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

Let us assume, to the contrary that √5 is irrational.

That is, we can find integers a and b (not equal 0) such that √5= a/b

Suppose a and b have a common factor other than 1, then we can divide by the common factor and assume that a and b are co-prime.

So, b √5=a

Squaring on both sides and rearranging, we get

5b²=a² => 5 divides a² => 5 divides a

Let a=5m,where m is an Integer.

Substituting a=5m in 5b² = a², will get

5b² = 25m² => b² = 5m²

5 divides b² and so 5 divides b.

Therefore a and b have at least 5 as a common factor and the conclusion contradicts the hypothesis that a and b have no common factors other than 1.

This contradiction has arisen because of our incorrect assumption that √5 is rational.

So, we conclude that √5 is irrational.