# 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.

## References

1. H.A. Heilbronn and E.H. Linfoot. On the imaginary quadratic corpora of class-number one. Quart. J. Math. (Oxford) 5 (1934): 293–301
2. Zbl. 10.337.
3. A. Hurwitz. Über die Komposition quadratischer Formen von beliebig vielen Variabein, Nachr. Gesell. Wiss. Göttingen 1898 309–316.Google Scholar
4. D.H. Lehmer. On imaginary quadratic fields whose class number is unity abst. #188 Bull Amer. Math. Soc. 39 (1933), 360.
5. Harold M. Stark. On complex quadratic fields with class number equal to one. Trans. Amer. Math. Soc. 122 (1966): 112–119
6. MR 33 #4043.