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Name: ___Notes___________________________________________
Number Theory
Pre-Algebra
A6
Lesson 6: Properties of Rational and Irrational Numbers
What happens when you add or multiply…
A. two rational numbers?
-10
R
1. –5 • 2 = ______________________
____
1
2
0.7333333333…
R
2. + = _______________________
_____
3
5
B. two irrational numbers?
2
R
1. √2 •√2 = ____________________
_____
6.283185307…
I
2. 𝜋 + 𝜋 = _____________________
_____
C. a rational number and an irrational number?
1. 5 • 𝜋 = ______________________
15.70796327…
_____
I
1
I
1.747546896…
2. + √2 = _____________________
_____
Rational
Irrational
Irrational
Both
3
D. Ms. Martin asked her class “Is the sum of 3.5 and √50 rational or irrational?” Mark
answered that the sum would always be irrational. State whether Mark is correct or
incorrect. Justify your reasoning.
Mark is correct because the sum of a rational and irrational always gives an
irrational
E. Ms. Bates asked her class “Is the product of two irrational numbers rational or
irrational?” Alexis answered that the product would always be irrational. State whether
Alexis is correct or incorrect. Justify your reasoning.
Alexis is incorrect because the product of two irrational numbers can be either
rational or irrational
Practice Problems
4
1. Ms. Rudd asked her class “Is the product of 5 and √121 rational or irrational?” Sam
answered that the product would always be rational. State whether Sam is correct or
incorrect. Justify your reasoning.
2. Ms. Martin asked her class “Is the sum of two irrational numbers rational or irrational?”
Karen answered that the sum would always be rational. State whether Karen is correct
or incorrect. Justify your reasoning.
3. Ms. Bates asked her class “Is the product of 4.2 and √2 rational or irrational?” Tom
answered that the product would be rational. State whether Tom is correct or
incorrect. Justify your reasoning.
Number Theory
A6
4. Determine if the difference of 7√4 and 5√81 is rational or irrational. Explain your answer
5. Determine if the quotient of 3√9 and √169 is rational or irrational. Explain your answer
6. For which value of A and B is A + B a rational number?
(1) A =
(2) A =
1
√2
1
√36
and B =
1
(3) A =
√4
and B =
1
(4) A =
√20
1
√4
and B =
1
√16
1
√10
and B =
1
√25
7. Given: A = 3√2
B = 2√3
C = 3√16 D = 2√9
Which expression results in a rational number?
(1) A + B
(3) C + D
(2) B + C
(4) D + A
8. (True/False) Determine if the following statements are true or false
a. __________ The sum of two rational numbers is always rational
b. __________ The product of two irrational numbers is always rational
c. __________ The sum of a rational number and an irrational number is always irrational
d. __________ The product of two irrational numbers is always irrational
e. __________ The product of two rational numbers is always rational
f. __________ The sum of two irrational numbers is always irrational
g. __________ The product of two rational numbers is always irrational
h. __________ The sum of two irrational numbers is always rational
i. __________ The sum of a rational number and an irrational number is always rational
j. __________ The product of a nonzero rational number and an irrational number
is always irrational
9. Given the following expressions:
5
3
I. – 8 + 5
III. (√5) • (√5)
II.
1
2
+ √2
IV. 3 • (√8)
Which expression(s) results in an irrational number?
(1) II, only
(2) III, only
(3) II and IV
(4) II, III, and IV
10. Given the following expressions:
1
22
I. – 2 + 7
III. (√13) • (√13)
II.
1
3
+ √9
IV. 2 • (√5)
Which expression(s) results in an irrational number?
(1) I and II
(2) II and III
(3) II and IV
(4) I and IV
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