Section 1.1 Number Systems
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
, or the
The set of
is the set
The notation in
is read “
is the set whose members are 1, 2, 3, and
so on.” The ellipsis (the three dots) at the end in
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
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
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
, we call
if there is another
. For example, 4 is a divisor of 12 (because 12 = 4
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.
If the only divisors of a natural number
are 1 and itself, then
is said to be
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.
If a natural number other than 1 is not prime, then we say that it is
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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In this textbook, deﬁnitions, equations, and other labeled parts of the text are numbered consecutively,
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
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.