Question

6. Prove that each of the following numbers is
irrational :
(ii) 3 - √2
(iii) √5 - 2​

Answer 1

Answer:

3 - √2- irrational

√5 - 2​ irrational

Step-by-step explanation:

Let 3 + √2 be rational.  Hence, 3 and 3 + √2 are rational.  

∴ 3 + √2 – 3 = √2 = rational [∵Difference of two rational is rational]  

This contradicts the fact that √2 is irrational.  

The contradiction arises by assuming 3 + √2 is rational.  

Hence, 3 + √2 is irrational.

√5 - 2​- irrational

Irrational. Sum of a rational and an irrational number is always irrational.

Proof by contradiction:

Let's say x is a rational no and y is an irrational no

Suppose: x+y is rational

Let x+y = z

So z is supposedly a rational no.

Now sum of 2 rational numbers is always rational.

Let's add -x to z, as x is rational so -x is also rational.

z-x = x+y-x = y

But we know that y is irrational.

Hence x+y must be irrational.

Hope this helps