1. Real Numbers and Number Lines
You will learn to find the distance between two points on a
number line.
1) Whole Numbers
2) Natural Numbers
3) Integers
4) Rational Numbers
5) Terminating Decimals
6) Nonterminating Decimals
7) Irrational Numbers
8) Real Numbers
9) Coordinate
10 Origin
11) Measure
12) Absolute Value

2. Real Numbers and Number Lines
Numbers that share common properties can be classified or grouped into sets.
Different sets of numbers can be shown on numbers lines.
This figure shows the set of _____________ .
0 1 2 3 4 5 6 7 8 9 10

3. Real Numbers and Number Lines
Numbers that share common properties can be classified or grouped into sets.
Different sets of numbers can be shown on numbers lines.
This figure shows the set of _____________ .
whole numbers
0 1 2 3 4 5 6 7 8 9 10

4. Real Numbers and Number Lines
Numbers that share common properties can be classified or grouped into sets.
Different sets of numbers can be shown on numbers lines.
This figure shows the set of _____________ .
whole numbers
0 1 2 3 4 5 6 7 8 9 10
The whole numbers include 0 and the natural, or counting numbers.

5. Real Numbers and Number Lines
Numbers that share common properties can be classified or grouped into sets.
Different sets of numbers can be shown on numbers lines.
This figure shows the set of _____________ .
whole numbers
0 1 2 3 4 5 6 7 8 9 10
The whole numbers include 0 and the natural, or counting numbers.
The arrow to the right indicate that the whole numbers continue _________.

6. Real Numbers and Number Lines
Numbers that share common properties can be classified or grouped into sets.
Different sets of numbers can be shown on numbers lines.
This figure shows the set of _____________ .
whole numbers
0 1 2 3 4 5 6 7 8 9 10
The whole numbers include 0 and the natural, or counting numbers.
indefinitely
The arrow to the right indicate that the whole numbers continue _________.

7. Real Numbers and Number Lines
This figure shows the set of _______ .
-5 -4 -3 -2 -1 0 1 2 3 4 5

8. Real Numbers and Number Lines
This figure shows the set of _______ .
integers
-5 -4 -3 -2 -1 0 1 2 3 4 5
positive integers

9. Real Numbers and Number Lines
This figure shows the set of _______ .
integers
-5 -4 -3 -2 -1 0 1 2 3 4 5
negative integers positive integers

10. Real Numbers and Number Lines
This figure shows the set of _______ .
integers
-5 -4 -3 -2 -1 0 1 2 3 4 5
negative integers positive integers
The integers include zero, the positive integers, and the negative integers.
The arrows indicate that the numbers go on forever in both directions.

11. Real Numbers and Number Lines
A number line can also show ______________.
-2 5 11 -1 1 0 3 2 1 4 13 2
3 8 5 8 3 3 8

12. Real Numbers and Number Lines
A number line can also show ______________.
rational numbers
-2 5 11 -1 1 0 3 2 1 4 13 2
3 8 5 8 3 3 8

13. Real Numbers and Number Lines
A number line can also show ______________.
rational numbers
-2 5 11 -1 1 0 3 2 1 4 13 2
3 8 5 8 3 3 8
a
A rational number is any number that can be written as a _______,
where a and b are integers and b cannot equal ____. b

14. Real Numbers and Number Lines
A number line can also show ______________.
rational numbers
-2 5 11 -1 1 0 3 2 1 4 13 2
3 8 5 8 3 3 8
a
A rational number is any number that can be written as a _______,
fraction
where a and b are integers and b cannot equal ____. b

15. Real Numbers and Number Lines
A number line can also show ______________.
rational numbers
-2 5 11 -1 1 0 3 2 1 4 13 2
3 8 5 8 3 3 8
a
A rational number is any number that can be written as a _______,
fraction
where a and b are integers and b cannot equal ____.
zero b

16. Real Numbers and Number Lines
A number line can also show ______________.
rational numbers
-2 5 11 -1 1 0 3 2 1 4 13 2
3 8 5 8 3 3 8
a
A rational number is any number that can be written as a _______,
fraction
where a and b are integers and b cannot equal ____.
zero b
The number line above shows some of the rational numbers between -2 and 2.
In fact, there are _______ many rational numbers between any two integers.

