Prove that (2 +√3) is irrational.

Question

Philoid · Accepted Answer

√3 is an irrational number and addition of a rational number and an irrational number is an irrational number.

∴, 2 + √3 is irrational, where √3 is an irrational number while 2 is a rational number.

Alternative - Assume that 2 + √3 is rational
2 + √3 = a/b, where a and b are integers.
⇒ √3 = a/b - 2
⇒ √3 = a/b – 2b/b
⇒ √3 = (a - 2b) / b
we know that a, b and 2 are integers and they are also rational {i.e RHS is rational}
therefore √3 will be rational.
but we know that √3 is irrational.
there is a contradiction
so, 2 + √3 is an irrational number