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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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 7RE
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**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
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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