Assume that 2√3 - 1 is rational
2√3 = a/b , where a and b are integers .
⇒ 2√3 = a/b + 1
⇒ 2√3 = a/b + b/b
⇒ √3 = (a + b) / 2b
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 - 1 is an irrational number.