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History of Imaginary Numbers

Date: 03/09/2001 at 13:57:56
From: Laurie Raley
Subject: Who Invented Imaginary Numbers?


I am a high school student Chicago, IL, and my class just started 
studying imaginary numbers. Someone asked my teacher who invented 
imaginary numbers, and my teacher had no idea, so she suggested we 
find out who did invent the theory. If you could please tell me or 
give me another place to look, I would be very thankful.

Thank you, 
Laurie A. Raley

Date: 03/09/2001 at 14:18:14
From: Doctor Rob
Subject: Re: Who Invented Imaginary Numbers?

Thanks for writing to Ask Dr. Math, Laurie.

This was an interesting question. After some research, I have found 
the following pertinent information.

At the MacTutor Math History archive in St. Andrews, I found

The fundamental theorem of algebra   

It says,

   "Cardan was the first to realise that one could work with 
quantities more general than the real numbers. This discovery was made 
in the course of studying a formula which gave the roots of a cubic 
equation.  The formula when applied to the equation x^3 = 15*x + 4 
gave an answer involving sqrt(-121) yet Cardan knew that the equation 
had x = 4 as a solution. He was able to manipulate with his 'complex 
numbers' to obtain the right answer yet he in no way understood his 
own mathematics."

Jeff Miller's page, 

Earliest Known Uses of Some of the Words of Mathematics (I)   


  "The terms IMAGINARY and REAL were introduced in French by Rene 
Descartes (1596-1650) in "La Geometrie" (1637): 

   '...neither the true roots nor the false are always real; sometimes
   they are, however, imaginary; namely, whereas we can always imagine 
   as many roots for each equation as I have predicted, there is still 
   not always a quantity which corresponds to each root so imagined.  
   Thus, while we may think of the equation x^3 - 6xx + 13x - 10 = 0 
   as having three roots, yet there is just one real root, which is 2, 
   and the other two, however, increased, diminished, or multiplied 
   them as we just laid down, remain always imaginary.' (page 380)

[The other two roots are 2 + i and 2 - i.]

  "An early appearance of the word imaginary in English is in "A 
treatise of algebra, both historical and practical" (1685) by John 
Wallis (1616-1703): 

   'We have before had occasion (in the Solution of some Quadratick 
   and Cubick Equations) to make mention of Negative Squares, and 
   Imaginary Roots, (as contradistinguished to what they call Real 
   Roots, whether affirmative or Negative)...  These *Imaginary* 
   Quantities (as they are commonly called) arising from *Supposed* 
   Root of a Negative Square, (when they happen) are reputed to imply 
   that the Case proposed is Impossible. 

  "The quotation above is from Chapter LXVI (p. 264), Of NEGATIVE 
SQUARES, and their IMAGINARY ROOTS in Algebra. This work is a 
translation of "De Algebra Tractatus; Historicus & Practicus" written 
in Latin in 1673. For the Latin edition of the latter consult "Opera 
mathematica", vol. II, Oxoniae, 1693. [Julio Gonzalez Cabillon] 

  "Some other terms that have been used to refer to imaginary numbers 
include "sophistic" (Cardan), "nonsense" (Napier), "inexplicable"
(Girard), "incomprehensible" (Huygens), and "impossible" (many 

The conclusion is that probably Girolamo Cardano (or Cardan) (1501-
1576) can be credited with the "discovery" of imaginary and complex 
numbers in the 16th century, but that the concept was not put on a 
firm footing until much later, especially in the work of Leonhard 
Euler (1707-1783) and Carl Friedrich Gauss (1777-1855).

Gauss once wrote,

  "That this subject [imaginary numbers] has hitherto been surrounded 
by mysterious obscurity, is to be attributed largely to an ill adapted
notation. If, for example, +1, -1, and the square root of -1 had been
called direct, inverse and lateral units, instead of positive, 
negative and imaginary (or even impossible), such an obscurity would 
have been out of the question."

It was Gauss who made the distinction between imaginary numbers a*i 
and complex numbers a + b*i (a and b real). Up until his work, both 
complex and imaginary numbers had been termed imaginary.

- Doctor Rob, The Math Forum   
Associated Topics:
High School History/Biography
High School Imaginary/Complex Numbers

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