Sunday, June 30, 2019
A proof that 22/7 - pi > 0 and more
My father was a High School English teacher who did not know much math. As I was going off to college, intending to major in math, he gave me the following sage advice:
1) Take Physics as well as Math since Physics and Math go well together. This was good advice. I took the first year of Physics for Physics Majors, and I later took a senior course in Mechanics since that was my favorite part of the first year course. Kudos to Dad!
2) π is exactly 22/7. I knew this was not true, but I also knew that I had no easy way to show him this. In fact, I wonder if I could have proven it myself back then.
I had not thought about this in many years when I came across the following:
Problem A-1 on the 1968 Putnam exam:
Prove 22/7 - π = ∫01 (x4(1-x)4)/(1+ x2 )dx
(I can easily do his by partial fractions and remembering that ∫ 1/(1+x^2) dx = tan-1x which is tan inverse, not 1/tan. See here.)
(ADDED LATER---I have added conjectures on getting integrals of the form above except with 4 replaced by any natural number. Be the first on your block to solve my conjectures! It has to be easier than the Sensitivity Conjecture!)
Let n ∈ N which we will choose later. By looking at the circle that is inscribed in a regular n-polygon (n even) one finds that
n tan(π/n) > π
So we seek an even value of n such that
n tan(π/n) < 22/7
Using Wolfram alpha the smallest such n is 92.
Would that convince Dad? Would he understand it? Probably not. Oh well.
Some misc points.
1) While working on this post I originally wanted to find tan(π/27) by using the half-angle formula many times, and get an exact answer in terms of radicals, rather than using Wolfram Alpha.
a) While I have lots of combinatorics books, theory of comp books, and more Ramsey Theory books than one person should own in my house, I didn't have a SINGLE book with any trig in it.
b) I easily found it on the web:
tan(x/2) = sqrt( (1-cos x)/(1+cos x) ) = sin x/(1+cos x) = (1-cos x)/(sin x).
None of these seems like it would get me a nice expression for tan(π/27). But I don't know. Is there a nice expression for tan(π/2k) ? If you know of one then leave a polite comment.
2) I assumed that there was a more clever and faster way to do the integral. I could not find old Putnam exams and their solutions the web (I'm sure they are there someplace! --- if you know then comment politely with a pointer). So I got a book out of the library The William Lowell Putnam Mathematical Competition Problems and Solutions 1965--1984 by Alexanderson, Klosinski, and Larson. Here is the clever solution:
The standard approach from Elementary Calculus applies.
Not as clever as I as hoping for.
3) I also looked at the integral with 4 replaced by 1,2,3,4,...,16. The results are in the writeup I pointed to before. It looks like I can use this sequence to get upper and lower bound on pi, ln(2), pi+2ln(2), and pi-2ln(2). I have not proven any of this. But take a look! And as noted above I have conjectures!
4) When I looked up INSCRIBING a circle in a regular n-polygon, Google kept giving me CIRCUMSCRIBING. Why? I do not know but I can speculate. Archimedes had a very nice way of using circumscribed circles to approximate pi. Its on youtube here. Hence people are used to using circumscribed rather than inscribed circles.