Question 7 of 26beginner🔧 ApplyShort Answer1 mark

Given that the decimal expansion of irrational numbers is non-terminating, non-recurring, and 17\sqrt{17} is irrational, what can we conclude about the decimal expansion of 17\sqrt{17}?

Correct Answer

We can conclude that the decimal expansion of 17\sqrt{17} is non-terminating, non-recurring.

Exercise: EXERCISE A1.2 | Q: 3 | (Page 6)
For More Understanding

Explanation

This question tests deductive reasoning. We are given two premises as true statements: (1) the decimal expansion of irrational numbers is non-terminating, non-recurring, and (2) 17\sqrt{17} is irrational. Applying deductive reasoning (modus ponens), since 17\sqrt{17} is irrational, it must follow the property of all irrational numbers. Therefore, its decimal expansion must be non-terminating and non-recurring. The textbook context emphasizes that in such problems, we assume the hypotheses are true and then apply deductive reasoning to reach a valid conclusion.

Solution Steps

  1. Step 1: Identify the given premises - irrational numbers have non-terminating, non-recurring decimal expansion, and 17\sqrt{17} is irrational.

  2. Step 2: Apply deductive reasoning - since 17\sqrt{17} is irrational, it must satisfy the property of irrational numbers.

  3. Step 3: Conclude that the decimal expansion of 17\sqrt{17} is non-terminating, non-recurring.

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