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algebra > abstract algebra > algebraic structure > associative algebra > dual number

algebra > differential algebra > dual number

algebra > commutative algebra > dual number

number > hypercomplex number > dual number

Terme préférentiel

dual number

In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form $a + bε$ , where $a$ and $b$ are real numbers, and $ε$ is a symbol taken to satisfy ${\\displaystyle \\varepsilon ^{2}=0}$ with ${\\displaystyle \\varepsilon \ eq 0}$ .
Dual numbers can be added component-wise, and multiplied by the formula

${\\displaystyle (a+b\\varepsilon )(c+d\\varepsilon )=ac+(ad+bc)\\varepsilon ,}$

which follows from the property $ε 2 = 0$ and the fact that multiplication is a bilinear operation.
The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Dual_number)

nombre dual

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RDF/XML TURTLE JSON-LD Date de création 23/08/2023, dernière modification le 22/09/2023