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:

  1. 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.
  2. √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.
  3. 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.
  4. √7 is rational because 7 is not a perfect square.

    • False: √7 is irrational, and the statement claims the opposite.
  5. 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."

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Follow-up questions:

  1. Why is a non-repeating, non-terminating decimal always irrational?
  2. How do perfect squares relate to the rationality of square roots?
  3. Can repeating decimals ever be irrational?
  4. Why is π classified as a transcendental number in addition to being irrational?
  5. 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