In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle j^{2}=1} , where j ≠ ± 1 {\displaystyle j\neq \pm 1} . A split-complex number has two real number components x and y, and is written z = x + y j . {\displaystyle z=x+yj.} The conjugate of z is z ∗ = x − y j . {\displaystyle z^{*}=x-yj.} Since j 2 = 1 , {\displaystyle j^{2}=1,} the product of a number z with its conjugate is N ( z ) := z z ∗ = x 2 − y 2 , {\displaystyle N(z):=zz^{*}=x^{2}-y^{2},} an isotropic quadratic form. The collection D of all split-complex numbers z = x + y j {\displaystyle z=x+yj} for x , y ∈ R {\displaystyle x,y\in \mathbb {R} } forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies N ( w z ) = N ( w ) N ( z ) . {\displaystyle N(wz)=N(w)N(z).} This composition of N over the algebra product makes (D, +, ×, *) a composition algebra. A similar algebra based on R 2 {\displaystyle \mathbb {R} ^{2}} and component-wise operations of addition and multiplication, ( R 2 , + , × , x y ) , {\displaystyle (\mathbb {R} ^{2},+,\times ,xy),} where xy is the quadratic form on R 2 , {\displaystyle \mathbb {R} ^{2},} also forms a quadratic space. The ring isomorphism
D → R 2 x + y j ↦ ( x − y , x + y ) {\displaystyle {\begin{aligned}D&\to \mathbb {R} ^{2}\\x+yj&\mapsto (x-y,x+y)\end{aligned}}}
is an isometry of quadratic spaces. Split-complex numbers have many other names; see § Synonyms 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
z = x + j y {\displaystyle z=x+jy}
where x and y are real numbers and the hyperbolic unit j, which is not a real number but an independent quantity, satisfies
j 2 = + 1 {\displaystyle j^{2}=+1}
In the field of complex numbers the imaginary unit i satisfies i 2 = − 1. {\displaystyle i^{2}=-1.} The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by
( x + j y ) + ( u + j v ) = ( x + u ) + j ( y + v ) ( x + j y ) ( u + j v ) = ( x u + y v ) + j ( x v + y u ) . {\displaystyle {\begin{aligned}(x+jy)+(u+jv)&=(x+u)+j(y+v)\\(x+jy)(u+jv)&=(xu+yv)+j(xv+yu).\end{aligned}}}
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
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