What is the main weakness in proving (\sqrt{3}) irrational using an approximate decimal?
Answer and explanation
Correct answer: It does not give a complete proof
An approximate decimal is useful for estimating the size of \(\sqrt{3}\), but it cannot prove whether the number is rational or irrational. Any finite decimal is rational, and a displayed decimal approximation leaves infinitely many possible later digits. Even a long nonterminating-looking calculation cannot establish an exact mathematical property by itself. A proof needs a conclusion that follows with certainty.
The standard proof assumes \(\sqrt{3}=p/q\) in lowest form and obtains an equation such as \(p^2=3q^2\). Divisibility by 3 then forces a corresponding divisibility conclusion for the integers, eventually making both numerator and denominator divisible by 3. That contradicts lowest form. Therefore option A is correct: approximation alone does not provide a complete proof.
Frequently asked questions
What is the correct answer to this question?
It does not give a complete proof
Why is this the correct answer?
An approximate decimal is useful for estimating the size of \(\sqrt{3}\), but it cannot prove whether the number is rational or irrational. Any finite decimal is rational, and a displayed decimal approximation leaves infinitely many possible later digits. Even a long nonterminating-looking calculation cannot establish an exact mathematical property by itself. A proof needs a conclusion that follows with certainty.
The standard proof assumes \(\sqrt{3}=p/q\) in lowest form and obtains an equation such as \(p^2=3q^2\). Divisibility by 3 then forces a corresponding divisibility conclusion for the integers, eventually making both numerator and denominator divisible by 3. That contradicts lowest form. Therefore option A is correct: approximation alone does not provide a complete proof.
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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