article Motor variable for functions of a split-complex number.
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