Question

Prove that √ is an irrational number​

Answer 1

Answer:

3*6=189,4*8=448,1*5=?

Explanation:

free point

Answer 2

Let us assume, to the contrary, that $\sqrt{x}$ here x is 2 is rational.

So, we can find integers r and s (≠ 0) such that 2 =

r

s .

Suppose r and s have a common factor other than 1. Then, we divide by the common

factor to get 2 , a

b $\sqrt{x}$ where a and b are coprime.

So, b 2 = a.

Squaring on both sides and rearranging, we get 2b2

= a2

. Therefore, 2 divides a2

.

Now, by Theorem 1.3, it follows that 2 divides a.

So, we can write a = 2c for some integer c.

Substituting for a, we get 2b2

= 4c2

, that is, b2

= 2c2

.

This means that 2 divides b2

, and so 2 divides b (again using Theorem 1.3 with p = 2).

Therefore, a and b have at least 2 as a common factor.

But this contradicts the fact that a and b have no common factors other than 1.

This contradiction has arisen because of our incorrect assumption that 2 is rational

This is done if root 2 is to be proved irrational for any other number do the same with different values