What is
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
What is r
![**Prove the following statement by contradiction:**
*If \(a\) and \(b\) are rational numbers, \(b \neq 0\), and \(r\) is an irrational number, then \(a + br\) is irrational.*
**Proof by contradiction:** Select an appropriate statement to start the proof.
- Suppose not. That is, suppose there exist irrational numbers \(a\) and \(b\) such that \(b \neq 0\), \(r\) is a rational number, and \(a + br\) is rational.
- Suppose not. That is, suppose there exist irrational numbers \(a\) and \(b\) such that \(b \neq 0\), \(r\) is an irrational number, and \(a + br\) is rational.
- Suppose not. That is, suppose there exist rational numbers \(a\) and \(b\) such that \(b \neq 0\), \(r\) is an irrational number, and \(a + br\) is rational.
- **Suppose not. That is, suppose there exist rational numbers \(a\) and \(b\) such that \(b \neq 0\), \(r\) is an irrational number, and \(a + br\) is rational.** ✔️
**Then by definition of rational:**
\[a = \frac{c}{d}, \quad b = \frac{i}{j}, \quad \text{and } a + br = \frac{m}{n} \]
where \(c, d, i, j, m, \) and \(n\) are **integers** ✔️ and \([d \neq 0, j \neq 0, \text{and } n \neq 0]\) ✔️. Since \(b \neq 0\), we also have that \(i \neq 0\). By substitution,
\[ \frac{c}{d} + \frac{i}{j}r = \frac{m}{n} \]
Solving this equation for \(r\) and representing the result as a single quotient in terms of \(c, d, i, j, m, \) and \(n\) gives that
\[ r = \]
Note that \(r\) **is** ✔️ a ratio of two integers because products and differences of integers **are** ✔️](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96c2bf0d-d0d0-4024-bc9b-590e3c9ad5ff%2F18f3d163-15f8-4a55-8860-7cc254533fb9%2F7erd41q_processed.png&w=3840&q=75)
Transcribed Image Text:**Prove the following statement by contradiction:**
*If \(a\) and \(b\) are rational numbers, \(b \neq 0\), and \(r\) is an irrational number, then \(a + br\) is irrational.*
**Proof by contradiction:** Select an appropriate statement to start the proof.
- Suppose not. That is, suppose there exist irrational numbers \(a\) and \(b\) such that \(b \neq 0\), \(r\) is a rational number, and \(a + br\) is rational.
- Suppose not. That is, suppose there exist irrational numbers \(a\) and \(b\) such that \(b \neq 0\), \(r\) is an irrational number, and \(a + br\) is rational.
- Suppose not. That is, suppose there exist rational numbers \(a\) and \(b\) such that \(b \neq 0\), \(r\) is an irrational number, and \(a + br\) is rational.
- **Suppose not. That is, suppose there exist rational numbers \(a\) and \(b\) such that \(b \neq 0\), \(r\) is an irrational number, and \(a + br\) is rational.** ✔️
**Then by definition of rational:**
\[a = \frac{c}{d}, \quad b = \frac{i}{j}, \quad \text{and } a + br = \frac{m}{n} \]
where \(c, d, i, j, m, \) and \(n\) are **integers** ✔️ and \([d \neq 0, j \neq 0, \text{and } n \neq 0]\) ✔️. Since \(b \neq 0\), we also have that \(i \neq 0\). By substitution,
\[ \frac{c}{d} + \frac{i}{j}r = \frac{m}{n} \]
Solving this equation for \(r\) and representing the result as a single quotient in terms of \(c, d, i, j, m, \) and \(n\) gives that
\[ r = \]
Note that \(r\) **is** ✔️ a ratio of two integers because products and differences of integers **are** ✔️
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)