Why π^π^π^π could be an integer (for all we know!).

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Here is Tim Gowers's reply to the original tweet:
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Start your Schanuel's Conjecture journey here:
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3^3^3^3 on wolfram alpha:
www.wolframalpha.com/input/?i...
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• All the Numbers - Numb...
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[@boysenbeary]

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

[@user_2793]

What an original joke

[@eatpant1412]

As an enginer I feel insulted. I use 355/113

[@petermarksteiner7754]

pi is exactly 3, because the bible says so: 1 Kings 7,23

[@chrismanuel9768]

Pi is 3.14. I don't need more accuracy than that.

[@nocturnhabeo]

Pi is 3+1 for a bit of room.

[@eccentriastes6273]

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.

[@palashverma3470]

Wait another 1000 years of course

[@mickelodiansurname9578]

And this sir is why you are not invited to such a party!

[@mixnewton5157]

@@palashverma3470 pi^pi^pi^pi^pi far away bigger than pi^pi^pi^pi it has 10^10^18 digit "10 followed by billion billion zero" zeros linearly, wait 10^10^18 year, second or blanck time, won't make a difference

[@JamesDavy2009]

I doubt that π^^5 ϵ *Z*

[@ribozyme2899]

Actually, if pi^pi^pi^pi is an integer, then pi^pi^pi^pi^pi is pi to an integer power, which cannot be an integer. (cause pi is transcendental)

[@4thalt]

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

[@gwynjudd]

Well Emma did feature in his "calculating pi by hand" video so they do know each other

[@JeremyRight-zi4yp]

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

[@enricocarrara8672]

For what it's worth, integers are also rational

[@kylewood4001]

Technically, the algebraic numbers include some imaginary numbers too, since the criterion is simply being a root of a polynomial with rational coefficients

[@terrariariley1643]

Look up why pi is transcendental ,because it is

[@ThomasWeinhart]

Three minutes in and already two math mistakes (transcendentals are irrationals and a^b^c=a^(b^c), not (a^b)^c, though he seems to be aware of the latter). It's not funny if he does not take his math seriously.

[@KevinJCoburn]

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

[@coolfred9083]

Hopefully that means there's something even cooler for then

[@yuvalne]

I mean, traditionally he's calculating pi in March 14

[@DapSchaf]

He was too hyped. Or there will be a super amazing video

[@leadnitrate2194]

March 14 is reserved for calculating pi using non-standard ways.

[@matthewclements6603]

March 14 is 14/03/21 in Britain.

[@absupinhere]

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

[@pXnTilde]

It was my only takeaway from this video

[@verrybrainie]

So it is klickbate?

[@columbus8myhw]

And, similarly, "irrationals" are called that because they're not ratios

[@rmsgrey]

Today, Matt Parker called me a nobody. I don't remember when, nor where, I made the connection between ratios and fractions and "rational".

I can't help but notice in order to understand spanish math you need to study english. In Spanish rational numbers = 'números racionales' but ratio = 'fracción'. You can pretty much see there's no real connection between the two in spanish. Always wondered why they were called 'racionales' and 'irracionales'.

[@diegog1853]

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.

[@kambuntschki6314]

Tbh im kinda disappointed because the entire point of the video was just "yeah we just cant know"

[@diegog1853]

@@kambuntschki6314 Yeah and now that I think about it, it really is a different thing to say: "we don't know what this number is" to say "this number *can* be any type of number". There are numbers that it obviously cannot be, like 0. So it might be the case that it is also impossible for it to be an integer, but we haven't develop that reasoning yet.

[@frankmerrill2366]

If using pi to (say) 8 places gives a number nearly midway between two integers (such as 87.54), it's pretty safe to assume that the answer is not an integer. Most probably even using just four decimal places may confirm the pi-to-pi-to-pi-to-pi is not an integer if the value falls far away from an integer. However, if the calculation comes out with something like 88.9999999997 (rounded), additional digits of pi may be necessary. If using pi = 3.14159265, one need only try 3.14159266 as well, and if there's no integer in the middle of the two calculations, this calculation cannot be an integer.

[@sk4lman]

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"!

[@soupisfornoobs4081]

"several corners" is one way to translate it, but it's understood to mean "several angles" by greek people

[@spiderjerusalem4009]

methylgon, ethylgon, propylgon, butylgon, amilgon, isopropylgon, isobutylgon, isoamilgon, sek-butylgon, tert-butylgon, sek-amilgon, tert-amilgon, etc, list goes on

[@akale2620]

Didn't they teach you what it means in school when you started it

[@lunlunnnnn]

​@@akale2620at my school they didn't teach us the etymology of the word; only that it has to do with triangles and how to use it

[@jebadavie]

​@@lunlunnnnnagreed. Sadly most schools did this. They just start with example problems and jump into the work. I was decent at math but didn't realize until my 30s that exponents 2(square) and 3(cubed) were called that because they formed that geometric shape out of the base unit.

