Question

Prove that

root 7

is irrational number.​

Answer 1

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$\implies$ 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 ------  (1)

p² is divisible by 7

p is divisible by 7

p = 7c  [c is a positive integer] [squaring on both sides ]

p² = 49 c² ---------   (2)

Subsitute p² in equ (1) we get,

7q² = 49 c²

q² = 7c²

=> q is divisible by 7

thus q and p have a common factor 7.

there is a contradiction

as our assumsion p & q are co prime but it has a common factor.

So that √7 is an irrational.

Answer 2

let us assume that √7 be rational. thus q and p have a common factor 7. as our assumsion p & q are co prime but it has a common factor. So that √7 is an irrational.

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