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Imagining Imaginary Numbers

  • John H. Conway
  • Richard K. Guy
Chapter

Abstract

Historically, complex numbers first arose from the solution of quadratic equations. You can solve the equation
$${x^2} - x - 6 = 0$$
either by factoring it as (x-3)(x+2) = 0, or by using the well-known formula, or by writing it as
$$ {\left( {x - \frac{1}{2}} \right)^2} = 6\frac{1}{4} = {\left( {2\frac{1}{2}} \right)^2}, $$
and you get the answers \( \frac{1}{2}\pm \;2\frac{1}{2} \), namely, 3 or -2; sensible answers and there’s no problem.

Keywords

Unique Factorization Gaussian Integer Imaginary Quadratic Field Algebraic Rule Infinite Precision 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. H.A. Heilbronn and E.H. Linfoot. On the imaginary quadratic corpora of class-number one. Quart. J. Math. (Oxford) 5 (1934): 293–301CrossRefzbMATHGoogle Scholar
  2. A. Hurwitz. Über die Komposition quadratischer Formen von beliebig vielen Variabein, Nachr. Gesell. Wiss. Göttingen 1898 309–316.Google Scholar
  3. D.H. Lehmer. On imaginary quadratic fields whose class number is unity abst. #188 Bull Amer. Math. Soc. 39 (1933), 360.zbMATHGoogle Scholar
  4. Harold M. Stark. On complex quadratic fields with class number equal to one. Trans. Amer. Math. Soc. 122 (1966): 112–119MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • John H. Conway
  • Richard K. Guy

There are no affiliations available

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