## happy pi day!

Is it really Pi Day again already? Happy 3.1415926535897932384626433832795028841971. Happy 3.1415926535897932384626433832795028841971, *everyone*. I considered watching the movie *Pi* (which I love, even though it doesn’t have much to do with the number), but I couldn’t find my DVD. So instead, I watched *Vanishing Point*, which has fast cars and a naked hippie chick on a motorcycle.

Anyway, here is a proof of the irrationality of π. (Proof of the transcendence of is *π* left as an exercise for the reader. Hint: first prove *e* is transcendental, then use Euler’s formula!)

In other news, the last few years of grad school have led me to the conclusion that advanced math is essentially a black hole into which time and self-esteem are sucked, and from which nothing good ever escapes.

Assume π = *a*/*b* with positive integers *a* and *b*.

Now, for some natural number *n* define the functions *f* and *F* as follows. Strictly speaking, *f* and *F* should each have *n* as an index as they depend on *n* but this would render things unreadable; remember that *n* is always the same constant throughout this proof.

Let

and let

*F(x)* = *f(x)* + … + (-1)^{j}*f*^{(2j)}*(x)* + … + (-1)^{n}*f*^{(2n)}*(x)*

where *f*^{(2j)} denotes the 2*j*-th derivative of *f*.

Then *f* and *F* have the following properties:

*f* is a polynomial with coefficients that are integer, except for a factor of 1/*n*!

*f(x)* = *f(π-x)*

0 <= *f(x)* <= π^{n}a^{n}/*n*! for 0 <= *x* <= π

For 0 <= *j* < *n*, the *j*-th derivative of *f* is zero at 0 and π.

For *n* <= *j*, the *j*-th derivative of *f* is integer at 0 and π

(inferred from (1.)).

*F*(0) and *F(π)* are integer (inferred from (4.) and (5.)).

*F* + *F* ” = *f*

(*F* ‘·sin – *F*·cos)’ = *f*·sin (inferred from (7.))

Hence, the integral over *f*·sin, taken from 0 to π, is integer.

For sufficiently large *n*, however, inequality (3.) tells us that this integral must be between 0 an 1. Hence, we could have chosen *n* such that the assumption is led *ad absurdum*.

*Blatantly stolen from here.*

This was written by

Eric, your haikuist. Posted on

Saturday, March 14, 2009, at 10:58 pm. Filed under

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