Split-complex number: Difference between revisions

Revision as of 03:55, 18 January 2025

Revision as of 21:21, 22 March 2025

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A similar algebra based on {{tmath|\R^2}} and component-wise operations of addition and multiplication, {{tmath|(\R^2, +, \times, xy),}} where {{mvar|xy}} is the [[quadratic form]] on {{tmath|\R^2,}} also forms a [[quadratic space]]. The [[ring isomorphism]]

<math display=block>\begin{align}

D &\to \mathbb{R}^2 \\

x + yj &\mapsto (x - y, x + y)

\end{align}</math>

is an [[quadratic space#isometry|isometry]] of quadratic spaces.

relates proportional quadratic forms, but the mapping is {{em|not}} an [[isometry]] since the multiplicative identity {{math|(1, 1)}} of {{tmath|\R^2}} is at a distance {{tmath|\sqrt 2}} from 0, which is normalized in {{mvar|D}}.

Split-complex numbers have many other names; see ''{{section link||Synonyms}}'' below. See the article ''[[Motor variable]]'' for functions of a split-complex number.