Johnny W. Yang

Johnny W. Yang

π Introduction π is irrational Approximation of π Use of π in culture

Known fact about π π is the ratio of the circumference of a circle to its diameter π is always the same number, no matter which circle you use to compute π is a very old number Known by the Egyptians (3.125) and the Babylonians (3.160) The symbol “π” was first used in 1706 by William Jones π was chosen as the letter to represent 3.141592 because π in Greek pronounce like our letter “p” means “perimeter”

What is π? is a mathematical constant that is the ratio of a circle circumference to its diameter In math, science, and engineering’s formula utilized π, which makes it one of the most important mathematical constant 2π in radian form is 360 degree so π is 180 degrees All the digits of π can never be fully known

π is Irrational 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 18th Century that π is irrational

Prove that “π is Irrational”

This proof is from a mathematician name Ivan Niven in 1947

The statement we need to prove is:S = The number π is irrational

The Inverse of the statement S is:T = The number π is rational

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.

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.

Now, if we make the variable transformation

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):

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

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.

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

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

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

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

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.

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

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.

So the statement T is false. 

Hence, statement S is true,

Therefore π is an irrational number.

Q.E.D

Approximation of π

Approximation of π

Some fractions are used such as 22/7 = 3.142857 142857 142857333/106 = 3.1415094339622641509355/113 = 3.141592920353982300852163/16604 = 3.14159238737653577451103993/33102 = 3.14159265301190260407

π = 3.1415926535897932384626433

* 355/113 known in the 5th Century in China *

π in Popular Culture

π in Popular Culture

-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”

- On the wall it has 707 of Pi- Based on 1853 calculation by

William Shanks.- It includes an error in the

528th

digits (detected in 1946, corrected in 1949)

π in Popular Culture

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!

•http://mathworld.wolfram.com/PiApproximations.html

•A proof that is irrational by Gilles Cazelais

•http://mathforum.org/library/drmath/view/57543.html

Sources