In the proof of √2, writing y ≠ 0 is necessary, but why does it not give the final contradiction?
Answer and explanation
Correct answer: It only keeps the fraction defined
The governing concept is the difference between a preliminary domain condition and the actual contradiction in a proof by contradiction. If √2 is represented as x/y, then y ≠ 0 is required simply because division by zero is undefined. This condition allows the fraction and the subsequent squaring step to make sense, but it does not imply anything about the parity or common factors of x and y. The decisive part comes later: from x² = 2y², one proves that x is even and then that y is even. That conflicts with choosing x/y in lowest terms, where gcd(x,y) = 1. Therefore option A is correct; the other options claim conclusions that do not follow from y ≠ 0.
Frequently asked questions
What is the correct answer to this question?
It only keeps the fraction defined
Why is this the correct answer?
The governing concept is the difference between a preliminary domain condition and the actual contradiction in a proof by contradiction. If √2 is represented as x/y, then y ≠ 0 is required simply because division by zero is undefined. This condition allows the fraction and the subsequent squaring step to make sense, but it does not imply anything about the parity or common factors of x and y. The decisive part comes later: from x² = 2y², one proves that x is even and then that y is even. That conflicts with choosing x/y in lowest terms, where gcd(x,y) = 1. Therefore option A is correct; the other options claim conclusions that do not follow from y ≠ 0.
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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