NUMBER SYSTEM
In mathematics, the numbers we use can be categorized into sets. Our number system has two
sets, the real numbers and the complex numbers. We will work with the real numbers in this
course.
DEFINITIONS:
REAL NUMBERS is the set of rational numbers and irrational numbers.
COUNTING NUMBERS OR NATURAL NUMBERS is the set of numbers defined by {1, 2,
3, 4, 5, ...}.
WHOLE NUMBERS is the set of numbers defined by {0, 1, 2, 3, 4, 5, ...}.
INTEGERS is the set of numbers defined by {..., -3, -2, -1, 0, 1, 2, 3, ...} or the set of all
positive and negative whole numbers.
RATIONAL NUMBERS is the set of numbers defined by
p
| p and q are integers, q ≠ 0}.
q
or
the set of numbers in which the decimal terminates or the decimal repeats.
{
examples:
These are all rational numbers.
1/2 = 0.5
terminated decimal
-2/3 = -0.6666...
repeating decimal
2/7 = 0.285714285... repeating decimal
9/4 = 2.25
terminated decimal
5 = 5.0
terminated decimal
-12/3 = 4.0
terminated decimal
√4 = 2.0
terminated decimal
IRRATIONAL NUMBERS is the set of numbers in which the decimal does not terminate and
does not repeat.
examples:
These are all irrational numbers.
2 (square root of 2) = 1.414213562... does not terminate nor repeats
= 3.141592654...
does not terminate nor repeats
3
= 0.3464101615... does not terminate nor repeats; also, the numerator is an
5
irrational and so the entire fraction will be irrational.
∏ (pi)
NOTE: Some numbers may fit into one or more sets. This is because the real numbers has
many different subsets.
See the chart below to give you a visual representation.
example:
Rational numbers is a subset of real Numbers.
Irrational numbers is a subset of real Numbers.
Integers is a subset of rational numbers and real numbers.
Whole numbers is a subset of integers, rational numbers, and real numbers.
Counting numbers is a subset of whole numbers, integers, rational numbers and real
numbers.
REAL NUMBERS
RATIONAL
NUMBERS
INTEGERS
WHOLE
NUMBERS
COUNTING
(NATURAL)
NUMBERS
IRRATIONAL
NUMBERS
EXERCISE 1:
Given the set of numbers
13
1
{17, , 3, 0,
, -49, 0.056725431… ,
2
8
List the numbers in the set that are:
a) integers
b) counting numbers
c) rational numbers d) irrational numbers
Answer:
a) The integers are 17, 3, 0, -49,
7 , -8.23,
11
, 2.333…,
8
64 }
e) real numbers
64 = 8
b) The counting numbers are 17, 3,
64 = 8
1
11
, 3, 0, -49, -8.23,
, 2.333…,
2
8
these convert to a decimal that terminates or repeats).
c) The rational numbers are 17,
13
, 0.056725431… ,
8
decimal that does not repeat and does not terminate).
d) The irrational numbers are
64 (because each of
7 (because each of these convert to a
e) The real numbers is the entire set.
EXERCISE 2:
State whether the following statements are true or false.
a) A real number is either a rational or irrational.
b) An integer is also a rational number.
c) Zero is not an integer.
d) 1.625 is an irrational number.
e) 0.333... is a rational number.
f) 3.31662479... is a irrational number.
g) 9/11 is not a whole number but it is a rational number.
Answer:
a) true. (by definition)
b) true. (an integer is a terminated decimal)
c) false. (zero is an integer because it is a positive or negative whole number)
d) false. (since the decimal terminates then it is a rational number)
e) true (the decimal repeats)
f) true (the decimal does not repeat and does not terminate)
g) true (9/11 converts into the decimal 0.8181... which repeats)