Slowly, over many years, Nash began to recover and
eventually returned to teaching at Princeton and contem-
plating mathematical work. He delivered a paper at the
10th World Congress of Psychiatry in 1996 describing
his illness. Along with the 1994 Noble Prize, Nash was
awarded the 1999 Leroy P. Steele Prize by the American
Mathematical Association. Nash describes his 45-page
dissertation as his “most trivial work” of his career.
natural number (counting number, whole number)
Any of the positive whole numbers 1, 2, 3, … is called a
natural number or a counting number. The natural num-
bers are used to count separate objects. The collection of
all counting numbers {1,2,3,…}, called the set of natural
numbers, is denoted N, or sometimes IN or Z+. Some
mathematicians choose to include the number
ZERO
in
this set. (There is no standard convention on the matter.)
The set of natural numbers is closed under
ADDI
-
TION
and
MULTIPLICATION
, that is, the sum or product
of any two natural numbers is again a natural number.
The set, however, is not closed under
SUBTRACTION
or
DIVISION
. For example, the result of subtracting 5 from
3 is no longer a natural number, nor is the result of
dividing 7 by 2.
An infinite set of objects whose elements can be
arranged in a list akin to the list of natural numbers is
said to be
COUNTABLE
. Not all infinite sets are countable.
See also
CLOSURE PROPERTY
;
DISCRETE
;
FIGURATE
NUMBERS
;
NUMBER
;
NUMBER THEORY
;
ORDINAL NUM
-
BERS
;
WHOLE NUMBER
.
necessary condition See
CONDITION
—
NECESSARY AND
SUFFICIENT
.
negation (not statement) In
FORMAL LOGIC
a state-
ment of the form “not p” is called the negation of the
statement p. For example, “Shakespeare did not write
Hamlet” is the negation of the statement that Shake-
speare did. In practice it is not always necessary for the
term not to occur. For example, the negation of x> yis
x≤y.
The negation of a statement pis denoted in sym-
bols as
¬
p(or sometimes as ~p, –p, or even –
p). The
negation of statement has truth-value opposite to that
of the original statement, and so has a
TRUTH TABLE
:
negative numbers Any
REAL NUMBER
less than
ZERO
is called a negative number. In practical applications,
negative numbers are used to denote quantities that are
below some reference point. For example, in the centi-
grade temperature scale, a temperature of –10°is 10°
below the freezing point of water.
The product of two negative numbers is a positive
quantity. For instance, (–1) ×(–1) = 1. This can be jus-
tified by making use of the
DISTRIBUTIVE PROPERTY
and
the fact that the product of any quantity with zero is
zero. Specifically, we have:
(–1) ×0 = 0
(–1) ×(1+ (–1)) = 0
(–1) ×1 + (–1) ×(–1) = 0
–1 + (–1) ×(–1) = 0
(–1) ×(–1) = 1
Negative numbers were generally viewed with sus-
picion throughout history. Ancient Egyptian and Baby-
lonian scholars ignored negative solutions to their
mathematical equations, as did ancient Greek scholars
(to whom “number” was directly associated with the
notion of physical length). Chinese scholars were com-
fortable working with negative quantities in intermedi-
ate steps toward solving a problem, but they never
permitted them as final solutions.
Seventh-century Indian scholar B
RAHMAGUPTA
is
noted as the first scholar to properly determine the
arithmetic of negative quantities. He deemed their exis-
tence as valid by equating positive quantities with pos-
sessions and negative quantities with debt. This work
was later expanded upon by scholar B
H
–
ASKARA
(ca.
1114–85). Although Arab scholars of this time read and
translated the Indian texts, they chose not to work with
negative quantities. As 12th-century European scholars
garnered much of their mathematical knowledge from
the Islamic world, familiarity with negative quantities
did not readily come to the West.
In the 16th century, prominent mathematicians such
as G
IROLAMO
C
ARDANO
, Michael Stifel, and F
RANÇOIS
V
IÈTE
adamantly rejected the notion of negative num-
348 natural number
p
¬
p
TF
FT