If c/d is not taken in lowest form in the proof of √2, which conclusion will not remain decisive?
Answer and explanation
Correct answer: Contradiction when both are even
The governing idea is the role of lowest terms in a contradiction proof. If √2 = c/d, squaring still gives c² = 2d² whether or not the fraction is reduced. The positivity statement √2 > 0 is also unaffected. However, the conclusion that both c and d are even is decisive only when gcd(c,d) = 1 was assumed at the start. Without lowest terms, a fraction may legitimately have an even numerator and denominator, such as 2/4, so their common divisibility is not itself a contradiction. Therefore option A is correct. The other choices describe steps that remain valid independently of reduction.
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What is the correct answer to this question?
Contradiction when both are even
Why is this the correct answer?
The governing idea is the role of lowest terms in a contradiction proof. If √2 = c/d, squaring still gives c² = 2d² whether or not the fraction is reduced. The positivity statement √2 > 0 is also unaffected. However, the conclusion that both c and d are even is decisive only when gcd(c,d) = 1 was assumed at the start. Without lowest terms, a fraction may legitimately have an even numerator and denominator, such as 2/4, so their common divisibility is not itself a contradiction. Therefore option A is correct. The other choices describe steps that remain valid independently of reduction.
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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