Pi is IRRATIONAL: simplest proof on toughest test

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In the last video of 2017 I showed you Lambert’s long but easy-to-motivate 1761 proof that pi is irrational. For today’s video Marty and I have tried to streamline an ingenious proof due to the famous French mathematician Charles Hermite into the hopefully simplest and shortest completely self-contained proof of the irrationality of pi. There are a few other versions of this proof floating around and we’ve incorporated the best ideas from these versions into what I’ll show you today; I’ll list some of these other versions below. I also talk about the problem of pi + e and pi x e being irrational at the end of the video, really nice stuff.

Articles to check out:

All these articles are great, but I think the one article that deserves most credit for having brought Hermite’s beautiful proof to the attention of the wider mathematical community is this article by Jan Stefvens: Zur Irrationalität von pi, Mitt. Math. Ges. Hamburg 18 (1999), 151-158. This one also has a very nice account of Lambert’s and Niven’s proofs.

In the video I mention that another version of this proof made an appearance in the toughest Australian maths exam in 2003; here is the link to this exam

As usual thanks to Marty and Danil for all their help with this video.

Enjoy!

[beirirangu]

"333 Only half evil" that's a great shirt, but I prefer:
"25.807... The root of all evil"

[DrThunder]

Host: "Everything so far has been pretty easy..."
Me: "IT HAS?!"

[Cubinator]

Proof that x^n/n! converges to zero (cheat version): The series e^x=sum(x^n/n!, n from 0 to infinity) converges so the sequence x^n/n! converges to zero. QED

[Mathologer]

In the last video of 2017 I showed you Lambert’s long but easy-to-motivate 1761 proof that pi is irrational. For today’s video Marty and I have tried to streamline an ingenious proof due to the famous French mathematician Charles Hermite into the hopefully simplest and shortest completely self-contained proof of the irrationality of pi. There are a few other versions of this proof floating around (see the description) and we’ve incorporated the best ideas from these versions into what I'll show you today, really nice stuff. Enjoy!
As usual if you'd like to help out with Mathologer please consider contributing subtitles in languages other than English and Russian :)

[ThomasClementAU]

Hey, fun fact, I got the highest mark on this exam in NSW. I enjoyed this question a lot!

It seemed funny / expected since the previous year's exam asked you to prove that e was irrational.

[markfrost]

I’m so glad that I have discovered your wonderful channel. As an economist who uses quite a bit of advanced mathematics, I not only find your videos edifying but exceptionally entertaining. Sometimes, I literally make some popcorn, and sit down and view them.

And I’m jealous of your shirts. :)

[michakuczynski]

Would really appreciate an 'algebra autopilot' for my exams ><

[Macieks]

I love math competitions, but when it has time constraints as small as 3 hours to do 8 problems it makes me vexed. For only one of the problems you'd have to do the thinking that took Mathologer 19 minutes to explain and then acutally write all of details that he quickly skipped and not make one mistake while doing it. All of it in under a half hour. Math in the real world is only constrained by time in the way that you only have limited time being alive.

[ais]

I know I'm a bit late but here's my take on lim as n-->inf of a^n/n!
And by "my" I mean "my memory of some guy on StackExchange's".

f(n)=a^n/n!
f(n+1)=a^(n+1)/(n+1)!

f(n+1)/f(n) = ( a^(n+1) *n!) / ( (n+1)! * a^n ) = ( a^n*a *n!) / ( (n+1)*n! * a^n ) = a/(n+1)

lim n-->inf a/(n+1) =0, therefore f(n) is decreasing in absolute value and it is decreasing until 0. Not sure if that last sentence has any logic holes in it but it looks good enough to me.

[BarakPearlmutter]

The proof of the corollary that either pi*e or pi+e is irrational did not appear to use any properties of e, so actually shows that either pi*x or pi+x is irrational, for any x. And the only property of pi that it used is that pi is not the solution of a quadratic polynomial with integer coefficients, which is true of any transcendental number. So this really establishes that for any transcendental number t, and any number x, with t*x or t+x must be irrational.

[i_am_anxious]

the reason x^n/n tends to zero is as follows;

x^n can only have factors of x. When n gets larger than x, it will constantly grow faster. You’ll have until x=n or until it’s x rounded down. as one fraction that will be a factor of all numbers that apply to x^n/n. Call that number z.

So, b steps after x rounded down plus one looks like this;



The bottom is larger terms than the top each time. As a result, whenever we add a new term after floor(x), it gets smaller, and thus after that point we have a constant times a decreasing number, ergo it tends to zero.

[TheQwerty]

If possible, i would like to watch a video where you read or analize a math paper. I think it can be interesting because if we had some more tools to understand those things, our self learning will increment

[daiconk]

What brought you to Australia? (Don't say "a plane." (If it was, I bet it was a complex plane.))

[timseguine]

The proof at the end: that generalises to any pair of irrational numbers where at least one is not algebraic, doesn't it?

[JacobGoodman]

The proof was quite beautiful and simple, but I agree that the motivation is very hard to decipher. I would love to see how Hermite constructed that sequence

[plasmaballin]

Why x^n/n! tends towards zero: When you add one to n, the numerator increases by a factor of x, but the denominator increases by a factor of whatever n now is (by the definitions of exponentiation and the factorial). Thus, adding ons to n is the same thing as multiplying the whole fraction by x/n. As n gets arbitrarily large, x/n approaches zero. This means the numbers in the sequence will get smaller everh time once n>x and approach zero.

[richardschreier]

Another magnificent piece of work from my favourite math mystic. You transform mathematics into performance art!  
I particularly like the argument that generalizes to "At least one of (pi+a) and (pi*a) is irrational, where a is any real number."  
I look forward to verifying the steps of Hermite's proof on my own!

[bernhardriemann]

I think that i'm able to proof that e/pi is irrational, is it possible to proof that e+pi or e*pi is irrational, if e/pi is irrational?

[peterflom]

You asked for interesting questions from exams. This was posed to me as an extra credit question

X + y + z = 1
X^2 + y ^2 + z^2 = 2
X^3 + y^3 + z^3 = 3

Either find x, y z or show it can't be done

[mattias]

I recently understood how big of a nerd I am when I started liking these videos