What is the purpose of taking r/s in lowest form in the proof that √3 is irrational?
Answer and explanation
Correct answer: To show a contradiction with the coprime assumption
The governing concept is proof by contradiction using a rational number in lowest terms. To assume √3 is rational, write it as r/s where r and s are integers, s is nonzero, and r and s are coprime. Squaring gives r² = 3s². From this relation, 3 divides r, so r = 3k; substitution then shows that 3 also divides s. Thus both r and s have 3 as a common factor, contradicting the original lowest-form condition that their HCF is 1. Option D correctly states this purpose. The aim is not to terminate a decimal, draw a diagram, or make the denominator zero. The lowest-form assumption is essential because without coprimality, common divisibility would not produce a contradiction.
Frequently asked questions
What is the correct answer to this question?
To show a contradiction with the coprime assumption
Why is this the correct answer?
The governing concept is proof by contradiction using a rational number in lowest terms. To assume √3 is rational, write it as r/s where r and s are integers, s is nonzero, and r and s are coprime. Squaring gives r² = 3s². From this relation, 3 divides r, so r = 3k; substitution then shows that 3 also divides s. Thus both r and s have 3 as a common factor, contradicting the original lowest-form condition that their HCF is 1. Option D correctly states this purpose. The aim is not to terminate a decimal, draw a diagram, or make the denominator zero. The lowest-form assumption is essential because without coprimality, common divisibility would not produce a contradiction.
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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