1. The Real NumberThe Real Number
SystemSystem

2. Real NumbersReal Numbers
Real numbers consist of all the rational and irrationalReal numbers consist of all the rational and irrational
numbers.numbers.
The real number system has many subsets:The real number system has many subsets:
Natural NumbersNatural Numbers
Whole NumbersWhole Numbers
IntegersIntegers

3. Natural NumbersNatural Numbers
Natural numbersNatural numbers are the set of countingare the set of counting
numbers.numbers.
{1, 2, 3,…}{1, 2, 3,…}

4. Whole NumbersWhole Numbers
Whole numbersWhole numbers are the set of numbers thatare the set of numbers that
include 0 plus the set of natural numbers.include 0 plus the set of natural numbers.
{0, 1, 2, 3, 4, 5,…}{0, 1, 2, 3, 4, 5,…}

5. IntegersIntegers
IntegersIntegers are the set of whole numbers and theirare the set of whole numbers and their
opposites.opposites.
{…,-3, -2, -1, 0, 1, 2, 3,…}{…,-3, -2, -1, 0, 1, 2, 3,…}

6. Rational NumbersRational Numbers
Rational numbersRational numbers are any numbers that can beare any numbers that can be
expressed in the form of , whereexpressed in the form of , where aa andand bb areare
integers, and bintegers, and b ≠ 0≠ 0..
They can always be expressed by usingThey can always be expressed by using
terminating decimals or repeating decimals.terminating decimals or repeating decimals.
b
a

7. Terminating DecimalsTerminating Decimals
Terminating decimals are decimals that containTerminating decimals are decimals that contain
a finite number of digits.a finite number of digits.
Examples:Examples:
 36.836.8
 0.1250.125
 4.54.5

8. Repeating DecimalsRepeating Decimals
Repeating decimals are decimals that contain a infiniteRepeating decimals are decimals that contain a infinite
number of digits.number of digits.
Examples:Examples:
 0.333…0.333…

 7.689689…7.689689…
FYI…The line above the decimals indicate that numberFYI…The line above the decimals indicate that number
repeats.repeats.
9.1

9. Irrational NumbersIrrational Numbers
Irrational numbersIrrational numbers are any numbers that cannot be expressedare any numbers that cannot be expressed
as .as .
They are expressed asThey are expressed as non-terminating, non-repeatingnon-terminating, non-repeating
decimalsdecimals; decimals that go on forever without repeating a; decimals that go on forever without repeating a
pattern.pattern.
Examples of irrational numbers:Examples of irrational numbers:
0.34334333433334…0.34334333433334…
45.86745893…45.86745893…
(pi)(pi)
b
a
π
2

10. Other Vocabulary Associated withOther Vocabulary Associated with
the Real Number Systemthe Real Number System
……(ellipsis)—continues without end(ellipsis)—continues without end
{ } (set)—a collection of objects or numbers. Sets are{ } (set)—a collection of objects or numbers. Sets are
notated by using braces { }.notated by using braces { }.
Finite—having bounds; limitedFinite—having bounds; limited
Infinite—having no boundaries or limitsInfinite—having no boundaries or limits
Venn diagram—a diagram consisting of circles orVenn diagram—a diagram consisting of circles or
squares to show relationships of a set of data.squares to show relationships of a set of data.

11. ExampleExample
Classify all the following numbers as natural, whole, integer,Classify all the following numbers as natural, whole, integer,
rational, or irrational. List all that apply.rational, or irrational. List all that apply.
a.a. 117117
b.b. 00
c.c. -12.64039…-12.64039…
d.d. -½-½
e.e. 6.366.36
f.f.
g.g. -3-3
π

12. To show how these number are classified, use the VennTo show how these number are classified, use the Venn
diagram. Place the number where it belongs on the Venndiagram. Place the number where it belongs on the Venn
diagram.diagram.
π
9
4
2
1
−
9
4
Rational Numbers
Integers
Whole Numbers
Natural
Numbers
Irrational Numbers
-12.64039…
117
0
6.36
-3

13. SolutionSolution
Now that all the numbers are placed where they belong in theNow that all the numbers are placed where they belong in the
Venn diagram, you can classify each number:Venn diagram, you can classify each number:
117 is a natural number, a whole number, an integer, and a117 is a natural number, a whole number, an integer, and a
rational number.rational number.
is a rational number.is a rational number.
0 is a whole number, an integer, and a rational number.0 is a whole number, an integer, and a rational number.
-12.64039… is an irrational-12.64039… is an irrational number.number.
-3 is an integer and a rational number.-3 is an integer and a rational number.
6.36 is a rational number.6.36 is a rational number.
is an irrational number.is an irrational number.
is a rational number.is a rational number.
π
9
4
2
1
−

14. FYI…FYI…
When taking the square root of any number that isWhen taking the square root of any number that is
not a perfect square, the resulting decimal will benot a perfect square, the resulting decimal will be
non-terminating and non-repeating. Therefore, thosenon-terminating and non-repeating. Therefore, those
numbers are always irrational.numbers are always irrational.