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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 ${\displaystyle \varepsilon \neq 0}$.

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 ε² = 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.

History

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.

Modern definition

In modern algebra, the algebra of dual numbers is often defined as the quotient of a polynomial ring over the real numbers ${\displaystyle (\mathbb {R} )}$ by the principal ideal generated by the square of the indeterminate, that is

${\displaystyle \mathbb {R} /\left\langle X^{2}\right\rangle .}$

It may also be defined as the exterior algebra of a one-dimensional vector space with ${\displaystyle \varepsilon }$ as its basis element.

Division

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 evaluate an expression of the form

${\displaystyle {\frac {a+b\varepsilon }{c+d\varepsilon }}}$

we multiply the numerator and denominator by the conjugate of the denominator:

${\displaystyle {\begin{aligned}{\frac {a+b\varepsilon }{c+d\varepsilon }}&={\frac {(a+b\varepsilon )(c-d\varepsilon )}{(c+d\varepsilon )(c-d\varepsilon )}}\\&={\frac {ac-ad\varepsilon +bc\varepsilon -bd\varepsilon ^{2}}{c^{2}+cd\varepsilon -cd\varepsilon -d^{2}\varepsilon ^{2}}}\\&={\frac {ac-ad\varepsilon +bc\varepsilon -0}{c^{2}-0}}\\&={\frac {ac+\varepsilon (bc-ad)}{c^{2}}}\\&={\frac {a}{c}}+{\frac {bc-ad}{c^{2}}}\varepsilon \end{aligned}}}$

which is defined when c is non-zero.

If, on the other hand, c is zero while d is not, then the equation

${\displaystyle {a+b\varepsilon =(x+y\varepsilon )d\varepsilon }={xd\varepsilon +0}}$

1. has no solution if a is nonzero
2. is otherwise solved by any dual number of the form b/d + yε.

This means that the non-real part of the "quotient" is arbitrary and division is therefore not defined for purely nonreal dual numbers. Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring) of the dual numbers.

Matrix representation

The dual number ${\displaystyle a+b\epsilon }$ can be represented by the square matrix ${\displaystyle {\begin{pmatrix}a&b\\0&a\end{pmatrix}}}$. In this representation the matrix ${\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}}$ squares to the zero matrix, corresponding to the dual number ${\displaystyle \varepsilon }$.

There are other ways to represent dual numbers as square matrices. They consist of representing the dual number ${\displaystyle 1}$ by the identity matrix, and ${\displaystyle \epsilon }$ by any matrix whose square is the zero matrix; that is, in the case of 2×2 matrices, any nonzero matrix of the form

${\displaystyle {\begin{pmatrix}a&b\\c&-a\end{pmatrix}}}$

with ${\displaystyle a^{2}+bc=0.}$^[1]

Differentiation

One application of dual numbers is automatic differentiation. Any polynomial

${\displaystyle P(x)=p_{0}+p_{1}x+p_{2}x^{2}+\cdots +p_{n}x^{n}}$

with real coefficients can be extended to a function of a dual-number-valued argument,

Zdroj:https://en.wikipedia.org?pojem=Dual_numbers
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