Question

prove that root 5 is an irrational number​

Answer 1

Answer:

let us assume that √5 is rational

then √5 = a/b

b√5 = a

squaring on both sides

a² = 5b²

if 5 divides a then 5 also divides b

a = 5c

squaring on both sides

a²=25c²

5b²=25c²

b²=5c²

here a,b,c have 5 as common factor

so,5 is rational number

but √5 is irrational

this contradiction has arrisen due to our incorrect assumption

this contadicts the fact that√5 is irrational

hope it helps u

Answer 2

Answer:

Step-by-step explanation:

Let √5 be a rational number,

∴ √5 = p/q

sq. both side, we get$5  = p^{2}/q^{2} \\5q^{2} = p^{2} \\\\p^{2} is \ divided\  by\ 5  \\\\\\p is \ also\ divided \ by 5....(1)\\\\p= 5r\\\\p^{2} = 25r^{2} \\5q^{2} = 25r^{2}\\\\q^{2} = 5r^{2} \\\\q^{2} \ is\ divided \ by\ 5\\q is \also\ divide\ by\ 5 ....(2)$

From (1) and (2) it is clear that 5 is common factor of p and q ,so  underroot 5 is a rational number, but it is a contradiction so our assumption is wrong.

Hence, given number is irrational.