Question

Prove that √3+√2 is an irrational number

Answer 1

Answer:Sol :

Suppose √3 - √2 is rational .

Let √3 - √2 = r       where r is a rational.

∴  (√3 - √2)2 = r2

∴  2 + 3 - 2√6 = r2

∴√6 = (5 - r2 ) / 2

Now , LHS = √6 is an irrational number .

RHS =  (5 - r2 ) / 2  But rational number cannot be equal to an irrational.

∴our supposition is wrong.

∴ √3 - √2 is irrational .

Answer 2

Let √3 is a rational number . So, two integers a and b can be found so that √3 = a/b
Assume that a and are co - prime.
therefore a = √3b
squaring both sides,
therefore a^2 = 3b^2
So, a^2 is divisible by 3 and it can be said that a is divisible by 3
Let a^2 = 3c , where c is an integer.
a^2 = 3b^2
therefore, (3c)^2 = 3b^2
b^2 = 3c^2
So,b^2 is divisible by 3 andbit can be said that b is divisible by 3
This means that a and b have 3 as a common factor which is a contradiction to fact that a and b are co-prime.
Hence,√3 cannnot be expressed a p/q or √3 is irrational.
Similarly,√2 is irrational. The sum of two iiratiobal numbers is an irrational number.
Therefore, √3 + √2 is irrational number.