section_path: History
section_name: History
paragraph_text: History
  
  The solution in radicals (without trigonometric functions) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called casus irreducibilis ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna, though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless". Cardano did use imaginary numbers, but described using them as "mental torture." This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.
  
  Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.
  
  Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.
  
  The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term 81 − 144 {\displaystyle {\sqrt {81-144}}} in his calculations, which today would simplify to − 63 = 3 i 7 {\displaystyle {\sqrt {-63}}=3i{\sqrt {7}}}. Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive 144 − 81 = 3 7. {\displaystyle {\sqrt {144-81}}=3{\sqrt {7}}.}
  
  The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (Niccolò Fontana Tartaglia and Gerolamo Cardano). It was soon realized (but proved much later) that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers is unavoidable when all three roots are real and distinct. However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic, trying to resolve these issues.
  
  The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature:
  
  ... sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.
  
  A further source of confusion was that the equation − 1 2 = − 1 − 1 = − 1 {\displaystyle {\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1} seemed to be capriciously inconsistent with the algebraic identity a b = a b {\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}}, which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity in the case when both a and b are negative, and the related identity 1 a = 1 a {\textstyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}}}, even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol i in place of − 1 {\displaystyle {\sqrt {-1}}} to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
  
  In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:
  
  (cos θ + i sin θ) n = cos n θ + i sin n θ. {\displaystyle (\cos \theta +i\sin \theta)^{n}=\cos n\theta +i\sin n\theta.}
  
  In 1748, Euler went further and obtained Euler's formula of complex analysis:
  
  e i θ = cos θ + i sin θ {\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }
  
  by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
  
  The idea of a complex number as a point in the complex plane (above) was first described by Danish – Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra.
  
  Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of − 1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology:
  
  If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, − 1 {\displaystyle {\sqrt {-1}}} positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.
  
  In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis.
  
  The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.
  
  Augustin-Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.
  
  The common terms used in the theory are chiefly due to the founders. Argand called cos φ + i sin φ the direction factor, and r = a 2 + b 2 {\displaystyle r={\sqrt {a^{2}+b^{2}}}} the modulus; Cauchy (1821) called cos φ + i sin φ the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used i for − 1 {\displaystyle {\sqrt {-1}}}, introduced the term complex number for a + bi, and called a + b the norm. The expression direction coefficient, often used for cos φ + i sin φ, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.
  
  Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.