Why are a and b assumed to be coprime in √3 = a/b?
Answer and explanation
Correct answer: Because the fraction is written in lowest form
If √3 were rational, it could be expressed as a/b, with a and b integers, b ≠ 0. Any rational fraction can be reduced by cancelling common factors, so we may choose a representation in lowest form. In that form, a and b are coprime: their greatest common divisor is 1. This condition is essential because the proof later shows that 3 divides a and then, from a² = 3b², also 3 divides b. That would give a and b a common factor 3, contradicting their assumed lowest form. Thus option A is correct. The other choices make unsupported claims and are not properties of every rational representation.
Frequently asked questions
What is the correct answer to this question?
Because the fraction is written in lowest form
Why is this the correct answer?
If √3 were rational, it could be expressed as a/b, with a and b integers, b ≠ 0. Any rational fraction can be reduced by cancelling common factors, so we may choose a representation in lowest form. In that form, a and b are coprime: their greatest common divisor is 1. This condition is essential because the proof later shows that 3 divides a and then, from a² = 3b², also 3 divides b. That would give a and b a common factor 3, contradicting their assumed lowest form. Thus option A is correct. The other choices make unsupported claims and are not properties of every rational representation.
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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