This document introduces several key concepts about real numbers and number lines: 1) Real numbers include both rational numbers (like integers and decimals) and irrational numbers. 2) A number line can be used to represent different sets of numbers, including whole numbers, integers, and rational numbers. 3) The coordinate of a point on a number line represents the real number associated with that point. The point where the number line crosses 0 is called the origin.

1. Real Numbers and Number Lines You will learn to find the distance between two points on a number line. What You'll Learn 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 Vocabulary

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 2 1 4 3 6 5 7 9 8 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 2 1 4 3 6 5 7 9 8 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 The whole numbers include 0 and the natural, or counting numbers. 0 2 1 4 3 6 5 7 9 8 10

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 The whole numbers include 0 and the natural, or counting numbers. The arrow to the right indicate that the whole numbers continue _________. 0 2 1 4 3 6 5 7 9 8 10

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 The whole numbers include 0 and the natural, or counting numbers. The arrow to the right indicate that the whole numbers continue _________. indefinitely 0 2 1 4 3 6 5 7 9 8 10

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

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

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

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

11. Real Numbers and Number Lines A number line can also show ______________. 0 2 1 -1 -2

12. Real Numbers and Number Lines A number line can also show ______________. rational numbers 0 2 1 -1 -2

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

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

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

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

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

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

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

20. Real Numbers and Number Lines Rational numbers can also be represented by ________. decimals Decimals may be __________ or _____________.

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

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

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

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.

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. Each rational number can be expressed as a terminating decimal or a nonterminating decimal with a repeating pattern. A bar over the repeating digits is used to indicate a repeating decimal.

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 0.101001000 . . . appear to be irrational numbers

30. Real Numbers and Number Lines ____________ include both rational and irrational numbers. 0 2 1 -1 -2

31. Real Numbers and Number Lines ____________ include both rational and irrational numbers. Real numbers 0 2 1 -1 -2

32. Real Numbers and Number Lines The number line above shows some real numbers between -2 and 2. ____________ include both rational and irrational numbers. Real numbers 0 2 1 -1 -2

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

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 _________ of the point. coordinate

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

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

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

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

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

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

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 between the points. Postulate 2-2 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 between the points. Postulate 2-2 Distance Postulate A B 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 between the points. Postulate 2-2 Distance Postulate A B 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 corresponding numbers. Postulate 2-3 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 between the points. Postulate 2-2 Distance Postulate A B 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 corresponding numbers. Postulate 2-3 Ruler Postulate measure = a – b B A a b

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

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

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

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

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

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

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

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

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?

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?

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?

57. End of Lesson Real Numbers and Number Lines