the real numbers 4-7 learn to determine if a number is rational or irrational
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The Real Numbers4-7
Learn to determine if a number is rational or irrational.
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
Example 1: Classifying Real Numbers
Identify as rational or irratioanl.
5
irrational
–12.75 is a terminating decimal.–12.75rational
16 2
rational
= = 24 2
16 2
A.
B.
C.
The Real Numbers4-7
Example 2
Write all names that apply to each number.
9
rational
–35.9 is a terminating decimal.–35.9rational
81 3
rational
= = 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
Example 3: Determining the Classification of All Numbers
A.
B.
The Real Numbers4-7
not a real number
Example 4: Determining the Classification of All Numbers
–4
4 9
rational
C.
D.
State if each number is rational, irrational, or not a real number.
The Real Numbers4-7
23
irrational
9 0
undefined, so not a real number
Example 5
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
C.
D.
Example 6
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
Example 7: 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
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
irrational rational
Possible answer –2 .5 8
The Real Numbers4-7
1. Identify rational, irrational, not real.
A. irrational
B. rational
C. Not real
D. All of the above
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