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### History of Imaginary Numbers

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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
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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):

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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Associated Topics:
High School History/Biography
High School Imaginary/Complex Numbers

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