What kind of beginning is used in the proofs of both √2 and √3?
Answer and explanation
Correct answer: They are first assumed rational
Both standard proofs use contradiction. To establish that √2 or √3 is irrational, the proof begins by assuming the opposite: the number is rational. A rational number can be represented as a/b, where a and b are integers, b is non-zero, and the fraction is in lowest form. For √2, this leads to a² = 2b²; for √3, it leads to a² = 3b². In each case, divisibility arguments eventually force both a and b to share a factor, contradicting their lowest-form condition. Hence option A is correct. The numbers are not initially assumed to be integers, zero, or negative; those descriptions do not express the required contrary assumption.
Frequently asked questions
What is the correct answer to this question?
They are first assumed rational
Why is this the correct answer?
Both standard proofs use contradiction. To establish that √2 or √3 is irrational, the proof begins by assuming the opposite: the number is rational. A rational number can be represented as a/b, where a and b are integers, b is non-zero, and the fraction is in lowest form. For √2, this leads to a² = 2b²; for √3, it leads to a² = 3b². In each case, divisibility arguments eventually force both a and b to share a factor, contradicting their lowest-form condition. Hence option A is correct. The numbers are not initially assumed to be integers, zero, or negative; those descriptions do not express the required contrary assumption.
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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