5.  Irrational number is any real number that cannot be expressed as a fraction p/q, where p is an integer and b is a non – zero integer.  Johann Heinrich Lambert proved in the 18 th Century that π is irrational

8. If statement T is true, then our original assertion in statement S is false. So, let us assume that statement T is true, and see if we can find a contradiction. If π is rational, then we can define integers a and b such that where a and b are integers with no common factors.

9. Now, let us define a function f(x) as follows: In this expression, n is another integer that we won’t specify at the present time – we’ll choose it later on.

10. Now, if we make the variable transformation

11. we can see that our function f(x) has the following property: Now let us define another function in terms of the derivatives of f(x):

12. Differentiating this function twice and adding, we can see that.

13. since derivatives of f(x) of order higher than 2n are zero. In order to proceed, we want to prove an important property about the function f(x) and its derivatives. Let’s begin by noting that we can write the function f(x) as follows: where the c coefficients take integral values.

14. Therefore, any derivative of f(x) lower than the n th derivative is zero at x = 0. The (n+J) derivative (J ≤ n), at x = 0, is simply equal to

15. We therefore conclude that the derivatives of f(x), evaluated at x = 0, are either zero or take integral values. This follows by noting that the quotient of the two factorials in the above expression is an integer. Further, from (1), we have that

16. We can therefore further conclude that the derivatives of f(x) evaluated at x = π are also either zero or take integral values. With all this in mind, we deduce that

17. Now by using very simple differential calculus, it is easy to show that and hence by integrating both sides of this expression, and noting that

18. we find that This has an integral value, as we have just shown that the two terms on the right-hand side of the above expression take integral values.

19. But, let us go back to our original definition of f(x). By inspection of the definition of f(x), it can be seen that and hence

20. Side note: This provides us with the contradiction that we seek. Look back at the integral (A). We’ve just shown that it takes an integral value. And yet, result (B) shows that the integrand can be made as small as we like, by choosing a sufficiently large value of n. Therefore, on the basis of (B), we can make the integral (A) as small as we like. The point is that we cannot have both situations at once, namely that the integral (A) is both an integer and arbitrarily small. This is an impossibility, and so the statement T is false. Hence, statement S is true, and π is an irrational number.

21. So the statement T is false. Hence, statement S is true, Therefore π is an irrational number. Q.E.D

23. Approximation of π Some fractions are used such as 22/7= 3.142857 142857 142857 333/106= 3.1415094339622641509 355/113= 3.1415929203539823008 52163/16604= 3.14159238737653577451 103993/33102= 3.14159265301190260407 π = 3.1415926535897932384626433 * 355/113 known in the 5 th Century in China *

25. -Widely known is the Pi – Day occurred every year on March 14 - Palais de la Découverte a science museum in Paris, contains a circular room called “Pi Room”Palais de la Découverte - On the wall it has 707 of Pi - Based on 1853 calculation by William Shanks. - It includes an error in the 528 th digits (detected in 1946, corrected in 1949)

26. MIT Cheer: I'm a Beaver, you're a Beaver, we are Beavers all. And when we get together, we do the Beaver call. E to the U du dx, E to the X dx. Cosine, secant, tangent, sine, 3.14159. Integral radical mu dv Slipstick, sliderule, MIT. Go Tech!

27. http://mathworld.wolfram.com/PiApproximations.html A proof that is irrational by Gilles Cazelais http://mathforum.org/library/drmath/view/57543.html