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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Mitternachtsformel or Mondscheinformel.
in German “entlernen”. This anecdote was reported to the author by Ernst Specker.
Alternatively, one may use \(x=iy\sqrt{\frac{c}{a}}\) to arrive again at the form \(y^{2}+dy-1=0\) like in the first case.
If x is the true value and \(\overset{~}{x}\) its numerical approximation, the relative error is defined as \(\frac{\overset{~}{x}-x}{x}=\frac{\overset{~}{x}}{x}-1\).
References
Ben-Menahem, A.: Historical encyclopedia of natural and mathematical sciences, vol. 1. Springer, Berlin, Heidelberg (2009)
Casselman, B.: x, y and z. Notices Amer. Math. Soc. 62(2), 147 (2015)
Jørgen, C., Gallardo, L., Vaserstein, L., Wheland, E.: Solving quadratic equations over polynomial rings of characteristic two. Publ. Mat. 42(1), 131–142 (1998)
Descartes, R.: Discours de la Méthode. Imprimerie de Ian Maire (1637)
Eells, W.C.: Greek methods of solving quadratic equations. Amer. Math. Monthly 18(1), 3–14 (1911)
Forsythe, G.E.: Pitfalls in computation, or why a math book isn’t enough. Amer. Math. Monthly 77(9), 931–956 (1970)
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
Brioschi, F.: Sulla risoluzione delle equazioni del quinto grado. Ann. Mat. Pura Appl. 1, 256 (1858). https://doi.org/10.1007/BF03197334
Herz-Fischler, R.: What are propositions 84 and 85 of Euclid’s data all about? Hist. Math. 11, 86–91 (1984)
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)
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.
Matthiessen, L.: Grundzüge der antiken und modernen Algebra der literalen Gleichungen. B.G. Teubner (1878)
Muller, D.E.: A method for solving algebraic equations using an automatic computer. Math. Tables Aids Comput. 10, 208–215 (1956)
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)
Rāshid, R., Morelon, R.: Encyclopedia of the history of Arabic science, vol. 2. Routledge, London, New York (1996)
Serway, R.A., Vuille, C., Faughn, J.S.: College physics. Brooks/Cole Cengage Learning, Belmont (2009)
van der Waerden, B.L.: Science awakening I. Noordhoff (1954)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hungerbühler, N. An alternative quadratic formula. Math Semesterber 67, 85–95 (2020). https://doi.org/10.1007/s00591-019-00262-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00591-019-00262-3