17. Real Numbers and Number Lines
A number line can also show ______________.
rational numbers
-2 5 11 -1 1 0 3 2 1 4 13 2
3 8 5 8 3 3 8
a
A rational number is any number that can be written as a _______,
fraction
where a and b are integers and b cannot equal ____.
zero b
The number line above shows some of the rational numbers between -2 and 2.
infinitely
In fact, there are _______ many rational numbers between any two integers.

18. Real Numbers and Number Lines
Rational numbers can also be represented by ________.

19. Real Numbers and Number Lines
decimals
Rational numbers can also be represented by ________.
3 2 0
0.375 0.666 . . . 0
8 3 7

20. Real Numbers and Number Lines
decimals
Rational numbers can also be represented by ________.
3 2 0
0.375 0.666 . . . 0
8 3 7
Decimals may be __________ or _____________.

21. Real Numbers and Number Lines
decimals
Rational numbers can also be represented by ________.
3 2 0
0.375 0.666 . . . 0
8 3 7
terminating
Decimals may be __________ or _____________.
0.375
terminating decimals.
0.49

22. Real Numbers and Number Lines
decimals
Rational numbers can also be represented by ________.
3 2 0
0.375 0.666 . . . 0
8 3 7
terminating nonterminating
Decimals may be __________ or _____________.
0.375
terminating decimals.
0.49
0.666 . . .
nonterminating decimals.
-0.12345 . . .

23. Real Numbers and Number Lines
decimals
Rational numbers can also be represented by ________.
3 2 0
0.375 0.666 . . . 0
8 3 7
terminating nonterminating
Decimals may be __________ or _____________.
0.375
terminating decimals.
0.49
0.666 . . .
nonterminating decimals.
-0.12345 . . .
The three periods following the digits in the nonterminating decimals indicate
that there are infinitely many digits in the decimal.

24. Real Numbers and Number Lines
Some nonterminating decimals have a repeating pattern.
0.17171717 . . . repeats the digits 1 and 7 to the right of the decimal point.

25. Real Numbers and Number Lines
Some nonterminating decimals have a repeating pattern.
0.17171717 . . . repeats the digits 1 and 7 to the right of the decimal point.
A bar over the repeating digits is used to indicate a repeating decimal.
0.171717 . . . 0.17

26. Real Numbers and Number Lines
Some nonterminating decimals have a repeating pattern.
0.17171717 . . . repeats the digits 1 and 7 to the right of the decimal point.
A bar over the repeating digits is used to indicate a repeating decimal.
0.171717 . . . 0.17
Each rational number can be expressed as a terminating decimal or a
nonterminating decimal with a repeating pattern.

27. Real Numbers and Number Lines
Decimals that are nonterminating and do not repeat
are called _______________.

28. Real Numbers and Number Lines
Decimals that are nonterminating and do not repeat
are called _______________.
irrational numbers

29. Real Numbers and Number Lines
Decimals that are nonterminating and do not repeat
are called _______________.
irrational numbers
6.028716 . . .
and appear to be irrational numbers
0.101001000 . . .

30. Real Numbers and Number Lines
____________ include both rational and irrational numbers.
-2 -1 0 3 1 2
1.8603 . . . 0.25 8 1.762 . . .
0.8 0.6

31. Real Numbers and Number Lines
Real numbers
____________ include both rational and irrational numbers.
-2 -1 0 3 1 2
1.8603 . . . 0.25 8 1.762 . . .
0.8 0.6

32. Real Numbers and Number Lines
Real numbers
____________ include both rational and irrational numbers.
-2 -1 0 3 1 2
1.8603 . . . 0.25 8 1.762 . . .
0.8 0.6
The number line above shows some real numbers between -2 and 2.

33. Real Numbers and Number Lines
Real numbers
____________ include both rational and irrational numbers.
-2 -1 0 3 1 2
1.8603 . . . 0.25 8 1.762 . . .
0.8 0.6
The number line above shows some real numbers between -2 and 2.
Postulate 2-1 Each real number corresponds to exactly one point on a number
line.
Number Line
Each point on a number line corresponds to exactly one real
Postulate number

34. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
_________ of the point.

35. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
coordinate
_________ of the point.

36. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
coordinate
_________ of the point.
On the number line below, __ is the coordinate of point A.
B C A x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

37. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
coordinate
_________ of the point.
On the number line below, 10 is the coordinate of point A.
__
B C A x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

38. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
coordinate
_________ of the point.
On the number line below, 10 is the coordinate of point A.
__
The coordinate of point B is __
B C A x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

39. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
coordinate
_________ of the point.
On the number line below, 10 is the coordinate of point A.
__
The coordinate of point B is __
-4
B C A x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

40. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
coordinate
_________ of the point.
On the number line below, 10 is the coordinate of point A.
__
The coordinate of point B is __
-4
Point C has coordinate 0 and is called the _____.
B C A x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

41. Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
coordinate
_________ of the point.
On the number line below, 10 is the coordinate of point A.
__
The coordinate of point B is __
-4
origin
Point C has coordinate 0 and is called the _____.
B C A x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

42. Real Numbers and Number Lines
The distance between two points A and B on a number line is found by
using the Distance and Ruler Postulates.
For any two points on a line and a given unit of measure, there is a
unique positive real number called the measure of the distance
Postulate 2-2 between the points.
Distance
Postulate

43. Real Numbers and Number Lines
The distance between two points A and B on a number line is found by
using the Distance and Ruler Postulates.
For any two points on a line and a given unit of measure, there is a
unique positive real number called the measure of the distance
Postulate 2-2 between the points.
Distance B A
Postulate
measure

44. Real Numbers and Number Lines
The distance between two points A and B on a number line is found by
using the Distance and Ruler Postulates.
For any two points on a line and a given unit of measure, there is a
unique positive real number called the measure of the distance
Postulate 2-2 between the points.
Distance B A
Postulate
measure
Points on a line are paired with real numbers, and the measure of
the distance between two points is the positive difference of the
Postulate 2-3 corresponding numbers.
Ruler
Postulate

45. Real Numbers and Number Lines
The distance between two points A and B on a number line is found by
using the Distance and Ruler Postulates.
For any two points on a line and a given unit of measure, there is a
unique positive real number called the measure of the distance
Postulate 2-2 between the points.
Distance B A
Postulate
measure
Points on a line are paired with real numbers, and the measure of
the distance between two points is the positive difference of the
Postulate 2-3 corresponding numbers.
Ruler B A
Postulate
b a
measure = a – b

46. Real Numbers and Number Lines
The measure of the distance between B and A is the positive difference
10 – 2, or 8.
B A x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
Another way to calculate the measure of the distance is by using
____________.

47. Real Numbers and Number Lines
The measure of the distance between B and A is the positive difference
10 – 2, or 8.
B A x
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10
Another way to calculate the measure of the distance is by using
____________.
absolute value
AB 10 2 BA 2 10
8 8
8 8

48. Real Numbers and Number Lines
Use the number line below to find the following measures.
A B C F x
-3 -2 -1 0 1 2
BA CF

49. Real Numbers and Number Lines
Use the number line below to find the following measures.
A B C F x
-3 -2 -1 0 1 2
5 8
BA CF
3 3
3
3
1

50. Real Numbers and Number Lines
Use the number line below to find the following measures.
A B C F x
-3 -2 -1 0 1 2
5 8
BA CF ( 1) 2
3 3
3
3
3 3
1

51. Real Numbers and Number Lines
Use the number line below to find the following measures.
A D E F x
-3 -2 -1 0 1 2
DA EF

52. Real Numbers and Number Lines
Use the number line below to find the following measures.
A D E F x
-3 -2 -1 0 1 2
1 8
DA EF
3 3
7
3
7
3

53. Real Numbers and Number Lines
Use the number line below to find the following measures.
A D E F x
-3 -2 -1 0 1 2
1 8 1
DA EF 2
3 3 3
7 5
3 3
7 5
3 3

54. Real Numbers and Number Lines
Traveling on I-70, the Manhattan exit is at mile marker 313.
The Hays exit is mile marker 154.
What is the distance between these two towns?
MH

55. Real Numbers and Number Lines
Traveling on I-70, the Manhattan exit is at mile marker 313.
The Hays exit is mile marker 154.
What is the distance between these two towns?
MH

56. Real Numbers and Number Lines
Traveling on I-70, the Manhattan exit is at mile marker 313.
The Hays exit is mile marker 154.
What is the distance between these two towns?
MH 313 154
159
159 miles

57. Real Numbers and Number Lines