Imagining Imaginary Numbers

  • John H. Conway
  • Richard K. Guy


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.


Unique Factorization Gaussian Integer Imaginary Quadratic Field Algebraic Rule Infinite Precision 
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  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

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