Dual numbers

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε² = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one dimensional vector space; the general case of n dimensions leads to the Grassmann numbers.

The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by ε is its only maximal ideal. Dual numbers form the coefficients of dual quaternions.

Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as ${\displaystyle \vartheta +d\varepsilon }$, where ${\displaystyle \vartheta }$ is the angle between the directions of two lines in three-dimensional space and ${\displaystyle d}$ is a distance between them. The n-dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century.

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