Question
Prove that √7 is an irrational number.
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Mathematics
10 months
2021-07-07T15:51:52+00:00
2021-07-07T15:51:52+00:00 2 Answers
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Answer:
Given √7
To prove: √7 is an irrational number.
Proof:
Let us assume that √7 is a rational number.
So it t can be expressed in the form p/q where p,q are co-prime integers and q≠0
√7 = p/q
Here p and q are coprime numbers and q ≠ 0
Solving
√7 = p/q
On squaring both the side we get,
=> 7 = (p/q)2
=> 7q2 = p2……………………………..(1)
p2/7 = q2
So 7 divides p and p and p and q are multiple of 7.
⇒ p = 7m
⇒ p² = 49m² ………………………………..(2)
From equations (1) and (2), we get,
7q² = 49m²
⇒ q² = 7m²
⇒ q² is a multiple of 7
⇒ q is a multiple of 7
Hence, p,q have a common factor 7. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number
√7 is an irrational number.
To prove :
is irrational.
Proof :
Assume that
is rational.
7 and b² is a factor of a², which means 7 is also a factor of a.
Substitute Eq. 2 in Eq. 1
7 and c² are factors of b². Which means 7 is a factor of b too.
7 is a common factor of a and b. But they were co prime numbers. This is a contradiction. The contradiction was arisen due to wrong assumption.
Hence proved,
is irrational.