are irrational. Prove that √11-√√7 is irrational." Prove indirectly: "It is known that √7 and √11 a
are irrational. Prove that √11-√√7 is irrational." Prove indirectly: "It is known that √7 and √11 a
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
![**Indirect Proof Exercise**
**Problem Statement:**
"It is known that √7 and √11 are irrational. Prove that √11 - √7 is irrational."
**Objective:**
Provide an indirect proof to show that the difference between the square roots of 11 and 7 is irrational.
**Key Concepts:**
- **Irrational Numbers:** Numbers that cannot be expressed as a fraction of two integers.
- **Indirect Proof:** Also known as proof by contradiction, is a proof technique that establishes the truth of a proposition by assuming that the proposition is false and then showing that this assumption leads to a contradiction.
**Instruction:**
To solve the problem using an indirect proof, follow these steps:
1. **Assume the Opposite:**
- Assume that √11 - √7 is rational.
2. **Express in Fractional Form:**
- Since we are assuming √11 - √7 to be rational, it can be expressed as a fraction of two integers, i.e., \( √11 - √7 = \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b ≠ 0 \).
3. **Manipulate the Assumption:**
- Rearrange the equation to isolate one of the square roots: \( √11 = √7 + \frac{a}{b} \).
4. **Square Both Sides:**
- Square both sides of the equation to eliminate the square roots:
\[
11 = (√7 + \frac{a}{b})^2
\]
- Expand the right-hand side:
\[
11 = 7 + \frac{2a√7}{b} + \frac{a^2}{b^2}
\]
5. **Isolate the Irrational Part:**
- Rearrange the equation to isolate the term containing √7:
\[
11 - 7 - \frac{a^2}{b^2} = \frac{2a√7}{b}
\]
\[
4 - \frac{a^2}{b^2} = \frac{2a√7}{b}
\]
6. **Contradiction:**
- Notice that the left side of the equation, \( 4 - \frac{a^2}{b^2} \), is rational](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46272ace-8a35-41ec-9b9a-0375bd37bb55%2F3ba729c7-1383-44ae-a57c-48cd45d49d14%2Fq0ielse_processed.png&w=3840&q=75)
Transcribed Image Text:**Indirect Proof Exercise**
**Problem Statement:**
"It is known that √7 and √11 are irrational. Prove that √11 - √7 is irrational."
**Objective:**
Provide an indirect proof to show that the difference between the square roots of 11 and 7 is irrational.
**Key Concepts:**
- **Irrational Numbers:** Numbers that cannot be expressed as a fraction of two integers.
- **Indirect Proof:** Also known as proof by contradiction, is a proof technique that establishes the truth of a proposition by assuming that the proposition is false and then showing that this assumption leads to a contradiction.
**Instruction:**
To solve the problem using an indirect proof, follow these steps:
1. **Assume the Opposite:**
- Assume that √11 - √7 is rational.
2. **Express in Fractional Form:**
- Since we are assuming √11 - √7 to be rational, it can be expressed as a fraction of two integers, i.e., \( √11 - √7 = \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b ≠ 0 \).
3. **Manipulate the Assumption:**
- Rearrange the equation to isolate one of the square roots: \( √11 = √7 + \frac{a}{b} \).
4. **Square Both Sides:**
- Square both sides of the equation to eliminate the square roots:
\[
11 = (√7 + \frac{a}{b})^2
\]
- Expand the right-hand side:
\[
11 = 7 + \frac{2a√7}{b} + \frac{a^2}{b^2}
\]
5. **Isolate the Irrational Part:**
- Rearrange the equation to isolate the term containing √7:
\[
11 - 7 - \frac{a^2}{b^2} = \frac{2a√7}{b}
\]
\[
4 - \frac{a^2}{b^2} = \frac{2a√7}{b}
\]
6. **Contradiction:**
- Notice that the left side of the equation, \( 4 - \frac{a^2}{b^2} \), is rational
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