Which option gives the correct final sentence in the proof of √2?
Answer and explanation
Correct answer: Therefore our rational assumption is false and √2 is irrational
The proof begins by assuming, for contradiction, that √2 can be written as a rational number p/q in lowest form, where p and q are integers, q ≠ 0, and gcd(p,q) = 1. Rearranging and comparing prime factors shows that both p and q must be divisible by 2. That contradicts the choice that the fraction was already in lowest form. In a proof by contradiction, the contradiction rejects the original assumption, not the valid algebraic steps or the fact that √2 is real and positive. Hence the correct conclusion is that √2 is irrational. Option A states the opposite, while C and D do not express the logical conclusion.
Frequently asked questions
What is the correct answer to this question?
Therefore our rational assumption is false and √2 is irrational
Why is this the correct answer?
The proof begins by assuming, for contradiction, that √2 can be written as a rational number p/q in lowest form, where p and q are integers, q ≠ 0, and gcd(p,q) = 1. Rearranging and comparing prime factors shows that both p and q must be divisible by 2. That contradicts the choice that the fraction was already in lowest form. In a proof by contradiction, the contradiction rejects the original assumption, not the valid algebraic steps or the fact that √2 is real and positive. Hence the correct conclusion is that √2 is irrational. Option A states the opposite, while C and D do not express the logical conclusion.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.
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