Section 1.1 Number Systems
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Version: Fall 2007
1.1 Number Systems
In this section we introduce the number systems that we will work with in the remainder
of this text.
The Natural Numbers
We begin with a deﬁnition of the
natural numbers
, or the
counting numbers
.
Deﬁnition 1.
The set of
natural numbers
is the set
N
=
{
1
,
2
,
3
,...
}
.
(2)
The notation in
equation (2)
2
is read “
N
is the set whose members are 1, 2, 3, and
so on.” The ellipsis (the three dots) at the end in
equation (2)
is a mathematician’s
way of saying “et-cetera.” We list just enough numbers to establish a recognizable pat-
tern, then write “and so on,” assuming that a pattern has been suﬃciently established
so that the reader can intuit the rest of the numbers in the set. Thus, the next few
numbers in the set
N
are 4, 5, 6, 7, “and so on.”
Note that there are an inﬁnite number of natural numbers. Other examples of nat-
ural numbers are 578,736 and 55,617,778. The set
N
of natural numbers is unbounded;
i.e., there is no largest natural number. For any natural number you choose, adding
one to your choice produces a larger natural number.
For any natural number
n
, we call
m
a
divisor
or
factor
of
n
if there is another
natural number
k
so that
n
=
mk
. For example, 4 is a divisor of 12 (because 12 = 4
×
3),
but 5 is not. In like manner, 6 is a divisor of 12 (because 12 = 6
×
2), but 8 is not.
We next deﬁne a very special subset of the natural numbers.
Deﬁnition 3.
If the only divisors of a natural number
p
are 1 and itself, then
p
is said to be
prime
.
For example, because its only divisors are 1 and itself, 11 is a prime number. On
the other hand, 14 is not prime (it has divisors other than 1 and itself, i.e., 2 and 7). In
like manner, each of the natural numbers 2, 3, 5, 7, 11, 13, 17, and 19 is prime. Note
that 2 is the only even natural number that is prime.
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If a natural number other than 1 is not prime, then we say that it is
composite
.
Note that any natural number (except 1) falls into one of two classes; it is either prime,
or it is composite.
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1
In this textbook, deﬁnitions, equations, and other labeled parts of the text are numbered consecutively,
2
regardless of the type of information. Figures are numbered separately, as are Tables.
Although the natural number 1 has only 1 and itself as divisors, mathematicians, particularly number
3
theorists, don’t consider 1 to be prime. There are good reasons for this, but that might take us too far
aﬁeld. For now, just note that 1 is not a prime number. Any number that is prime has exactly two
factors, namely itself and 1.