Question

Prove that ✓2+✓3 is irrational number

Answer 1

m/n = √2 + √3
Square both sides:
m^{2} / n^{2} = 5 + 2√6
"Solve" for √6:
√6 = (m^2 - 5n^2) / (2n^2)
so if √2 + √3 is rational, then so is √6.

Let a and b be the integers with gcd(a,b) = 1 such that
a/b = √6
Square both sides and multiply by b^2:
a^2 = 6b^2
Now, the right side is divisible by 2, so a^2 is divisible by 2, which then implies that a is divisible by 2 (since 2 is prime). Therefore we can write a=2k for some integer k:
4k^2 = (2k)^2 = 6b^2
Divide by 2:
2k^2 = 3b^2
Now the left side is divisible by 2, so 3b^2 is divisible by 2, from which it follows that b is divisible by 2.
However, this would mean that 2 divides gcd (a,b) = 1. Contradiction.

Therefore √6 is irrational, and therefore √2 + √3 is irrational also.
Hence, proved...


Thanks..