Consider the statement: "The sum of any two rational numbers is a rational number." Is it true or false? If True, prove it directly. If False, provide a conterexample.
Consider the statement: "The sum of any two rational numbers is a rational number." Is it true or false? If True, prove it directly. If False, provide a conterexample.
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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![**Title:** Understanding Rational Numbers
**Statement:** "The sum of any two rational numbers is a rational number." Is it true or false? If true, prove it directly. If false, provide a counterexample.
**Analysis:**
To determine the truth of this statement, recall the definition of a rational number. A rational number is any number that can be expressed in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
**Proof:**
1. Let's consider two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \).
2. To find their sum, use the formula for adding fractions:
\[
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
\]
3. Here, \( ad + bc \) and \( bd \) are both integers (since they are results of integer operations: addition and multiplication).
4. Therefore, \( \frac{ad + bc}{bd} \) is a rational number because both the numerator and the denominator are integers and the denominator is not zero.
**Conclusion:**
The sum of any two rational numbers is indeed a rational number. The statement is true, as demonstrated by the above proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7803273c-2063-4cef-bfbb-565e22329fee%2Ff0bda1e7-6b03-4c44-bd2e-c1145e3c832b%2Fd7ad6dc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title:** Understanding Rational Numbers
**Statement:** "The sum of any two rational numbers is a rational number." Is it true or false? If true, prove it directly. If false, provide a counterexample.
**Analysis:**
To determine the truth of this statement, recall the definition of a rational number. A rational number is any number that can be expressed in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
**Proof:**
1. Let's consider two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \).
2. To find their sum, use the formula for adding fractions:
\[
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
\]
3. Here, \( ad + bc \) and \( bd \) are both integers (since they are results of integer operations: addition and multiplication).
4. Therefore, \( \frac{ad + bc}{bd} \) is a rational number because both the numerator and the denominator are integers and the denominator is not zero.
**Conclusion:**
The sum of any two rational numbers is indeed a rational number. The statement is true, as demonstrated by the above proof.
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