course 3 4-7 the real numbers n1.h how do i distinguish between rational and irrational numbers?...
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Course 3
4-7 The Real NumbersN1.h How Do I Distinguish Between Rational And Irrational Numbers?
Course 3
Lesson Presentation
Course 3
4-7 The Real Numbers
Learn to determine if a number is rational or irrational.
Course 3
4-7 The Real Numbers
real numberrational numberirrational number
Vocabulary
Course 3
4-7 The Real Numbers
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= 0.62
31.44 = 1.2
Course 3
4-7 The Real Numbers
Irrational numbers cannot be written as a fraction & can only be written as decimals that do not terminate or repeat (NON-TERMINATING/NON-REPEATING DECIMALS. The square root of a non-perfect square is an irrational number.
For example,
2 ≈1.4142135623730950488016…
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!
Course 3
4-7 The Real Numbers
The set of real numbers consists of the set of rational numbers and the set of irrational numbers.
Irrational numbersRational numbers
Real Numbers
Integers
Wholenumbers
Course 3
4-7 The Real Numbers
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.
Course 3
4-7 The Real Numbers
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
Course 3
4-7 The Real Numbers
State if each number is rational, irrational, or not a real number.
21
irrational
0 3
rational
0 3
= 0
Example 2: Determining the Classification of All Numbers
A.
B.
Course 3
4-7 The Real Numbers
not a real number
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.
Course 3
4-7 The Real Numbers
23 is a whole number that is not a perfect square.
23
irrational
9 0
not a number, so not a real number
Check It Out: Example 2
A.
B.
State if each number is rational, irrational, or not a real number.
Course 3
4-7 The Real Numbers
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.
Course 3
4-7 The Real Numbers
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.
Course 3
4-7 The Real Numbers
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
Course 3
4-7 The Real Numbers
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
Course 3
4-7 The Real NumbersLesson 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