Why is divisibility reasoning necessary instead of approximate decimal in the proof of (\sqrt{3})?
Answer and explanation
Correct answer: Because divisibility reasoning gives a complete proof
An approximate decimal tells us only that a number is close to a displayed value. It does not describe all of its digits and therefore cannot establish an exact claim about rationality. To prove that \(\sqrt{3}\) is irrational, we need an argument that works for every possible fraction representing it, not just a numerical estimate. Divisibility gives that exact structure.
Assume \(\sqrt{3}=p/q\) in lowest form, where \(q\neq0\). Squaring gives \(p^2=3q^2\), so 3 divides \(p^2\), which implies that 3 divides \(p\). Substituting \(p=3k\) then shows that 3 divides \(q\) as well. Both numbers would share 3, contradicting lowest form. Thus option A is correct.
Frequently asked questions
What is the correct answer to this question?
Because divisibility reasoning gives a complete proof
Why is this the correct answer?
An approximate decimal tells us only that a number is close to a displayed value. It does not describe all of its digits and therefore cannot establish an exact claim about rationality. To prove that \(\sqrt{3}\) is irrational, we need an argument that works for every possible fraction representing it, not just a numerical estimate. Divisibility gives that exact structure.
Assume \(\sqrt{3}=p/q\) in lowest form, where \(q\neq0\). Squaring gives \(p^2=3q^2\), so 3 divides \(p^2\), which implies that 3 divides \(p\). Substituting \(p=3k\) then shows that 3 divides \(q\) as well. Both numbers would share 3, contradicting lowest form. Thus option A is correct.
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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