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Numbers, Complex
Complex numbers are ubiquitous in modern
science, yet it took mathematicians a long
time to accept their existence. They are
numbers of the form
biaz
+
=
where a and b
are real numbers, and
i
is a symbol called the
imaginary unit which satisfies the seemingly
impossible equation .1
2
−=i The numbers a
and b are called the real and imaginary parts
of z, respectively. The imaginary unit can be
thought of as the square root of –1 and is also
written
.1−=i
In fact, any negative number
has a complex square root; for example, the
square root of –15 is the complex number
.1515 i⋅=− Today, science students
routinely encounter complex numbers, for
instance as solutions to quadratic equations.
In mathematics, complex numbers form
an independent area of research but are also
used to prove theorems in other areas of
mathematics; examples are Machin's formula
and the Prime number theorem. In the natural
sciences, complex numbers often simplify
calculations, for example in the theory of
relativity where the distance between points
in space-time can be computed using
imaginary time coordinates. The complex
exponential function is used in electrical
engineering as a convenient way of
simultaneously describing the amplitude and
phase of an alternating current, and in chaos
theory, the complex plane is the scene of
computer-generated fractals such as the
Mandelbrot set, named after Benoît
Mandelbrot.
Unlike natural numbers which are used
for counting, and real numbers which are used
for measuring distances, complex numbers
have no obvious real-life interpretation. For
this reason, it remained a controversial topic
for three centuries after their discovery in the
16th century what complex numbers really
are. The term “imaginary numbers” for non-
real complex numbers was coined by René
Descartes in 1637 to indicate that they do not
really exist, a view later shared by Isaac
Newton. About 1765, Leonhard Euler
characterized square roots of negative
numbers as impossible quantities, and as late
as 1831, Augustus De Morgan objected to the
absurd nature of complex as well as negative
numbers. Only in 1837 did William Rowan
Hamilton give a proper construction of the
complex numbers, thereby indisputably
proving their inner consistency. Nevertheless,
it was their usefulness, the beauty of their
simplicity, and the ability to visualize them
rather than Hamilton’s proof that eventually
outweighed the objections against complex
numbers and led to their universal acceptance
by the end of the 19th century.
Algebra and Geometry of Complex
Numbers
Complex numbers appeared for the first time
in Gerolamo Cardano’s Ars Magna from
1545. In this famous book containing the
formulas for solving cubic and quartic
equations, Cardano also showed that the
equations 10
=
+
yx and
40
=
⋅
yx
have the
common solution
155 −+=x and
.155 −−=y
Cardano, however, dismissed these complex
numbers as useless and did not pursue the
matter further. Rafael Bombelli undertook a
more systematic investigation in L’Algebra
from 1572 where he demonstrated how
complex numbers can be added, subtracted,
multiplied, and divided using the usual rules
of algebra and the equation .1
2
−=i For
example,
iii 43)2()31(
+
=
+
+
+
,
iii 21)2()31(
+
−
=
+
−
+
,
iiiiii 71362)2()31(
2
+−=+++=+⋅+ .
Division is slightly more complicated; it is
most easily performed by multiplying both
numerator and denominator by the conjugate
of the denominator:
i
i
ii
ii
i
i
+=
+
=
−⋅+
−
⋅
+
=
+
+
1
5
55
)2()2(
)2()31(
2
31 .
Using these operations, Bombelli showed
how real solutions to cubic equations can be
found even when square roots of negative
numbers appear in Cardano’s formula for
cubic equations. Bombelli’s brilliant use of
complex numbers for solving polynomial
equations eventually led to the Fundamental
theorem of algebra according to which every
polynomial equation of positive degree has a
complex solution. The first essentially correct
proof of this result, which had been
anticipated already in the 17th century, was
given by Carl Friedrich Gauss in 1799 in his
doctoral dissertation.
Complex numbers can be represented
geometrically as points in the complex plane,
invented in 1797 by Caspar Wessel. Shortly
afterwards, it was independently conceived
and popularized by Gauss who used it
implicitly in his proof of the Fundamental
theorem of algebra. This concrete geometric
interpretation of complex numbers was
instrumental in the struggle to come to terms
with their nature. In the complex plane, points
on the x-axis correspond to real numbers,
points on the y-axis to so-called purely
imaginary numbers, and in general the point
with coordinates (a, b) corresponds to the
complex number
.bia
+
Viewing complex
numbers as points in the complex plane gives
a new geometric understanding of Bombelli’s
rules of addition and multiplication. Also, the
numerical value |z| of a complex number
biaz
+
=
is defined geometrically as the
distance between the points (0, 0) and (a, b),
or
.
22
baz +=
Machin's Formula and the Computation
of π
John Machin in 1706 discovered the formula
−
⋅=
239
1
arctan
5
1
arctan4
4
π
and used it together with the Taylor series
L
−+−+−=
9
7
5
3
)arctan(
9753
xxxx
xx
to compute π to 100 decimal places, a world
record at the time. Although Machin's formula
involves only real numbers, it has a
surprisingly simple and elegant proof using
the following identity of complex numbers,
thus illustrating their utility in other areas of
mathematics:
)1(2
239
)5(
4
i
i
i
+⋅=
+
+.
Exponential and Trigonometric Functions
The exponential function e
x
and the
trigonometric functions cos(x) and sin(x) are
well-known functions of a real variable x.
They can be expressed as Taylor series as
follows:
L
+++++=
!
4
!
