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

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 _____________ . 2 1 4 3 6 5 7 9 8 10

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 2 1 4 3 6 5 7 9 8 10

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

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

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

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

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

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

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

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

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

Real Numbers and Number Lines A number line can also show ______________. rational numbers 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 ____.

Real Numbers and Number Lines A number line can also show ______________. rational numbers 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 ____. fraction

Real Numbers and Number Lines A number line can also show ______________. rational numbers 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 ____. fraction zero

Real Numbers and Number Lines A number line can also show ______________. rational numbers 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 ____. 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.

Real Numbers and Number Lines A number line can also show ______________. rational numbers 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 ____. 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

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

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

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

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

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.

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. The three periods following the digits in the nonterminating decimals indicate that there are infinitely many digits in the decimal.

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.

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.

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. Each rational number can be expressed as a terminating decimal or a nonterminating decimal with a repeating pattern.

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

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

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

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

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

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

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

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

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

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. x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B C A

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 x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B C A

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 The coordinate of point B is __ x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B C A

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 The coordinate of point B is __ -4 x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B C A

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 The coordinate of point B is __ -4 Point C has coordinate 0 and is called the _____. x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B C A

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 The coordinate of point B is __ -4 Point C has coordinate 0 and is called the _____. origin x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B C A

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. Postulate 2-2 Distance Postulate 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.

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. Postulate 2-2 Distance Postulate 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. A B measure

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. Postulate 2-2 Distance Postulate 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. A B measure Postulate 2-3 Ruler Postulate 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.

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. Postulate 2-2 Distance Postulate 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. A B measure Postulate 2-3 Ruler Postulate 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. B A a b measure = a – b

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

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

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

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

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

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

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

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

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?

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?

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?

Real Numbers and Number Lines End of Lesson