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The Real Numbers3-7
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson QuizzesLesson Quizzes
The Real Numbers3-7
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 Numbers3-7
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 Numbers3-7
Learn to determine if a number is rational or irrational.
The Real Numbers3-7
irrational numberreal numberDensity Property
Vocabulary
The Real Numbers3-7
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 Numbers3-7
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 Numbers3-7
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 Numbers3-7
The set of real numbers consists of the set of rational numbers and the set of irrational numbers.
The Real Numbers3-7
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 Numbers3-7
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 Numbers3-7
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 Numbers3-7
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 Numbers3-7
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 Numbers3-7
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 Numbers3-7
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 Numbers3-7
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 Numbers3-7
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
