Chapter 1

COMPLEX NUMBERS

1.1 Cubic equations and imaginary numbers.

Imaginary numbers appeared in algebra during the Reinassance, as a tool forsolving the cubic equation1. At the time the problem of searching the solutionsof quadratic equations like x2 + 1 = 0 was considered meaningless. However, acubic equation always has a real solution, but the available method of solutioneventually provided it as a sum of terms with imaginary numbers; the puzzlewas solved by Raffaele Bombelli.

Before the modern era, the solution of the cubic equation was obtained bygeometric means. The Persian poet and scientist Omar Khayyam (IX cent.)discussed the solution of the cubic equation as the intersection of a parabolaand a hyperbola, by methods that fore ran Cartesian geometry. The priority forthe algebraic solution of the cubic equation is uncertain: it was probably knownto Scipione del Ferro, a professor in Bologna, and Nicolo Fontana (Tartaglia).The general solution2 was published in the book Ars Magna (1545) by GerolamoCardano (Pavia 1501, Rome 1576). It is based on the algebraic identity

(t − u)3 + 3tu(t− u) = t3 − u3 (1.1)

which he obtained by geometrical construction3. After setting x = t − u, the

1Refs: Jacques Sesiano, An Introduction to the Hystory of Algebra, Mathematical World27, AMS; Morris Kline, Mathematical Thought from Ancient to Modern Times, 3 vol, OxfordUniversity Press 1972; Carl B. Boyer, A History of Mathematics, Princeton University Press1985. A precious source of hystorical news and pictures is the Mathematics Genealogy Project

[www.genealogy.ams.org].2The use of letters to denote parameters of equations was introduced by Francois Viete

few years later; Cardano solved examples of cubic equations, with all possible signs.3Consider a cube with edge length t. If three concurring edges are partitioned in segments

of lengths u and t−u, the cube is cut into two cubes and four parallepipeds. The total volumeis t3 = u3 + (t − u)3 + 2tu(t − u) + u2(t − u) + u(t − u)2; simple algebra gives the identity(W. Dunham, Journey through Genius, the Great Theorems of Mathematics, Wiley ScienceEd. 1990).

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identity becomes the reduced cubic equation

x3 + 3px + q = 0 (1.2)

with tu = p and t3 − u3 = −q. Therefore, the solution x = t − u of (1.2) isobtained by solving the quadratic equations for t3 and u3, in terms of p and q.Raffaello Bombelli (Bologna 1526, Rome? 1573) in his treatise Algebra was thefirst to regard imaginary numbers as a necessary detour to produce real solutionsfrom real cubic equations. He studied the equation x3 − 15x − 4 = 0, withreal solution x = 4. Cardano’s method works as follows: from tu = −5 andt3 − u3 = 4 obtain t3 + 125/t3 − 4 = 0 i.e. t6 − 4t3 + 125 = 0 with solutionst3 = 2 ±

√−121. Imaginary terms appear:

x = (2 +√−121)1/3 + (2 −

√−121)1/3.

Bombelli showed that 2±√−121 = (2±

√−1)3, so that imaginary terms cancel,

and the simple real result x = 4 is recovered.

Exercise 1.1.1. Show that a cubic equation z3 + a1z2 + a2z + a3 = 0 can

be brought to the standard form w3 ± 3w + q = 0 by a linear transformation

z = aw + b. For q real, obtain the condition for having three real roots. Find

the three roots through the substitution w = s ∓ 1/s.

1.2 The quartic equation.

The Ars Magna also contains the solution of the quartic equation, due to Car-dano’s disciple Ludovico Ferrari (1522, 1565). In modern language, a linearvariable change puts it in the form x4 = ax2 + bx + c. The great idea is theintroduction of an auxiliary parameter y in the equation:

(x2 + y)2 = (a + 2y)x2 + bx + y2 + c;

The parameter is chosen in order to make also the right hand side (r.h.s.) of theequation a perfect square of a binomial in x, so that square roots of both sidescan then be taken. The condition for (a + 2y)x2 + bx + y2 + c to be a perfectsquare is the vanishing of the discriminant: 0 = b2 − 4(a + 2y)(y2 + c). Thecubic equation can be solved by Cardano’s formula. The value of y is enteredin the quartic equation, (x2 + y)2 = (a + 2y)[x + b/2(a + 2y)]2, and a squareroot brings it to a couple of quadratic equations in x [Boyer].

