Dual numbers
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A variety of dualities in mathematics are listed at duality (mathematics).
For the dual grammatical number found in some languages see dual grammatical number.
In abstract algebra, the dual numbers are a particular two-dimensional commutative unital associative algebra over the real numbers, arising from the reals by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). Every dual number has the form z = a + bε with a and b uniquely determined real numbers. The plane of all dual numbers is an "alternative complex plane" that complements the ordinary complex number plane C and the plane of split-complex numbers. The "unit circle" of dual numbers consists of those with a = 1 or -1 since these satisfy z z * = 1 where z * = a - bε.However, note that exp(bε) = 1 + bε, so the exponential function applied to the ε-axis covers only half the "circle".
This construction can be carried out more generally: for a commutative ring R one can define the dual numbers over R as the quotient of the polynomial ring R[X] by the ideal (X2): the image of X then has square equal to zero and corresponds to the element ε from above. This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials (purely algebraic differential forms).
Over any ring R, the dual number a + bε is a unit (i.e. multiplicatively invertible) if and only if a is a unit in R. In this case, the inverse of a + bε is a-1 - ba-2ε. As a consequence, we see that the dual numbers over any field (or any commutative local ring) form a local ring.
One application of dual numbers is to automatic differentiation. Consider the real dual numbers above. Given any real polynomial P(x)=p0+p1x+p2x2+...+pnxn, it is straightforward to extend the domain of this polynomial from the reals to the dual numbers. Then we have this result: P(a+bε)=P(a)+bP'(a)ε, where P' is the derivative of P. By computing over the dual numbers, rather than over the reals, we can use this to compute derivatives of polynomials. More generally we may define division of dual numbers and then go on to define transcendental functions of dual numbers by defining f(a+bε)=f(a)+bf'(a)ε. By computing compositions of these functions over the dual numbers and examining the coefficient of ε in the result we find we have automatically computed the derivative of the composition.