Lecture 50

History of Complex Numbers

The problem of complex numbers dates back to the 1st century, when
Heron of Alexandria (c. 1070 AD) attempted to find the volume of a frus-
tum of a pyramid which required of computing the square root of 81−144,
though negative numbers were not conceived in the Hellenistic world. Later,
around 850 AD, the Indian mathematician Mahavira Acharya wrote “as
in the nature of things, a negative (quantity) is not a square (quantity),
it has therefore no square root”. We also have the following quotation
from Bhaskara Acharya (1114-85), a Hindu mathematician: “The square
of a positive number, also that of a negative number, is positive: and the
square root of a positive number is two–fold, positive and negative; there is
no square root of a negative number, for a negative number is not square”.
In 1545 the Italian mathematician, physician, gambler, and philosopher
Girolamo Cardano (1501–76) published his Ars Magna (The Great Art), in
which he described algebraic methods for solving cubic and quartic equa-
tions. This book was a great event in mathematics. In fact, after 3000 years
it was the first major achievement in algebra, when the Babylonians showed
how to solve quadratic equations. Cardano, also dealt with quadratics in
his book. One of the problems which he called “manifestly impossible” is
the following: Divide 10 into two parts whose product is 40, i.e., find
the solution of x + y = 10, xy = 40, or equivalently, the solution of the
2
quadratic √ equation 40 − x(10
√ − x) = x − 10x + 40 = 0, which has √ the
roots 5 √+ −15 and 5 − −15. Cardano formally multiplied 5 + −15
by 5 − −15 and obtained 40; however, to calculations he said “putting
aside the mental tortures involved”. He did not pursue the matter but
concluded that the result was “as subtle as it is useless”. This event was
historic, since it was the first time the square root of a negative number
was explicitly written down. For the cubic x3 = ax + b so–called Cardano
formula is
v s  v s 
u u
2 2
u3 b b  a  3 u
3 b b  a 3
x = + − + − − .
t t
2 2 3 2 2 3

When applied to the historic example x3 = 15x + 4, the formula yields

√ √
q q
3 3
x = 2 + −121 + 2 − −121.

Although Cardano claimed that his general formula for the solution
√ of the
cubic was inapplicable in this case (because of the appearance of −121),
322 Lecture 50

square roots of negative numbers could no longer be so lightly dismissed.

Whereas for the quadratic (e.g., x2 + 1 = 0) one could say that no
solution exists, for the cubic x3 = 15x + 4, a real solution,√ namely
x = 4, does exist; in fact, the two other solutions, − 2 ± 3, are
also real. p
It now remainedp to reconcile the formal and “meaningless” solu-
√ √
tion x = 2 + −121 + 2 − −121 of x3 = 15x + 4, found by using
3 3

Cardano’s formula, with the solution x = 4, found by inspection. The

task was undertaken by the hydraulic engineer Rafael Bombelli (1526–73)
about thirty years after the publication of Cardano’s work.
√
√ had the “wild thought” that since the radicals 2 + −121
Bombelli
and 2 − −121 differ only in sign, the same might be true of their cube
roots. Thus, he let
q
3 √ √ q
3 √ √
2 + −121 = a + −b and 2 − −121 = a − −b

and proceeded to solve for a and b by manipulating these expressions

according to the established rules for real variables. He deduced that a = 2
and b = 1 and thereby showed that, indeed,
√ √ √ √
q q
3 3
2 + −121 + 2 − −121 = (2 + −1) + (2 − −1) = 4.

Bombelli had thus given meaning to the “meaningless”. This event signaled
the birth of complex numbers. Breakthrough was achieved by thinking the
unthinkable and daring to present it in public. Thus, the complex numbers
forced themselves in connection with the solutions of cubic equations rather
than the quadratic equations.

To formalize his discovery, Bombelli developed a calculus of operations

with complex numbers. His rules, in our symbolism, are (−i)(−i) = − 1
and
(±1)i = ± i, (+i)(+i) = − 1, (−i)(+i) = + 1,
(±1)(−i) = ∓ i, (+i)(−i) = + 1.
He also considered examples involving addition and multiplication of com-
plex numbers, such as 8i + (−5i) = + 3i and
√ √ √
q  q  q
3 3 3
4 + 2i 3 + 8i = 8 + 11 2i.

Bombelli thus laid the foundation stone of the theory of complex num-
bers. However, his work was only the beginning of the saga of complex
numbers. Although his book l’Algebra was widely read, complex numbers
were shrouded in mystery, little understood, and often entirely ignored. In
fact, for complex numbers Simon Stevin (1548-1620) in 1585 remarked that
“there is enough legitimate matter, even infinitely much, to exercise oneself
History of Complex Numbers 323

without occupying oneself and wasting time on uncertainties”. John Wal-

lis (1616-1703) had pondered and puzzled over the meaning of imaginary
numbers in geometry. He wrote, “These Imaginary Quantities (as they are
commonly called) arising from the Supposed Root of a Negative Square
(when they happen) are reputed to imply that the Case proposed is Im-
possible”. Gottfried Wilhelm von Leibniz (1646–1716) made the following
statement in the year 1702: “The imaginary numbers are a fine and won-
derful refuge of the Divine Sprit, almost an amphibian between being and
non–being.” Christiaan Huygens (1629-95) a prominent Dutch mathemati-
cian, astronomer, physicist, horologist, and writer of early science fiction,
was just as puzzled as was Leibniz, in reply
p to a query
p he wrote to√Leibniz
√ √
“One would never have believed that 1 + −3 + 1 − −3 = 6 and
there is something hidden in this which is incomprehensible to us”. Leon-
hard Euler (1707–83) √was candidly
√ astonished by the remarkable fact that
expressions such as −1, −2, etc., are neither nothing, nor greater
than nothing, nor less than nothing, which necessarily constitutes them
imaginary or impossible.
p √ √In √fact, he was confused with the absurdity
(−4)(−9) = 36 = 6 6= −4 −9 = (2i)(3i) = 6i2 = −6.

