“We set pi equal to 3” Engineers: *applause*

The year is 3021. Computing power has finally advanced to the point that we can confirm that pi to the power of pi to the power of pi to the power of pi is not in fact an integer. The Intergalactic Society of Mathematics is hosting a party to celebrate. Suddenly, someone speaks up from the back of the room. "But what about pi to the power of pi to the power of pi to the power of pi _to the power of pi_ ? Is that an integer?" The room falls silent.

2:00 - calling them "irrationals" is indirect, since π or e are irrationals as well. Numbers like √2 are algebraic, an antonym to transcendental.

6:48 I love how Matt just casually referred to the two people as Emma and Timothy like if they were close friends

Why don't we calculate it in base π? π in base π is just 10, an integer! The only problem is that the good old integers are now transcendental.

I'm surprised that you didn't save this for March 14.

I remember the moment I realized what the word trigonometry meant..! I started looking at the word "polygon", meaning "several corners". I then thought of what a triangle would be called, "probably Tri-gon". Then it absolutely struck me, "Tri-gono-metry = The measurement of triangles"!

Great video. Although I expected some kind of argument for why we would expect this number to be an integer. But as I understand it, there is no reason to believe that it is anything in particular. We simply don't know. Although I am inclined to think it is probably not an integer, it is true that you can get integers or rational by operating irrationals and transcendentals in certain ways. But there is always, I think, a good explanation for it, it seems that you have to be deliberate about it. Kind of like when trying to convert rationals into integers, if you multiply randomly, you will fail in even a vast majority of cases, when multiplying by the inverse for instance, you succeed. But of course, I don't know much about it, it is just the impression I got from watching the video. Pretty interesting question.

This reminds me of 8 year old me trying to repeatedly multiply 9999 to itself in my calculator. I too was limited by the technology of my time.

My math teacher used to say, “if you don’t like natural logarithms just e-raise it. Then you don’t have to deal with it”

It's remarkable how modern mathematics can produce amazingly powerful and accurate results for physics, engineering, computing and essentiatially all fields of applied science, yet remarkly simple statements in number theory, combinatorics, transcendental number theory and other pure math branches are not only unproven but seem to be utterly unpproachable by every mean know to mathematicians today and many see no progress for decades, sometimes more ...

Thank you for making complicated math concepts fun and entertaining. Peace and Love Matt

Matt, can you please get closed captioning? I really appreciate your presentations and cannot tell what you are saying. The deaf community would benefit so much!

“Everyone remembers where they were when they noticed that” Ah, yes. This takes me back to two seconds ago.

This was such a fun video to watch. Definitely one of my favorites from Matt.

This video was amazing. So many fascinating thoughts. Absolutely loved it!❤

"We set pi equal to 3” I felt a great disturbance in the force.

"Say what you want about 3, at least we know it exactly. It's equal... to 3." This is what we call high-quality educational content.

Actually, we can apply number theory to this, in particular, Fermat's Little Theorem. We have methods of calculating the nth digit of pi in binary without having to calculate all the previous digits. In the appropriately chosen modulus, this is all you need to determine if the number is integer or not.

This is amazing. I love that you led with Tim Gowers' response, to reassure all the mathematicians in the audience: this isn't as simple as it might look, keep watching! 😅

√2 is the only irrational number in existence, now confirmed

Never before have I seen someone have so much fun with a stock studio audience, and I love it so much

I’ve watched this video 3 or 4 times since it came out. Great quality and fun video

Enjoy this channel immensely. Most of us need tutoring when it comes to mathematics.

As soon as I saw the title, I went to WolframAlpha, haha!

"How about we start by setting pi equal to 3..." What is this, stand-up engineering?

I wonder if this could be approached geometrically. I'm not sure what it would mean to raise a unit circle to the power of a unit circle, but with such of a conceptual tool, maybe it would be easier to figure out if it's sensible for pi^^3 to be an integer or not. If a unit circle raised to a power of itself, however that conceptualization presents itself, in some way increases its approximate proximity to a shape of non-transcendental volume, then it's conceivable that pi up-up-arrow x is an integer for some value of x. If the complexity of the resulting shape increases, and does so again when again raised to the power of the unit circle, perhaps we could conclude that it is not sensible for any x to yield pi ^^ x = integer.

I think it would be better to refer to the “irrationals” from the beginning of the video as constructables or algebraic instead of irrational, because transcendental numbers are also irrational but they aren’t constructable nor algebraic.

