What is "split-complex number"

n. (context mathematics English) An element of the two-dimensional (l/en: algebra) over the (l/en: field) of (l/en real number real numbers) which contains an element ȷ ≠ 1 satisfying ȷ² = 1.

Split-complex product

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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

x + y j,

where x and y are real numbers. The number j is similar to the imaginary unit i, except that

j = +1.

As an algebra over the reals, the split-complex numbers are the same as the direct sum of algebras under the isomorphism sending to ). The name split comes from this characterization: as a real algebra, the split-complex numbers split into the direct sum . It arises, for example, as the real subalgebra generated by an involutory matrix.

Geometrically, split-complex numbers are related to the modulus in the same way that complex numbers are related to the square of the Euclidean norm . 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 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 real matrix as a complex number.

Split-complex numbers have many other names; see the synonyms section below. See the article Motor variable for functions of a split-complex number.