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If x ER is irrational and r E Q, show that r + x ER is irrational.

If x ER is irrational and r E Q, show that r + x ER is irrational.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** If \( x \in \mathbb{R} \) is irrational and \( r \in \mathbb{Q} \), show that \( r + x \in \mathbb{R} \) is irrational. **Explanation:** In this problem, you are given that \( x \) is an irrational number and \( r \) is a rational number. The goal is to prove that the sum of an irrational number \( x \) and a rational number \( r \) is also irrational. ### Key Points: 1. **Real Numbers (\(\mathbb{R}\))**: The set of all rational and irrational numbers. 2. **Rational Numbers (\(\mathbb{Q}\))**: Numbers that can be expressed as the quotient of two integers (i.e., \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)). 3. **Irrational Numbers**: Numbers that cannot be expressed as a simple fraction, and their decimal expansion is non-terminating and non-repeating. ### Proof Strategy: - Assume the contrary: Suppose \( r + x \) is rational. - Use properties of rational and irrational numbers to derive a contradiction. This type of problem is significant in real analysis and number theory, and it demonstrates the properties and behaviors of different types of numbers when combined in various operations.
Transcribed Image Text:**Problem Statement:** If \( x \in \mathbb{R} \) is irrational and \( r \in \mathbb{Q} \), show that \( r + x \in \mathbb{R} \) is irrational. **Explanation:** In this problem, you are given that \( x \) is an irrational number and \( r \) is a rational number. The goal is to prove that the sum of an irrational number \( x \) and a rational number \( r \) is also irrational. ### Key Points: 1. **Real Numbers (\(\mathbb{R}\))**: The set of all rational and irrational numbers. 2. **Rational Numbers (\(\mathbb{Q}\))**: Numbers that can be expressed as the quotient of two integers (i.e., \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)). 3. **Irrational Numbers**: Numbers that cannot be expressed as a simple fraction, and their decimal expansion is non-terminating and non-repeating. ### Proof Strategy: - Assume the contrary: Suppose \( r + x \) is rational. - Use properties of rational and irrational numbers to derive a contradiction. This type of problem is significant in real analysis and number theory, and it demonstrates the properties and behaviors of different types of numbers when combined in various operations.
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