Question

prove that root 5is irrational​

Answer 1

Step-by-step explanation:

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.

hope it helped u :)

Answer 2

Answer- The above question is from the chapter 'Real Numbers'.

Given question: Prove that √5 is an irrational number.

Solution: Let us suppose that √5 is a rational number.

√5 = p/q (where p,q are co-primes, q≠0)

Transposing q to L.H.S., we get,

√5q = p

Squaring both sides, we get,

5q² = p² --- (1)

5 is a factor of p².

Using Fundamental Theorem of Arithmetic, we get,

5 is also a factor of p.

Put p = 5c.

Put the value of p in equation 1, we get,

5q² = 25p²

⇒ q² = 5p²

5 is a factor of q².

Using Fundamental Theorem of Arithmetic, we get,

5 is also a factor of q.

⇒ √5  is a rational number.

This contradicts the statement that p and q are co-primes and √5 is a rational number.

∴ We arrive at a wrong result due to our wrong assumption that √5  is a rational number.

∴ √5 is not an irrational number.