In the proofs of √2 and √3, what should not be treated as the basis of proof?
Answer and explanation
Correct answer: Approximate decimal value
A mathematical proof requires an exact chain of justified statements, not merely numerical evidence. An approximate decimal value can suggest that √2 or √3 is not an integer, but any finite approximation can be close to a rational number and therefore cannot establish irrationality. The rigorous proofs assume a lowest-term rational fraction and then use equations such as x² = 2y² or h² = 3k² together with prime divisibility to obtain a contradiction. Thus option A is correct: the approximate decimal is useful for intuition, not as the logical basis. Options B, C, and D are essential structural components of the standard proofs.
Frequently asked questions
What is the correct answer to this question?
Approximate decimal value
Why is this the correct answer?
A mathematical proof requires an exact chain of justified statements, not merely numerical evidence. An approximate decimal value can suggest that √2 or √3 is not an integer, but any finite approximation can be close to a rational number and therefore cannot establish irrationality. The rigorous proofs assume a lowest-term rational fraction and then use equations such as x² = 2y² or h² = 3k² together with prime divisibility to obtain a contradiction. Thus option A is correct: the approximate decimal is useful for intuition, not as the logical basis. Options B, C, and D are essential structural components of the standard proofs.
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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