Similar doubts concerning the meaning and legitimacy of complex num-

bers persisted for two and a half centuries. Nevertheless, during the same
period complex numbers were extensively used and a considerable amount
of theoretical work was done by such distinguished mathematicians as René
Descartes (1596–1650) (who coined the term imaginary number, before him
the these numbers √ were called sophisticated or subtle); Euler (who was the
first to designate −1 by i); Abraham de Moivre (1667–1754) in 1730
noted that the complicated identities relating trigonometric functions of an
integer multiple of an angle to powers of trigonometric functions of that
angle could be simply reexpressed by the following well–known formula
(cos θ + i sin θ)n = cos nθ + i sin nθ, and many others. Complex numbers
also found applications in map projection by Johann Heinrich Lambert
(1728–77), and by Jean le Rond d’Alembert (1717–83) in hydrodynamics.

The desire for a logically satisfactory explanation of complex numbers

became manifest in the later part of the 18th century, on philosophical, if
not on utilitarian grounds. With the advent of the Age of Reason, when
mathematics was held up as a model to be followed not only in the natural
sciences but also in philosophy as well as political and social thought, the
inadequacy of a rational explanation of complex numbers was disturbing.
By 1831, the great German mathematician Karl Friedrich Gauss (1777–
1855) had overcome his scruples concerning complex numbers (the phrase
complex numbers is due to him) and, in connection with a work on number
theory, published his results on the geometric representation of complex
numbers as points in the plane. However, from Gauss’ diary, which was left
among his papers, it is clear that he was already in possession of this inter-
pretation by 1797. Through this representation Gauss clarified the “true
324 Lecture 50

metaphysics of imaginary numbers” and bestow on them complete fran-

chise in mathematics. Similar representations by the Norwegian surveyor
Casper Wessel (1745–1818) in 1797 and by the Swiss clerk Jean–Robert Ar-
gand (1768-1822) in 1806 went largely unnoticed. The concept modulus of
complex numbers is due to Argand, and absolute value, for modulus, is due
to Karl Theodor Wilhelm Weierstrass (1815-97). The Cartesian coordinate
system called the complex plane or Argand diagram is also named after the
same Argand. Mention should also be made of an excellent little treatise by
C.V. Mourey (1828), in which the foundations for the theory of directional
numbers are scientifically laid. The general acceptance of the theory is not
a little due to the labors of Augustin Louis Cauchy (1789-1857) and Niels
Henrik Abel (1802-1829), and especially the latter, who was the first to
boldly use complex numbers with a success that is well known.

Geometric applications of complex numbers appeared in several mem-

oirs of prominent mathematicians such as August Ferdinand Möbius (1790–
1868), George Peacock (1791–1858), Giusto Bellavitis (1803–80), Augus-
tus De Morgan (1806–71), Ernst Kummer (1810–93), Leopold Kronecker
(1823–91). In the next three decades further development took place. Spe-
cially, in 1833 William Rowan Hamilton (1805–65) gave an essentially rig-
orous algebraic definition of complex numbers as pairs of real numbers.
However, a lack of confidence in them persisted, for example, the English
mathematician and astronomer George Airy (1801–92) declared “I have not
the smallest confidence in any result which is essentially obtained by the
use of imaginary√symbols”. The English logician George Boole (1815–64)
in 1854 called −1 an “uninterpretable symbol”. The German mathe-
matician Leopold Kronecker believed that mathematics should deal only
with whole numbers and with a finite number of operations, and is credited
with saying, God made the natural numbers; all else is the work of man.
He felt that irrational, imaginary, and all other numbers excluding the pos-
itive integers were man’s work and therefore unreliable. However, French
mathematician Jacques Salomon Hadamard (1865–1963) said the shortest
path between two truths in the real domain passes through the complex
domain. By the latter part of the 19th century all vestiges of mystery and
destruct of complex numbers could be said to have disappeared, although
some resistance continued among few textbook writers well into the 20th
century. Nowadays, complex numbers are viewed in the following different
ways:

1. Points or vectors in the plane;

2. Ordered pairs of real numbers;
3. Operators (i.e., rotations of vectors in the plane);
4. Numbers of the form a + bi, with a and b real numbers;
5. Polynomials with real coefficients modulo x2 + 1;
History of Complex Numbers 325
 
a b
6. Matrices of the form , with a and b real numbers;
−b a
7. An algebraically closed, complete field (a field is an algebraic structure
which has the four operations of arithmetic).
The foregoing descriptions of complex numbers are not the end of the
story. Various developments in the 19th and 20th centuries enabled us to
gain a deeper insight into the role of complex numbers in mathematics (al-
gebra, analysis, geometry, and the most fundamental work of Peter Gustav
Lejeune Dirichlet (1805–59) in number theory); engineering (stresses and
strains on beams, resonance phenomena in structures as different as tall
buildings and suspension bridges, control theory, signal analysis, quantum
mechanics, fluid dynamics, electric circuits, air–craft wings, and electro-
magnetic waves); physics (relativity, fractals, and Schrödinger equation).
Although scholars who employ complex numbers in their work today do
not think of them as mysterious, these quantities still have an aura for the
mathematically naive. For example, the famous 20th–century French intel-
lectual
√ and psychoanalyst Jacques Lacan (1901-81) saw a sexual meaning
in −1.