Question

prove that √5 is irrational.​

Answer 1

Let us assume that √5 is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are intezers.

so, √5 = p/q

p = √5q

we know that 'p' is a rational number. so √5 q must be rational since it equals to p

but it doesnt occurs with √5 since its not an intezer

therefore, p =/= √5q

this contradicts the fact that √5 is an irrational number

hence our assumption is wrong and √5 is an irrational number.

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Answer 2

Answer:

let us assume that√5 is a rational number.

then we can write itin the form of p/q where q is not= to 0

also p and q are co-primes

so, √5=p/q

=> 5= p^2/q^2. [on squaring both the sides]

=>5q^2= p^2. -------eqn 1

here p is divisible by 5

Now let p=5a. [where a= +ve integer]

Again,

on squaring both sides,

p^2=(5a)^2

=25a^2. --------eqn 2

now equating eqn 1 n 2

=> 5q^2=25a^2

=> q^2=5a^2

q is divisible by 5

from this we can say that p and q have common factor i.e.5

this contradicts our assumption that p and q are co-primes

Hence,√5 is an irrational number

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