"For simplicity's sake, why don't we start with setting pi equal to 3." Engineers everywhere rejoiced

Matt: "It is complex..." Me: "Okay, explain it." Matt: "...literally." Me: "Oh."

Last year when I viewed this video, I brushed off jane street like I do with any ad I see in any video. Today, Jane Street is my absolute dream job and I would absolutely do anything for a job there. It is truly an amazing company. Lesson here, ads are not always that terrible.

A proof that there are no integers in the sequence π, π^π, π^π^π, … would certainly be interesting. A proof that there are integers might be even more interesting.

To calculate pi^pi^pi^pi more easily, mathematicians should just work in base pi....

Fun fact: Even though we don't know for sure if pi+e and pi*e are irrational, we know that at least one of them is. Otherwise, if pi+e and pi*e were both rational, then the solutions (namely pi and e) to the equation x^2 - (pi+e)x + pi*e = 0 would be quadratic irrationals, but we know this is not the case.

It be pretty wild if any power tower turned out to be an integer. It'd mean that using higher order inverse tetration you could define pi in terms of integers. This would be categorically like pi turning out to be sqrt3. It can be defined using finite algebra (though extended from what we usually arbitrarily limit ourselves to.

I don't understand almost anything in English, but thanks to the pictures I get the gist. It's gorgeous, I'm thrilled, thank you very much!

*Me putting the expression in a calculator to see if it's an integer before watching the video*

"We set pi to 3" Astrophysicists: Wait what, thought the approximation was 10?

This video has the best intro. I often come back to this just for the first 10 seconds. And then stick around for the whole vid, obv

Fun fact ! The short way to describe this (which unfortunately doesn't have accepted notation) is to say pi tetrated 4, sometimes you can use ^^ to indicate power towers instead, so it would be pi^^4 but many syntax structures use that for exponents instead so its not universal.

"Pie to the pie to the pie to the pie" My doctor didn't like this diet plan

I know what pi to the pi to the pi to the pi is. Its "Error: Overflow"

4:44 Mathematics dictator

“RATIOnal. Everyone remembers where they were when they first noticed that.” I was here, watching this video.

Worried that the pandemic is finally getting to Matt and he's building an army of imaginary audience friends

Don't give that virtual audience CG tomatoes, whatever you do.

Still the best opening to any UKposts video

Eu gosto disso! Boa explicação detalhada!

"Everyone remembers where they were, the first time they noticed that" Yeah, on the toilet about 10 seconds ago, what a beautiful moment that was

I like how he wrote that 11^6/13 is rational as a callback to a video he did about why an advanced casio calculator said that 11^6/13=156158413*pi/3600

you are doing gods work my friend

For my whole life including mechanical engineering degree I was sure that pi is rational and it equals 22/7, I even used 22/7 instead of 3.14 in my calculations and I was happy that IM 100% ACCURATE and now I acknowledged that this is not exactly pi… man, that hurt…

I'm surprised there was no mention of the fact that e^(i.pi) = -1 Transcendental AND imaginary numbers combined to produce an integer.

Matt: lets set Pi equal to 3 Everyone: boooooo Engineers: this is my time to shine...

What is rather more bewildering, and simultaneously maddening, is that we use the speed of light to define time, and we use time to define the speed of light. It makes me head hurt whenever I try to think about it/them.

The only surprise here for me is that I have never really thought about how to *practically* store the results of greater and greater calculations of π (so that they are easily and more or less instantly usable to others, say across a network.) Storing the data as one byte per Digit would be a (by current standards) fairly substantial and yet fairly commonplace storage of 50 terabytes. That would be as a potential BigNum of one byte per digit. Edit: The problem of how to make enormously precise Pi calculations more easily accessible has me wondering, what about efforts to improve on 22/7? For every next big leap in Pi-cision, are we keeping up with some effort to maintain a series of ratios that can fill in segments of the digits (and/or correct the imprecisions of the previous approximation?) For example, for a given precision of Pi, there may be n/m that serves to adjust the precision by: 22/7 ± n/m ("adjust" similarly to correcting a trajectory or other vector.) Alternately, you might use a ratio that gives you accurate digits to a point, discard the rest and add to that another ratio that merely provides several more digits and then raise that ratio to an inverse power of ten to drop those digits into their slot.

"We know 3, beacuse it is equal to 3" Yes the floor here is made of floor

Let me tell you: PyPy to the PyPI results in a lot of incompatible libraries. (Thankfully, the most important ones are compatible.)

I love the 2010 powerpoint themes used in your titles! haha jk love everything but that stood out for a comment for me for some reason

Still in the adverts but I can tell I’m gonna love this channel.

