History of Imaginary Numbers
Date: 03/09/2001 at 13:57:56 From: Laurie Raley Subject: Who Invented Imaginary Numbers? Hello, 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 http://www-history.mcs.st-and.ac.uk/history/HistTopics/Fund_theorem_of_algebra.html 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) http://jeff560.tripod.com/i.html says, "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 authors). 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 http://mathforum.org/dr.math/
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