Dual number

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of n dimensions leads to the Grassmann numbers. In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of n dimensions leads to the Grassmann numbers. The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by ε is its only maximal ideal.Dual numbers form the coefficients of dual quaternions. Like the complex numbers and split-complex numbers, the dual numbers form an algebra that is 2-dimensional over the field of real numbers. 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 + d ε {displaystyle vartheta +dvarepsilon } , where ϑ {displaystyle vartheta } is the angle between the directions of two lines in three-dimensional space and d {displaystyle d} is a distance between them. The n-dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century. Using matrices, dual numbers can be represented as The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication; both operations are commutative and associative within the algebra of dual numbers. This correspondence is analogous to the usual matrix representation of complex numbers.However, it is not the only representation with 2 × 2 real matrices, as is shown in the profile of 2 × 2 real matrices. 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 so the exponential map applied to the ε-axis covers only half the 'circle'.