[@parmparm9341]

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.

[@wumbowumbo1688]

SAME LOL 😂

[@retvolution]

Lmao same

[@asheep7797]

Same too

[@ianc8266]

You can remember more digits than that with "I need a drink, alcoholic of course, after the heavy lectures."

[@Shreyy17]

What I did (although a decade later) was using all 12 digits of the calc by 999.... and then multiplying by itself lol

[@spankasheep]

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

[@ThomasSMuhn]

Well, the Bible says that pi equals 3; and the Bible also says that the Bible is never wrong. QED.

[@efulmer8675]

How about when Indiana almost legally declared pi is equal to 3.2?

[@ThomasSMuhn]

@@efulmer8675 'Cause godless heathens they are down there?

[@efulmer8675]

@@ThomasSMuhn It was in the late 1800s and the Indiana State Legislature brought in a mathematician to help settle the issue. They settled the issue by throwing out the bill All-0. Still, it is a hilarious collision of math and reality.

[@DonReba]

I'm pretty sure this is only allowed under martial law.

[@omaanshkaushal3522]

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

[@KuhWristChin]

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

[@vermiformappendix]

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”

[@troodon1096]

That totally sounds like a joke a math teacher would tell.

[@rysea9855]

Kinda genius ngl

[@Simon-nx1sc]

@@troodon1096 Damnit, apparently, I'm destined to become a math teacher

[@thelivingcube]

heh

[@math_the_why_behind]

Haha!

[@Xenophilius]

"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.

[@tomc.5704]

I'll gladly take his word for it, but I have never seen a proof

[@afuzzycreature8387]

to be fair, we have harvard grad students who will argue against this

[@rosepinkskyblue]

LMAO 🤣

[@NerdTheBox]

tetris person poggers

[@spl420]

It's more than we know about 0.1+0.2

[@EquuleusPictor]

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 ...

[@hyperbaroque]

I think it's because the material world is a bridge itself between solutions. Physical reality serves as an "elegant solution" that solves the identities of all transcendental numbers in one instant. By working with physical reality we get to experience the subtleties we are missing by using this bridgework without knowing all the underlying equations. Oh, did this bridge we made using the bridgework of physical reality twist itself apart in a mind-bending way? We study it and find an underlying equation involving harmonics, and work to contramand that equation as a point of ethics in bridge-building. (And so on.) So to paraphrase Newton and Hawking regarding "standing on shoulders", with physical reality we are standing on unknown shoulders of unknown giants. (And to finish the thought: mathematics is the blind study of the anatomy of those shoulders, in hopes of discovering something about those giants.)

[@joleneonyoutube]

what an absolutely stunning comment and quote, I hadnt heard or seen that finished thought before, thank you for sharing@@hyperbaroque

[@carlhopkinson]

Infinities are infinitely harder to deal with.

[@dusk_and_dawn2187]

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

[@flan1591]

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

[@longpham-sj5sv]

you look sus ngl

[@ScormGaming]

@@longpham-sj5sv Now that was the comment I was looking for

[@grahamsayle]

When the pretender is mistrustful

[@DeadPool-fx3sq]

Unliked this comment due to the likes beong 456

[@davidmenn8771]

Icarly? Sam Puckett?

[@gregoryburns4821]

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!

[@frankjosephjr3722]

Captions take a few hours to show up on videos

[@EcceJack]

@@frankjosephjr3722 Does it? I've only ever uploaded videos (not on this account!) that didn't need an immediate release, and found I could easily add subtitles before "publishing" the video - and then they appeared immediately. I suppose it's possible that - if you're trying to upload immediately - these things take a while to process..?

[@aaronjulien7331]

@@frankjosephjr3722 yeah, atuo-generated ones

[@dovecat]

@@EcceJack they may be referring to the KZbin auto generated captions

[@MrZerRap]

It Might be a good idea to allow for the comunnity to caption the videos, Matt! I'm willing to volunteer in doing Portuguese subtitles if you want!

[@JohnSmith-ut5th]

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.

[@stargazer7644]

But we aren’t calculating pi here, we’re calculating pi to a power.

[@sethkunert6234]

​@@stargazer7644that is still an nth of pi

[@WhiteGandalfs]

@@stargazer7644 We are looking after nothing else than: HAS pi something behind the comma or has it not? And for that, we are allowed to use modulus. modulus 1, to be specific. Which makes things drastically easy. And then concerning accuracy: We only need enough accuracy to get the first few (maybe just for satisfaction the first three or so) digits after the comma correctly, all others are just overhead. Hint: The digits will not be zero (or 9) behind the comma. If they were zero (or 9), we first would gather a bit more accuracy. Only if there after a lot more zeros would make their debut, would we need to invest in thoughts about proving anything. But since the digits behind the comma will for sure not be around zero, all other thoughts about proving integer-ness are invalid anyways.

