Question

Prove that √3+√2is an irrational number

Answer 1

The proof for root 2 being irrational is widely available, so I’ll take that as a given.

If 3 + root2 is rational then it is equal to a/b where a,b are integers (3+root2 = a/b)

Rearrange to get b*root2 = a - b*3. Integers are closed under subtraction and multiplication, so set c = a-b*3 as an integer.

Rearrange again to get root2 = c/b is a rational number, but root2 is irrational.

Proof by contradiction: 3 + root2 is irrational.

Answer 2

if possible let assume that √2+√3is irrational which is equal to a , where a is a rational no .

since,√3+√2=a

squaring both sides
a²=(√3+√2)²
a²=(√3)²+(√2)²+2×√3×√2
a²=3+2+2√6
a²-5/2=√6


since we know that difference and quotients of any rational number is rational but √6 is irrational.

which is contradiction hence ,√3+√2 is irrational