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Concept
Dual number
Summary
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 with .
Dual numbers can be added component-wise, and multiplied by the formula
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.
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 θ + dε, where θ is the angle between the directions of two lines in three-dimensional space and d is a distance between them. The n-dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century.
In modern algebra, the algebra of dual numbers is often defined as the quotient of a polynomial ring over the real numbers by the principal ideal generated by the square of the indeterminate, that is
It may also be defined as the exterior algebra of a one-dimensional vector space with as its basis element.
Division of dual numbers is defined when the real part of the denominator is non-zero. The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the non-real parts.
Therefore, to divide an equation of the form
we multiply the numerator and denominator by the conjugate of the denominator:
which is defined when c is non-zero.
If, on the other hand, c is zero while d is not, then the equation
has no solution if a is nonzero
is otherwise solved by any dual number of the form b/d + yε.
Official source
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