Math Problem Statement
Select all the statements that are true about π and √7, relating to their rationality and decimal properties.
Solution
Let's analyze each statement from the image:
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π is irrational because π is a repeating decimal.
- False: π is irrational, but not because it's a repeating decimal. π is non-repeating and non-terminating, which makes it irrational.
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√7 is irrational because 7 is not a perfect square.
- True: The square root of a non-perfect square (like 7) is irrational. Perfect squares, like 9 or 16, have rational square roots, but √7 does not.
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7.1234... is irrational because it is a nonterminating, nonrepeating decimal.
- True: Non-terminating and non-repeating decimals are the defining characteristics of irrational numbers.
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√7 is rational because 7 is not a perfect square.
- False: √7 is irrational, and the statement claims the opposite.
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π is irrational because π is not a repeating decimal.
- True: This is the correct reason why π is irrational—it's a non-terminating, non-repeating decimal.
Correct Answers:
- Statement 2: "√7 is irrational because 7 is not a perfect square."
- Statement 3: "7.1234... is irrational because it is a nonterminating, nonrepeating decimal."
- Statement 5: "π is irrational because π is not a repeating decimal."
Let me know if you'd like further details!
Follow-up questions:
- Why is a non-repeating, non-terminating decimal always irrational?
- How do perfect squares relate to the rationality of square roots?
- Can repeating decimals ever be irrational?
- Why is π classified as a transcendental number in addition to being irrational?
- How does the concept of rational vs irrational numbers apply to other roots like cube roots?
Tip:
A repeating decimal (like 1/3 = 0.333...) is always rational, while non-repeating decimals indicate irrational numbers.
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Math Problem Analysis
Mathematical Concepts
Irrational Numbers
Rational Numbers
Square Roots
Decimals
Formulas
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Theorems
Definition of Irrational Numbers
Non-repeating, Non-terminating Decimals
Square Root Properties
Suitable Grade Level
Grades 9-11