3
!
2
1
432
xxx
xe
x
L
−+−+−=
!
8
!
6
!
4
!
2
1)cos(
8642
xxxx
x
L
−+−+−=
!
9
!
7
!
5
!
3
)sin(
9753
xxxx
xx
Using these expressions, the complex
functions e
z
, cos(z), and sin(z) are defined for
a complex variable z. With these definitions
and the fundamental equation ,1
2
−=i Euler
in 1748 proved a formula which reveals a
surprising kinship between these seemingly
unrelated functions:
)sin()cos( xixe
ix
⋅+= .
This result, known as Euler's formula,
generalizes a formula found by Abraham de
Moivre in 1730:
).sin()cos())sin()(cos( nxinxxix
n
⋅+=⋅+
Inserting x = π into Euler's formula gives
Euler's identity:
.1−=
πi
e
This identity combines the three most
important mathematical constants, π, e, and i,
into one single expression of striking
simplicity and beauty. A 1988 poll of readers
of Mathematical Intelligencer voted Euler's
identity “the most beautiful theorem in
mathematics,” ahead of the infinitude of
primes, the transcendence of π, and the four-
color theorem.
Complex Analysis
Complex analysis is the study of complex
functions, i.e., functions f(z) defined on some
subset U of the set of complex numbers and
with complex values. After initial
contributions by Euler and Gauss, complex
analysis was systematically investigated by
Augustin-Louis Cauchy in the 1820s. Later in
the 19th century, the theory was further
developed by Bernhard Riemann and Karl
Weierstrass. The set of definition U is called a
domain if it is open and connected, and f(z) is
called holomorphic if it satisfies the condition
of complex differentiability. Contrary to what
the name suggests, complex analysis is in
many ways simpler than real analysis since
complex differentiability is a much stricter
property than real differentiability. For
example, every holomorphic function satisfies
the so-called Cauchy-Riemann equations
which have no analogue in the realm of real
functions. Also, the Identity theorem states
that two holomorphic functions f(z) and g(z)
defined on the same domain U are identical if
only they agree on a line segment, a result
very far from being true in real analysis.
Other theorems and conjectures in complex
analysis are concerned with other types of
complex functions such as entire and
meromorphic functions.
An entire function is a holomorphic
function defined on the entire complex plane.
The complex exponential function e
z
and the
complex trigonometric functions cos(z) and
sin(z) are examples of entire functions.
Liouville's theorem, named after Joseph
Liouville, states that every non-constant entire
function is unbounded. This is considerably
strengthened by Picard's little theorem,
named after Charles Picard, which states that
every non-constant entire function takes every
complex value with at most one exception.
For example, cos(z) and sin(z) both take every
complex value, whereas e
z
takes every
complex value except 0.
A meromorphic function is a quotient of
two holomorphic functions defined on a
domain U where the denominator is not
identically zero. The zeros of the denominator
are called singularities; they can be either
removable singularities or poles. The
complex tangent function tan(z) = sin(z) /
cos(z) is an example of a meromorphic
function; it has zeros at 0, ±π, ±2π etc. and
poles at ±π/2, ±3π/2, ±5π/2 etc. The
mysterious Riemann zeta function ζ(z) is
another example; it has a single pole at z = 1.
The Riemann conjecture, arguably the most
important unsolved problem in all of
mathematics, states that all non-real zeros of
ζ(z) have real part equal to one half. The
Riemann conjecture is one of the seven
Millennium Prize Problems for whose
solution the Clay Mathematics Institute has
offered a prize of one million dollars.
Hamilton's Quaternions as Extensions of
Complex Numbers
The complex numbers form an extension of
the real numbers, just as the real numbers
form an extension of the rational, integral, and
natural numbers. It is therefore natural to ask
if there are further numbers extending the
complex numbers. This question was
answered in the affirmative by Hamilton in
1843 when he discovered the quaternions. A
quaternion is a number of the form
dkcjbiaq
+
+
+
=
where a, b, c, and d are
real numbers, and i, j, and k are symbols
satisfying
.1
222
−=⋅⋅=== kjikji
Quaternions, however, do not satisfy the
commutative law of multiplication. For
example, the product of i and j depends on the
order of the factors: i · j ≠ j · i.
The numerical value of a quaternion q is
defined as
.
2222
dcbaq +++=
A quaternion with numerical value |q| = 1 is
called a unit quaternion. Each unit quaternion
corresponds in a certain way to a rotation of
three-dimensional space. For this reason,
quaternions have important applications in
computer graphics. It happens that each unit
quaternion q corresponds to the same rotation
as its negative –q. This mathematical subtlety
explains one of the most surprising
phenomena in quantum mechanics, namely,
that the state of an electron is changed if the
electron is rotated 360 degrees; only a rotation
of 720 degrees leaves the electron unchanged.
SEE ALSO:
Mathematics: Discovery or Invention; Pi;
Renaissance; Trigonometry; Vectors.
FURTHER READINGS:
Mazur, Barry. Imagining Numbers
(Particularly the Square Root of Minus
Fifteen). New York: Farrar, Straus and
Giroux, 2003.
Nahin, Paul J. Dr. Euler's Fabulous Formula:
Cures Many Mathematical Ills. Princeton:
Princeton University Press, 2006.
Stewart, Ian and David Tall. Complex
Analysis. Cambridge: Cambridge University
Press, 1983.
David Brink, Ph.D.
University College Dublin, Ireland