Dual numbers

In abstract algebra, the dual numbers are a particular two-dimensional commutative associative algebra over the real numbers, arising from the reals R by adjoining one new element ε with the property ε² = 0. Every dual number has the form a + bε with a and b uniquely determined real numbers.

This construction can be carried out more generally: for a commutative ring R one can define the dual numbers over R as the quotient R[X]/(X²): the image of X then has square equal to zero. This ring and its generalisations play an important part in the algebraic theory of derivations and differential forms.

The dual numbers over any field form a commutative local ring; the maximal ideal consists of classes of the form a + bX where a ≠ 0.

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The word "dual" in mathematics is used in several other meanings as well, see for instance dual space and dual polyhedron.