Functions of a spacetime variable

Mathematics and Computer Education, Fall 2002 by Borota, Nicolae A, Osler, Thomas J

1. INTRODUCTION

Like the familiar complex numbers x iy, spacetime numbers x jt are a two dimensional extension of the one-dimensional real numbers x. In our previous paper [2], we explained how to add, subtract, multiply and divide spacetime numbers. We saw that some features of this arithmetic are identical to arithmetic with complex numbers (like addition and subtraction), but other manipulations (like multiplication and division) were very different. In this paper we introduce a subject of importance in applications of mathematics that has previously only been presented at a level appropriate for researchers or graduate students.

We begin with a brief review of key concepts from [2], then introduce the spacetime version of Euler's Formula and use it to represent spacetime numbers. We end by showing how to extend any familiar function of the real variable x to the spacetime variable x jt. We avoid a rigorous definition-lemma-theorem style in favor of a self-discovery approach. Rigorous presentations are easily found in the references. This paper can be presented to students in calculus, real analysis or complex analysis. The minimum requirement is a familiarity with complex numbers, complex arithmetic, series and Euler's formula for the complex exponential.

Problems are given throughout the paper with selected answers appearing in the final section.

6. COMMENTS ON THE REFERENCES

For further study we suggest the excellent paper by Garret Sobczyk [16]. We strongly recommend this paper even though it does use the terminology of modern abstract algebra. We also recommend the paper by Fjelstad [4]. In this paper Fjelstad explains how he and his students rediscovered spacetime numbers (he calls them perplex numbers). The references by Band [1], Majernik [9] and Ronveaux [11] are letters in response to the paper of Fjelstad and we think the reader will find these interesting. Readers familiar with fractals and the Mandelbrot set will find the papers of Senn [12] and Metzier [10] interesting because they show what the Mandelbrot set looks in the spacetime plane. The paper by Lambert [7] is an amusing critique of this entire enterprise (in French). References [5] and [6] are graduate level textbooks. The remaining references are research papers in the theory and application of spacetime numbers and other related number systems.

Our list of references represents only a small part of the available literature. Note that any material on Clifford algebras is of potential interest since spacetime numbers are a special case of these.

REFERENCES

1. W. Band, "Comments on `Extending relativity via the perplex numbers' [American Journal of Physics, Vol. 54, p. 416 (1986)]", American Journal of Physics, Vol. 56, p. 469 (1988).

2. N. Borota, E. Flores, T. Osler, "Spacetime Numbers The Easy Way", Mathematics and Computer Education, Vol. 34, No. 2, pp.159-168 (2000).

3. W. K. Clifford, "Applications of Grassman's extensive algebra", American Journal of Mathematics, Vol. 1, pp. 350-358 (1878).

4. P. Fjelstad, "Extending relativity via the perplex numbers", American Journal of Physics, Vol. 54, No.5, pp. 416-422 (1986).

5. D. Hestenes, SpaceTime Algebra, Gordon and Breach, Cooper Station, NY (1966). 6. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Kluwer, Dordrecht, The Netherlands (1992).

7. Dominique Lambert,"Les nombres complexes hyperboliques: des complexes qui nous laissent perplexes", [Hyperbolic complex numbers: complexes that leave us perplexed] Revue-des-Questions-Scientifiques [Review-Questions-Scientific] Vol. 166, No. 4, pp. 383-400 (1995).

8. V. Majernik, "Basic space-time transformations expressed by means of two component number systems", Acta Physics, A. Polon, Vol. 86, No. 5, pp. 291-295 (1994). 9. V. Majernik, "The perplex numbers are in fact the binary numbers", American Journal of Physics, Vol. 56, No. 8, p. 763 (1988).

10. Metzier, Wolfgang, "The 'mystery' of the quadratic Mandelbrot set", American Journal of Physics, Vol. 62, pp. 813-814 (1994).

11. Andre Ronveaux, "About `perplex numbers' ", American Journal of Physics, Vol. 55, p. 392 (1987).

12. Peter Senn, "The Mandelbrot set for binary numbers", American Journal of Physics, Vol. 58, p. 1018 (1990).

13. G. Sobczyk, "Spacetime vector analysis", Physics Letters, 84A, p. 45 (1981). 14. G. R. Miller, A. B. Thaheem, "Derivatives of matrix order", Arabic-Journal of

Mathematical and Sciences, Vol. 3, No. 1, pp. 21-36 (1997).

15. Hungshan Ren, The hyperbolic quasifield of numbers on the plane and analytic functions on it, part I, Heilongjiang-Daxue-Ziran-Kexue-Xuebao, Natural Science Journal of Heilongjiang University, No. 2, pp. 9-13 (1988).

16. G. Sobczyk, "The hyperbolic number plane", The College Mathematics Journal, Vol. 26, pp.268-280 (1995).

Nicolae A. Borota, Office of Assessment New Jersey Department of Education Trenton, New Jersey 08625-0500

Thomas J. Osler, Department of Mathematics Rowan University Glassboro, New Jersey 08028 Osler@rowan. edu