Question

Show that √2 + 1 is not a rational number.

Answer 1

Let root(2) + 1 be rational say x/y where x,y belongs to Integers with HCF of (x,y) = 1.

so, root(2) + 1 = x/y

thus root(2) = (x-y)/y.

Also 'x-y' be some other integer 'a' and 'y' be 'b' such that the HCF of (a,b) = 1

thus, root(2) = a/b

Squaring both sides,

2 = a^2/b^2

Clearly, since a^2 and b^2 have a common factor of '2', thus HCF of a^2 and b^2 cannot be 1, which violates our initial assumption

hence root(2) +1 is not a rational number.

Answer 2

let √2+1 be rational number

√2+1=a/b

√2=a-b/b

as we know, √2=a/b

squaring both sides,

2=a²/b²

since a²and b² have a common factor of 2, thus HCF of a² and b² cannot be 1.

so, √2+1 is irrational number

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