"What kind of clickbait is this???" A seriously nerdy kind

I feel like everyone is thinking in circles here.

7:50 wrong 3 is infinite

I guess it theoretically could be possible, but isn't it safe to assume that the probability that pi^^4 is a perfect integer is basically 0? Like is there any other case of a transcendental raised to a transcendental being an integer other than raising one to it's operational inverse (like e^ln2) ?

I can never hear "three to the three to the three" without having bad flashbacks to Graham's Number

I was somewhat curious. Using some log calculations, the whole digit part (or the whole number I guess should pi^pi^pi^pi be an integer) would require ~245 petabytes of information. Surprisingly, while no computer has that kind of storage capacity, quite a few cloud storage have quite a bit more than that amount. So we may not be able to process that number, but we could store it if some alien gave that number to us.

11^6/13 as a rational is a nice touch at 1:43

"negative 1/12" I see what you're getting at. It's good to see a fellow believer.

e^(i*pi) is an integer. I'm surprised you didn't mention it. Great video!

7:00 Timothy was so concerned with whether or not he could break the 31.4 trillion digit record for pi that he never stopped to wonder if he SHOULD... guess we now just need to wait for a hero to get to 314 trillion

I love how Douglas Adams' 42 always gets a reference 🙂

The fact that he pauses just before saying each number makes me think he's actually calculating them all in his head

I was under the impression that "irrational" included "transcendental", and that things like root-2 were more specifically "algebraic".

This is the mathematical content I’ve been waiting for

the insertion of studio audience for example is simply revolutionary.

Correction: Irrationals include transcendental numbers. "Things that are a solution to a nice polynomial equation" are called algebraic numbers (2:05)

0:09 : "An integer?" *Someone puts a hat on it* "Perry the integer?!"

10:30 Python supports arbitrary precision decimals via the `decimal` library, and there's an example on the docs page to calculate Pi to an arbitrary number of digits.

"They are any number that can be written as a ratio, in fact, it's in the name" That should have come naturally to me...... I mean rationally...

Wouldn't "π + e" just be "pie"?

Would mechanical or quantum computing be able to solve this problem in the future? Especially mechanical, as the real world doesn't have digits.

so it's kinda like multiplying by a negative number where you can know if it's negative or positve based on if you're multiplying an odd or even number of times?

1:53 I remember where I was when I noticed that. I was sat in my chair watching a video about how pi^pi^pi^pi could be an integer

Arithmetic alert! At 8+ min, while you're showing powers of 3 (mod 1000), 3⁹ (mod 1000) is shown as 618, which is clearly impossible (it has to be an odd number!). The actual value is 683 (3⁹ = 19683). 3²⁷ (mod 1000) is, however, correctly shown as 987. Still a great video! Fred

Hey Matt just a little heads up: the Guile virtual machine has infinite precision numbers support. Have fun!

Many thanks for this good video.

"We know that God exists because mathematics is consistent and we know that the devil exists because we cannot prove the consistency." -- Andre Weil (an agnostic)

Matt: "What type of click bait is this?" Me (who clicked the video): "The good type, obviously."

Wouldn’t there be a way using p-adic numbers? Assuming we have a p-adic representation of pi or something sufficiently accurate?

1:42 I was todays years old, when I finally learned about why it is called rational numbers

“I’m gonna do what’s called an engineer move” *Sets Pi equal to 3*

"we set pi equal to 3" Pappa flammy has entered the chat

Presumably a π^π^π^π wasn't chosen at random. It would have been interesting to know a bit more about *why* mathematicians think it _might_ be an integer.

Yes, I love the final "digts" of powers of 3, too. ;-)

Who else here is not subscribed to this channel but is so much of a nerd that the UKposts algorithm showed this video at the top of your recommendations?

This is like trying to rebuild after a hurricane by sending three more hurricanes through

My 1st thought we need a function that describes the distance to the nearest int and then try limits

”Let’s set π equal to 3.” I sense some Graham’s Numbery stuff approaching 😨.

When I saw integers, rationals, irrational, and transcendental numbers in a 2x2 grid with the idea that transcendental numbers were badly behaved, I immediately thought that we were assigning D&D alignments to types of numbers. Integers are Lawful Good, Rationals are Lawful Evil, Root 2 is Chaotic Good (the Robin Hood of numbers, if you will), and Pi is Chaotic Evil. When it made a certain sense to my brain, I went and had a lie down.

pi^pi^pi = Dream's luck when speedrunning.