Dual numbers and oriented lines

Dual numbers are expressions of the form

a+ b e

where a,b are real numbers and e satisfies the identity e²=0. A general reference is [1]. Their formal resemblance to complex numbers a+ bi is more than superficial.

(1) Explain how dual and complex numbers are related.

One can do arithmetic with dual numbers in a manner analogous to that carried out with complex numbers. The concepts of conjugate and modulus are also defined. There are however some fundamental differences between complex and dual numbers, essentially because the dual number system contains zero divisors.

(2) Explain the main features of the arithmetic of dual numbers.

Once again, like complex numbers, dual numbers have a geometric interpretation. A dual number represents an oriented line in the plane.

(3) Describe this geometric interpretation.

It is then possible to visualize arithmetic operations as motions of oriented lines.

(4) Write down the most general motion of oriented lines. Exhibit examples.

Finally, or not so, the theory of dual numbers can be effectively used to prove numerous geometric theorems relating to points, lines, and circles. This should be profusely illustrated.

References

[1]

I. M. Yaglom, Complex Numbers in Geometry, Academic Press, New York, 1968.

[2]

H. Schwerdtfeger, Geometry of complex numbers, University of Toronto Press, Toronto, 1962.