In both proofs in what form is the number first written?
Answer and explanation
Correct answer: In lowest fraction form
To prove that a square root such as \(\sqrt{2}\) or \(\sqrt{3}\) is irrational, the proof begins by assuming the opposite: that the number is rational. Every rational number can be written as a fraction \(\frac{m}{n}\), where m and n are integers, n is nonzero, and the fraction is in lowest terms. The lowest-terms condition means m and n have no common factor.
This form is essential because the later argument shows that both m and n must be divisible by the same number, usually 2 for \(\sqrt{2}\) or 3 for \(\sqrt{3}\). That contradicts their being coprime. A decimal or percentage form does not provide this useful coprime condition. Therefore option A, the simplest fraction form, is correct.
Frequently asked questions
What is the correct answer to this question?
In lowest fraction form
Why is this the correct answer?
To prove that a square root such as \(\sqrt{2}\) or \(\sqrt{3}\) is irrational, the proof begins by assuming the opposite: that the number is rational. Every rational number can be written as a fraction \(\frac{m}{n}\), where m and n are integers, n is nonzero, and the fraction is in lowest terms. The lowest-terms condition means m and n have no common factor.
This form is essential because the later argument shows that both m and n must be divisible by the same number, usually 2 for \(\sqrt{2}\) or 3 for \(\sqrt{3}\). That contradicts their being coprime. A decimal or percentage form does not provide this useful coprime condition. Therefore option A, the simplest fraction form, 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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