Split-complex number
In abstract algebra, the split-complex numbers (or hyperbolic numbers, also perplex numbers, and double numbers) are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the form
where x and y are real numbers. The number j is similar to the imaginary unit i, except that
As an algebra over the reals, the split-complex numbers are the same as the direct sum of algebras R ⊕ R under the isomorphism sending x + y j to (x + y, x − y). The name split comes from this characterization: as a real algebra, the split-complex numbers split into the direct sum R ⊕ R. It arises, for example, as the real subalgebra generated by an involutory matrix.
Geometrically, split-complex numbers are related to the modulus (x2 − y2) in the same way that complex numbers are related to the square of the Euclidean norm (x2 + y2). Unlike the complex numbers, the split-complex numbers contain nontrivial idempotents (other than 0 and 1), as well as zero divisors, and therefore they do not form a field.
In interval analysis, a split complex number x + y j represents an interval with midpoint x and radius y. Another application involves using split-complex numbers, dual numbers, and ordinary complex numbers, to interpret a 2 × 2 real matrix as a complex number.
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