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An alternative quadratic formula

  • Norbert HungerbühlerEmail author
Mathematik in der Lehre
  • 2 Downloads

Abstract

One would usually expect that a subject such as the quadratic equation which is known since Babylonian times would not offer any interesting new aspect today. It is, however, a feature of mathematics that one can always gain new insights by looking at an old topic from a new angle. A look back at the history reveals that the quadratic equation has indeed been repeatedly investigated in all epochs and cultures. The solution formulas for this equation are correspondingly numerous, although most of them are only little known. It may come as a surprise that here a further, particularly symmetric solution formula can be added to the catalogue of quadratic formulae.

Keywords

Quadratic equation Quadratic formula 

Mathematics Subject Classification

01A99 97H30 51N99 12D99 

Notes

Acknowledgements

The author would like to thank Hans Peter Dreyer for pointing out to him the nice exercise of the falling stone, Jacques Gélinas for the remark about the numerical stability, and Klaus Volkert for pointing out the wonderful book [12] of Mattheissen. The author is also grateful for the helpful remarks and hints of the referee.

References

  1. 1.
    Ben-Menahem, A.: Historical encyclopedia of natural and mathematical sciences, vol. 1. Springer, Berlin, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Casselman, B.: x, y and z. Notices Amer. Math. Soc. 62(2), 147 (2015)Google Scholar
  3. 3.
    Jørgen, C., Gallardo, L., Vaserstein, L., Wheland, E.: Solving quadratic equations over polynomial rings of characteristic two. Publ. Mat. 42(1), 131–142 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Descartes, R.: Discours de la Méthode. Imprimerie de Ian Maire (1637)zbMATHGoogle Scholar
  5. 5.
    Eells, W.C.: Greek methods of solving quadratic equations. Amer. Math. Monthly 18(1), 3–14 (1911)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Forsythe, G.E.: Pitfalls in computation, or why a math book isn’t enough. Amer. Math. Monthly 77(9), 931–956 (1970)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Halbeisen, L., Hungerbühler, N., Lazarovich, N., Lederle, W., Lischka, M., Schumacher, S.: Forms of choice in ring theory. Results Math. (2019).  https://doi.org/10.1007/s00025-018-0935-1 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brioschi, F.: Sulla risoluzione delle equazioni del quinto grado. Ann. Mat. Pura Appl. 1, 256 (1858).  https://doi.org/10.1007/BF03197334 CrossRefGoogle Scholar
  9. 9.
    Herz-Fischler, R.: What are propositions 84 and 85 of Euclid’s data all about? Hist. Math. 11, 86–91 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Høyrup, J.: Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Sources and studies in the history of mathematics and physical sciences. (2002)CrossRefGoogle Scholar
  11. 11.
    Klein, F.: Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Birkhäuser Verlag, Basel (1993). Reprint of the 1884 original, edited, with an introduction and commentary by Peter Slodowy.CrossRefGoogle Scholar
  12. 12.
    Matthiessen, L.: Grundzüge der antiken und modernen Algebra der literalen Gleichungen. B.G. Teubner (1878)zbMATHGoogle Scholar
  13. 13.
    Muller, D.E.: A method for solving algebraic equations using an automatic computer. Math. Tables Aids Comput. 10, 208–215 (1956)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in Fortran 77, 2nd edn. Fortran numerical recipes, vol. 1. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  15. 15.
    Rāshid, R., Morelon, R.: Encyclopedia of the history of Arabic science, vol. 2. Routledge, London, New York (1996)CrossRefGoogle Scholar
  16. 16.
    Serway, R.A., Vuille, C., Faughn, J.S.: College physics. Brooks/Cole Cengage Learning, Belmont (2009)Google Scholar
  17. 17.
    van der Waerden, B.L.: Science awakening I. Noordhoff (1954)Google Scholar

Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsETH ZürichZürichSwitzerland

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