182 Algebra of Complex Numbersthe final result was a real number; for one example of this kind, see Exer-
cise 4.1.5 on page 185 below. It still took some time to get used to the new
concept, and to develop the corresponding theory. In fact, the term "imag-
inary" in connection with the complex numbers was introduced around
1630 by Descartes, who intended the term to be derogatory. In 1777, Euler
suggested the symbol i for y/— 1, and the complex numbers started to get
the respect they deserve. The foundations of modern complex analysis were
laid during the first half of the 19th century. By the end of the 19th cen-
tury, it was already impossible to imagine mathematics without complex
numbers. Part of the reason could be that, as the French mathematician
JACQUES SALOMON HADAMARD (1865-1963) put it, the shortest path be-
tween two truths in the real domain passes through the complex domain.
In the following sections, we will see plenty of examples illustrating this
statement.
Since solving polynomial equations was the main motivation for the
introduction of complex numbers, let us say a bit more about these equa-
tions. Given a polynomial, the objective is to find an algebraic formula
for the roots, that is, an expression involving a finite number of additions,
subtractions, multiplications, divisions, and root extractions, performed on
the coefficients of the polynomial. The formula x = (—b± \/b^2 — 4ac)/(2a)
for the roots of the quadratic equation ax^2 + bx + c = 0 was apparently
known to ancient Babylonians some 4000 years ago. The formulas of Car-
dano for equations of degree three and four are much more complicated but
still algebraic. Ever since the discovery of those formulas, various mathe-
maticians tried to extend the results to equations of degree five or higher,
until, around 1820, the Norwegian mathematician NIELS HENRIK ABEL
(1802-1829) proved the non-existence of such algebraic representations for
the solutions of a general fifth-degree equation. In 1829, the French math-
ematician EVARISTE GALOIS (1811-1832) resolved the issue completely by
proving the non-existence of an algebraic formula for the solution of a gen-
eral polynomial equation of degree five or higher, and also describing all
the equations for which such a formula does exist. Note that both Abel
and Galois were under 20 years of age when they made their discoveries.
The solutions of a polynomial equation can exist even without an alge-
braic formula to compute them; the fundamental theorem of algebra ensures
that a polynomial of degree n and with coefficients in C has exactly n roots
in C, and there are many ways to represent the roots using infinitely many
operations of addition, subtraction, multiplication, and division, performed
on the coefficients. Such representations lead to various numerical methods