is root 7 a rational or irrational number
Answers
Step-by-step explanation:
ROOT 7 IS IRRATIONAL NUMBER
Answer:
Let us assume that √7 be rational.
Then it must in the form of p / q [q ≠ 0] [p and q are co-prime]
√7 = p / q
=> √7 x q = p
[Squaring on both sides]
=> 7q² = p²
=> 7 | p²
=> 7 | p ----------------(1)
Since, p is divisible by 7 there would exist a natural number c such that p = 7c [c is a positive integer]
p = 7c
[Squaring on both sides]
p² = 49 c²
Subsituting the above equation in p²=7q²,
=> 49c² = 7q²
[Dividing by 7 on both sides]
=> 7c² = q²
=> 7 | q²
=> 7 | q --------------------(2)
From (1) and (2), we find that q and p have a common factor(i.e 7).
But this is contrary to our assumption that p & q are co prime but it has a common factor.
So our initial assumption is wrong and √7 is irrational.
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