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dual number
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Contents
Idea
A dual number is given by an expression of the form a + ϵ b a + \epsilon b , where a a and b b are real numbers and ϵ 2 = 0 \epsilon^2 = 0 (but ϵ ≠ 0 \epsilon \ne 0 ). The set of dual numbers is a topological vector space and a commutative algebra over the real numbers.
We can generalise (at least the algebraic aspects) from ℝ \mathbb{R} to any commutative ring R R .
Interpretations
This can be thought of as:
the vector space ℝ 2 \mathbb{R}^2 made into an algebra by the rule( a , b ) ⋅ ( c , d ) = ( a c , a d + b c ) ; (a, b) \cdot (c, d) = (a c, a d + b c) ;
the subalgebra of those 2 2 -by-2 2 real matrices of the form( a b 0 a ) ; \left(\array { a & b \\ 0 & a } \right);
the polynomial ring ℝ [ x ] \mathbb{R}[\mathrm{x}] modulo x 2 \mathrm{x}^2 ;
the parabolic 2 2 -dimensional algebra of hypercomplex number s;
the algebra of functions on the infinitesimal interval (the smallest of the infinitesimally thickened points ) in synthetic differential geometry .
if ϵ \epsilon is regarded as being of degree 1 and ℝ ⊕ ϵ ℝ \mathbb{R} \oplus \epsilon \mathbb{R} is regarded accordingly as a superalgebra then this is the algebra of functions on the odd line ℝ 0 | 1 \mathbb{R}^{0|1} .
the square-0-extension corresponding to the ℝ \mathbb{R} -module (see there) given by ℝ \mathbb{R} itself.
We think of ℝ \mathbb{R} as a subset of 𝔻 \mathbb{D} by identifying a a with a + 0 ϵ a + 0 \epsilon .
Properties
𝔻 \mathbb{D} is equipped with an involution that maps ϵ \epsilon to ϵ ¯ = − ϵ \bar{\epsilon} = -\epsilon :
a + ϵ b ¯ = a − ϵ b . \overline{a + \epsilon b} = a - \epsilon b .
𝔻 \mathbb{D} also has an absolute value :
| a + ϵ b | = | a | ; {|a + \epsilon b|} = {|a|} ;
notice that the absolute value of a dual number is a non-negative real number, with
| z | 2 = z z ¯ . {|z|^2} = z \bar{z}.
But this absolute value is degenerate, in that | z | = 0 {|z|} = 0 need not imply that z = 0 z = 0 .
Some concepts in analysis can be extended from ℝ \mathbb{R} to 𝔻 \mathbb{D} , but not as many as work for the complex numbers . Even algebraically, the dual numbers are not as nice as the real or complex numbers, as they do not form a field .
Last revised on March 10, 2015 at 13:28:47.
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