Question

prove that root 5 is irrational and also give that statement ​

Answer 1

5 is the common factor of p and q,

tfore out assumption was wrong

tfore root 5 is irrational

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Answer 2

Let us assume to contrary that √5 is rational number.

i.e., we can find a and b(≠0) such two positove coprime integers so that,

a/b = √5

a = √5b

squaring both sides,

a² = 5b² -(1)

now,

a²/5 = b²

clearly a² is divisible by 5

it means a is also divisible by 5

so,

a is divisible by 5 -(2)

we can find any integer q such that,

a = 5q

squaring both sides,

a² = 25q²-(3)

from (1) and (3),

25q² = 5b²

5q² = b²

b²/5 = q²

clearly,

b² is divisible by 5

it means,

b is divisible by 5-(4)

from (2) and (4),

a and b both are divisible by 5

it means a and b atleast must have 5 as common factor,

But, this contradicts the fact that a and b are co prime integers.

This contradiction has arisen due to our incorrect assumption that √5 is rational.

So, we conclude that √5 is irrational.