Example 1.2.1. To solve the quartic equation x4 − 3x2 − 2x + 5 = 0 introduce

the parameter y and rewrite the equation as (x2 +y)2 = (3+2y)x2 +2x−5+y2.

Choose y such that r.h.s. is a perfect square in x, i.e. 2y3 + 3y2 − 10y − 16 = 0with a solution y = −2. Then the equation is (x2 − 2)2 = −x2 + 2x − 1, i.e.

x2 − 2 = ±i(x − 1). The two quadratic equations give the four solutions of the

quartic.

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The achievement started a great effort to solve higher order equations. Van-dermonde and especially Giuseppe Lagrange (Torino 1736, 1813) emphasizedthe role of the permutation group and symmetric functions. In 1770 Lagrangeobtained a new method of solution of the quartic equation

z4 + a1z3 + a2z

2 + a3z + a4 = 0 (1.3)

Since it is instructive, we give a brief description of it. The coefficients of theequation are symmetric functions of the roots,

a1 = −∑

i

zi, a2 =∑

i<j

zizj , a3 = −∑

i<j<k

zizjzk, a4 = z1z2z3z4.

Any other symmetric function of the roots is expressible in terms of them. Hethen considered the combination of the roots s1 = z1z2 + z3z4, which is notsymmetric under all 4! permutations. By doing all permutations he obtainedonly two new combinations: s2 = z1z3 + z2z4 and s3 = z1z4 + z2z3. The p = 3quantities A1 = s1 + s2 + s3, A2 = s1s2 + s1s3 + s2s3 and A3 = s1s2s3 areinvariant under permutations of the roots zi and are expressible in terms ofthe coefficients ai: A1 = a2, A2 = a1a3 − 4a4, and A3 = a2

3+ a2

1a4 − 4a2a4.

Therefore, si are the solutions of the cubic equation

s3 − a2s2 + (a1a3 − 4a4)s − a2

3+ a2

1a4 − 4a2a4 = 0,

and can be evaluated by Cardano’s method. Once the roots s1, s2 and s3 areobtained, one solves the quadratic equation s1(z1z2) = (z1z2)

2 + a4 to get z1z2

and z3z4. In the similar way one gets z1z3, z2z4 and the roots are easily found.

1.3 Beyond the quartic.

For the fifth-order equation Lagrange hardly tried to guess a polynomial com-bination of the roots that, under the 5! permutations, could produce at mostp = 4 different combinations s1 ... s4. They would solve a quartic equation. Hisdisciple Ruffini (Modena 1765, 1822) showed that such a polynomial should beinvariant under 5!/p permutations of the roots zi, and does not exist.The norwegian mathematician Niels Henrik Abel (1802, 1829) put the last wordin the memoir On the algebraic resolution of equations published in 1824. Heproved that no rational solutions involving radicals and algebraic expressions ofthe coefficients exist for general equations of order higher than four. The iden-tification of the equations that can be solved by radicals was done by EvaristeGalois, by methods of group theory to which he much contributed. An equationis solvable by radicals if and only if, given two roots, the others depend ratio-nally on them (1830)4.In 1789 E. S. Bring (1786), by exploiting an earlier method by Tschirnhausen(1683), showed that any equation of fifth degree can be brought to the amazingly

4for a presentation of Galois theory, see V. V. Prasolov, Polynomials, Springer (2004)

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simple form z5+q4z+q5 = 0, and then to z5+5z+a = 0 if q4 6= 0 (more generally,an equation xn+a1x

n−1+. . .+an = 0 can be reduced to yn+q4yn−4+. . .+qn = 0

by means of the variable change y = p0 +p1x+ . . .+p4x4 and solving equations

of degree 2 and 3).Charles Hermite succeeded in obtaining a solution to the quintic equation, interms of elliptic functions (1858). Such functions generalize trigonometric ones,which are useful to solve the cubic equation5. Soon after Leopold Kroneckerand Francesco Brioschi6 gave alternative derivations7.In 1888 the solution of the general sixth order equation was obtained by Brioschiand Maschke, in terms of hyperelliptic functions.Of course no one would solve even a quartic by the methods described, as effi-cient numerical methods yield the roots with the desired accuracy.

