to represent an (“imaginary”) solution to the equation
x2+ 1 = 0, the
COMPLEX NUMBERS
Care born. Surpris-
ingly, as shown by the
FUNDAMENTAL THEOREM OF
ALGEBRA
, the introduction of this single number is all
that is needed to solve any
POLYNOMIAL
equation anxn
+…+ a1x+ a0= 0. Thus the complex numbers represent
a system of numbers that is algebraically closed in the
sense that the construction of no new type of number is
needed to solve arithmetic equations.
On a conceptual level, the notion of “number” is
intimately connected with the act of counting. Simple
counting systems of ancient times used tally marks to
record numbers, and over the millennia this basic
numeration scheme evolved to the sophisticated
PLACE
-
VALUE SYSTEM
we use today. (The ancient Egyptians of
around 3000
B
.
C
.
E
. were perhaps the first to move
from the use of tally marks alone.) It was a great intel-
lectual achievement for mankind when the notion of
“number” was removed from the specific objects being
counted, recognizing, for instance, that two cows, two
houses, and two days all share a common property of
“two-ness.” (Even today we sometimes use different
words to count different types of “two.” For instance,
the words twins, couple, and pair cannot be used inter-
changeably to represent two people.) This simple recog-
nition of an abstract commonality between sets of
objects was exploited by German mathematician
G
EORG
C
ANTOR
(1845–1918) who, in the late 1800s,
developed a general notion of
CARDINALITY
. With it,
Cantor extended the notion of “number” to include
counts of sets of infinite size. He established, for
instance, that there are an infinite number of different
types of infinity and managed to develop a meaningful
system of arithmetic for his transfinite numbers.
Irish mathematician S
IR
W
ILLIAM
R
OWAN
H
AMIL
-
TON
(1805–65) followed a different route and worked
to extend the notion of “number” to represent opera-
tions on n-dimensional space. A
N
A
RGAND DIAGRAM
shows that the complex numbers have a natural repre-
sentation as points on a plane. Hamilton sought to give
meaning to an arithmetic for points in three- and
higher-dimensional space. Although he did not succeed
in accomplishing this goal for three-dimensional space,
his invention of the
QUATERNIONS
shows this feat can
be done in four-dimensional space. (The octonions pro-
vide an arithmetic for eight-dimensional space.)
The following diagram illustrates the relationship
between the number systems described:
N⊂Z⊂Q⊂R⊂C⊂ quaternions
See also A
RABIC MATHEMATICS
; B
ABYLONIAN
MATHEMATICS
;
BASE OF A NUMBER SYSTEM
; C
HINESE
MATHEMATICS
;
DECIMAL REPRESENTATION
; E
GYPTIAN
MATHEMATICS
; G
REEK MATHEMATICS
;H
INDU
-A
RABIC
NUMERALS
; I
NDIAN MATHEMATICS
; M
AYAN MATHEMAT
-
ICS
; R
OMAN NUMERALS
;
ZERO
.
number line (real line) A straight line, usually hori-
zontal, for which each point on the line represents a
REAL NUMBER
is called a number line. One assumes
that the line extends indefinitely both to the left and to
the right. A single point Oon the line, called the origin,
corresponds to the number
ZERO
in the real number
system, and it is conventional to assume that a point a
distance aunits to the right of Orepresents the positive
real number aand a point bunits to the left of Othe
negative real number –b. The integers are thus repre-
sented as evenly spaced points, one unit apart, along
the line. A number line is a one-dimensional C
ARTE
-
SIAN COORDINATE
system.
The theory of
CARDINALITY
shows that there are
just as many points on the number line as there are
points in a two-dimensional plane. The
DIAGONAL
ARGUMENT
shows that the set of
RATIONAL NUMBERS
(fractions) take up absolutely no space on the num-
ber line.
See also
DIMENSION
.
number systems See
BASE OF A NUMBER SYSTEM
.
number theory (higher arithmetic) The study of the
arithmetic properties of numbers is called number the-
ory. The fact that many simple statements about num-
bers can be extraordinarily difficult to prove, if at all
possible, makes this topic an alluring and stimulating
subject for mathematicians. (G
OLDBACH
’
S CONJEC
-
TURE
, for instance, remains unsolved.) C
ARL
F
RIEDRICH
G
AUSS
(1777–1855), charmed by the subject and its
“inexhaustible wealth,” called number theory the
“queen of mathematics.”
Elementary number theory is the study of those
topics in number theory that utilize only the basic tech-
niques of
ARITHMETIC
and high-school mathematics in
number theory 359