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The Real Numbers4-7
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson QuizzesLesson Quizzes
The Real Numbers4-7
Warm UpEach square root is between two integers. Name the two integers.
Use a calculator to find each value. Round to the nearest tenth.
10 and 11
–4 and –3
1.4
–11.1
1. 119
2. – 15
3. 2
4. – 123
The Real Numbers4-7
Problem of the Day
The circumference of a circle is approximately 3.14 times its diameter. A circular path 1 meter wide has an inner diameter of 100 meters. How much farther is it around the outer edge of the path than the inner edge?
6.28 m
The Real Numbers4-7
Learn to determine if a number is rational or irrational.
The Real Numbers4-7
irrational numberreal numberDensity Property
Vocabulary
The Real Numbers4-7
AnimalsVertebrates
Biologists classify animals based on shared characteristics. The cardinal is an animal, a vertebrate, a bird, and a passerine.
You already know that some numbers can be classified as whole numbers, integers, or rational numbers.
BirdsPasserines
The Real Numbers4-7
Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat.
3 = 3.84 5
= 0.623
1.44 = 1.2
The Real Numbers4-7
Irrational numbers can only be written as decimals that do not terminate or repeat. If a whole number is not a perfect square, then its square root is an irrational number.
2 ≈1.4142135623730950488016…
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!
The Real Numbers4-7
The set of real numbers consists of the set of rational numbers and the set of irrational numbers.
The Real Numbers4-7
Additional Example 1: Classifying Real Numbers
Write all names that apply to each number.
5 is a whole number that is not a perfect square.
5
irrational, real
–12.75 is a terminating decimal.–12.75rational, real
16 2
whole, integer, rational, real
= = 24 2
16 2
A.
B.
C.
The Real Numbers4-7
Check It Out: Example 1
Write all names that apply to each number.
9
whole, integer, rational, real
–35.9 is a terminating decimal.–35.9rational, real
81 3
whole, integer, rational, real
= = 39 3
81 3
A.
B.
C.
9 = 3
The Real Numbers4-7
State if each number is rational, irrational, or not a real number.
21
irrational
0 3
rational
0 3
= 0
Additional Example 2: Determining the Classification of All Numbers
A.
B.
The Real Numbers4-7
not a real number
Additional Example 2: Determining the Classification of All Numbers
–4
4 9
rational
2 3
=2 3
4 9
C.
D.
State if each number is rational, irrational, or not a real number.
The Real Numbers4-7
23 is a whole number that is not a perfect square.
23
irrational
9 0
undefined, so not a real number
Check It Out: Example 2
A.
B.
State if each number is rational, irrational, or not a real number.
The Real Numbers4-7
not a real number
–7
64 81
rational
8 9
=8 9
64 81
C.
D.
Check It Out: Example 2
State if each number is rational, irrational, or not a real number.
The Real Numbers4-7
The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3.
The Real Numbers4-7
Additional Example 3: Applying the Density Property of Real Numbers
2 5
3 + 3 ÷ 23 5
There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.
5 5
= 6 ÷ 21 2
= 7 ÷ 2 = 3
31 2
3 3 31 5
2 5 43 33
54 5
Find a real number between 3 and 3 .
3 5
2 5
A real number between 3 and 3 is 3 .3 5
2 5
1 2
The Real Numbers4-7
Check It Out: Example 3
3 7
4 + 4 ÷ 24 7
There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.
7 7= 8 ÷ 2
1 2
= 9 ÷ 2 = 4
41 2
4 44 4 4 42 7
3 7
4 7
5 7
1 7
6 7
Find a real number between 4 and 4 .
4 7
3 7
A real number between 4 and 4 is 4 .4 7
3 7
1 2
The Real Numbers4-7
Standard Lesson Quiz
Lesson Quizzes
Lesson Quiz for Student Response Systems
The Real Numbers4-7Lesson Quiz
Write all names that apply to each number.
1. 2. –
State if each number is rational, irrational, or not a real number.
3. 4.
Find a real number between –2 and –2 .3 8
3 4
5.
2
4 • 9
16 2
25 0
not a real number rational
real, irrational real, integer, rational
Possible answer –2 .5 8
The Real Numbers4-7
1. Identify all names that apply to .
A. real, irrational
B. real, rational
C. not real, irrational
D. not real, rational
Lesson Quiz for Student Response Systems
The Real Numbers4-7
2. Identify the name that applies to .
A. irrational
B. rational
C. not a real number
D. none
Lesson Quiz for Student Response Systems
The Real Numbers4-7
3. Identify a real number between .
A. –4
B.
C.
D.
Lesson Quiz for Student Response Systems