1.4 The Complex Field

Complex numbers were used in the early XVIII century by Leibnitz, who de-scribed them as halfway between existence and nonexistence. Jean Bernoulli,Abraham De Moivre and, above all, the genius Leonhard Euler (1707, 1783)discovered several relations involving trigonometric, exponential and logarith-mic functions with imaginary arguments.The great improvement in the perception of complex numbers as well definedentities was their visualization as vectors, by the norwegian cartographer CasparWessel (1797) and the mathematician Jean Robert Argand (1806), or as pointsin the plane, by Carl Frederich Gauss. It was Gauss’ authority and investi-gations since 1799, that gave complex numbers a status in analysis. Wessel’swork was recognized only a century later, after its translation in French, andArgand’s paper was much criticized.In 1833 the Irish mathematician Willian R. Hamilton (1805, 1865) presentedbefore the Irish Academy an axiomatic setting of the complex field C as a for-mal algebra on pairs of real numbers (Chapter 2).In 1867 Hankel proved that the algebra of complex numbers is the most generalone that fulfills all fundamental laws of arithmetic [Boyer].

5for example, the equation y3− 3y + 1 = 0 can be solved with the aid of trigonometric

tables: put y = 2 cos x and solve 0 = 2 cos(3x) + 1. Then 3x = 2π/3 and 3x = 4π/3 i.e.y1 = 2 cos(2π/9) and y2 = 2 cos(4π/9); the other solution is y3 = −y1 − y2.

6F. Brioschi (1824, 1897) taught mechanics in Pavia. He then founded Milan’s Politecnico(1863), where he taught hydraulics. He participated in Milan’s insurrection, and becamemember of the Parliament. Among his students (in Pavia): Giuseppe Colombo (inauguratedthe first electric power generator in Milan and Europe, succeeded to Brioschi in the directionof the Politecnico), Eugenio Beltrami (non Euclidean geometry, singular values of a matrix,Laplace Beltrami operator in curved space) and Luigi Cremona (painter).

7See website http://wapedia.mobi/en/Bring radical for a discussion of quintic equationand its reduction to the standard form. V. Barsan, Physical applications of a new method

of solving the quintic equation, arXiv:0910.2957v2; G. Zappa, Storia della risoluzione delle

equazioni di V e VI grado .., Rend. Sem. Mat. Fis. Milano (1995).

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Figure 1.1: Leonhard Euler (1707, 1783) belongs to an impressive genealogyof mathematicians, rooted in Leibnitz and the Bernoullis. Euler spent manyyears in St. Petersburgh, at the birth of the russian mathematical school. Hediscovered several important formulae for complex functions, and establishedmuch of the modern notation. His student Joseph Lagrange was the advisor ofFourier and Poisson. Poisson’s students Dirichlet and Liouville were mentorsof illustrious mathematicians that contributed to the advancements of complexanalysis in Paris (on the side of Dirichlet: Darboux, and then Borel, Cartan,Goursat, Picard, and then Hadamard, Julia, Painleve ...; on the side of Liouville:Catalan and then Hermite, Poincare, Pade, Stieltjes ...).

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Figure 1.2: Carl Friederich Gauss (1777, 1855) became a celebrity aftercomputing the orbit of the first asteroid Ceres, discovered and lost of sight bypadre Piazzi in Palermo (1801). To interpolate the best orbit from observedpoints, Gauss devised the Least Squares method. The orbital elements placedCeres in the region were astronomers were searching for the fifth planet, thatfitted in Titius and Bode’s law. Gauss proved the “fundamental theorem ofalgebra”: a polynomial of degree n has n zeros in the complex plane. Heanticipated several results of complex analysis, which he did not publish. Hisgenealogy contains venerable scientists as the astronomers Bessel and Encke, andmathematicians: Dedekind, Sophie Germain, Gudermann, and Georg Riemann.Among Gauss’ “nephews” are Ernst Kummer and Karl Weierstrass.

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