show that √Π+1 is irrational
If it were rational, then it follows that pi must be rational. However, we know that pi is irrational, therefore sqrt(pi) + 1 must be irrational.
is there any proof
Show that the reciprocal of 3+2√2 is an irrational number
To show that √π+1 is irrational, we need to prove that it cannot be expressed as a fraction of two integers.
One approach is to use proof by contradiction. Assume that √π+1 is rational, which means it can be written as a fraction in the form p/q, where p and q are integers with no common factors other than 1 and q ≠ 0.
We can square both sides of the equation (√π+1)^2 = (p/q)^2:
π + 2√π + 1 = p^2/q^2
π + 1 = p^2/q^2 - 2√π
Now, let's consider the right-hand side of the equation. The term p^2/q^2 is rational since it is the ratio of two integers. Since √π is irrational, 2√π is also irrational. Therefore, p^2/q^2 - 2√π is a sum of a rational number and an irrational number, which must be an irrational number.
Now, look at the left-hand side of the equation. We know that π is irrational, and 1 is rational. The sum of an irrational number and a rational number is irrational.
Since the right-hand side is irrational and the left-hand side is irrational, we have a contradiction. Our assumption that √π+1 is rational must be false.
Therefore, √π+1 is irrational.