[@fewwiggle]

@@WhiteGandalfs "HAS pi something behind the comma or has it not?" Do you mean the decimal point? Regardless, the problem is π^π^π^π You need to know the EXACT value of MANY, MANY digits of π to know if the 'last' digit is an integer.

[@gwillen]

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! 😅

[@blackpenredpen]

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

[@captainsnake8515]

Blackpenredpen: “do not trust wolfram alpha, trust algebra” Also blackpenredpen:

[@michaelwu9892]

you are our favorite pokemon math youtuber

[@rogo7330]

You had to fight evil Not join it!

[@cezarcatalin1406]

Michael Wu COMRADE !

[@nahometesfay1112]

@@captainsnake8515 I trust wolfram alpha with my life Well really my school work, but that's pretty much my life right now. Yay college!

[@CharlesJrPike]

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

[@sykes1024]

Eh, even an engineer'd probably use 22/7. Setting pi to 3, is closer to what a theoretical astro physicist would do. Though, maybe they'd just set pi to 1.

[@kindlin]

@@sykes1024 I love the joke, but in actuality, Natural Units make perfect sense. We have set all of our units to be useable with day-to-day activities, like driving a car or baking a cake. If you set the units to be most useful for theoretical astrophysicists, then you get a lot of 1's, and all the equations become a lot easier to work with, on a theoretical basis. You only need to bring back in all the powers of c and h if you want to make an experimental prediction in numbers that make any sense to us hoomans.

[@pdorism]

As a physicist, I always have pi = e = 3. We don't use calculators, we just look at the first digit and the order of magnitude

[@Aeronwor]

that seems right, but you do need to include 30% safety factor and round up to the next standard size.

[@davidmarshall2399]

@@Aeronwor or use 4. Depends which side is conservative

[@benjiboy1337]

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.

[@usof75756]

This is all way beyond me but if I had to make something up I would guess that circle^circle would be a sphere. So taking that all the way to the end would be a 5d circle. Granted I have no idea what I'm talking about and there's no way it's that simple.

[@benjiboy1337]

@@usof75756 I'm not actually sure what operation turns a unit circle into a unit sphere, but a unit circle raised to the power of a unit circle would be something like pi^2 unit circles, projected into four dimensions... I think. Since you're basically multiplying every point on the unit circle by another unit circle, the area should be (pi*r^2)^(pi*r^2), giving us pi^2*r^4. Plugging in 1 for r, we get just pi^2... so this might not be that useful of a line of inquiry after all. Using geometry for higher maths is mind bending, because we live in a 3-spatial one-temporal dimensional reference frame. There's some precedent for transcendental numbers to "cancel out" to an integer, though I only know of one actual case in Euler's formula. There's probably a Nobel or equivalent prize waiting for whoever discovers an equally beautiful formula in mathematics.

[@peepock7796]

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.

[@jacksonsmith2955]

Integers are rational numbers too. To be more precise he could have labeled the groups "integers", "non-integer rationals", and "non-transcendental irrationals" but he got the point across which is what really matters.

[@laikahusky6358]

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

[@peteranon8455]

.... and cried....

[@billwhoever2830]

Trust me, I'm an engineer: π=3

[@persilious81]

@@billwhoever2830 But for some reason my wheels always fall off

[@themushroom2130]

@@persilious81 “I want a refund”

[@michalnemecek3575]

I'm not an engineer because I always use at least 3.14 (unless I'm using a calculator, which always uses about 3.14159265359)

[@NightiNerd]

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.

[@stevanmiladinovic4007]

Base-Pi that'd still be a ten-billion-digit number.

[@yyattt]

pi^pi^pi^pi = 10^10^10^10 base pi 10^10^10^10 is an integer therefore pi^pi^pi^pi is an integer if we work in base pi. QED

[@NightiNerd]

@snarl banarl Hmmm, that's true. Now I have another idea: π^π^π^π is 10 in base π^π^π^π. It's an integer! We leave the proof for other bases to the interested reader.

[@electricengine8407]

10 in base pi is NOT an integer

[@cucen24601]

This is a galaxy brain meme lol

[@encyclical]

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

[@hbxit1888]

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.

[@Milkymalk]

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

[@BattousaiHBr]

fear not, the complexity is merely _imaginary_

[@Kanzu999]

*Applause from crowd*

[@michaelvstemerman]

@@BattousaiHBr Quarternions be like:

[@slkjvlkfsvnlsdfhgdght5447]

@@BattousaiHBr boo! boo!

[@kennarajora6532]

@@BattousaiHBr only part of it is. The other is the real part.

[@KirillTsukanov]

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

[@Luca_5425]

I was surprised as well

[@usernamenotfound80]

π^π^π^π is rational. Proof: It isn't √2.

[@cpotisch]

@@Luca_5425 You know he was joking, right?

[@Luca_5425]

@@cpotisch of course

[@Chisito23]

@@usernamenotfound80 QED 😎👌

[@perplexedon9834]

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.

[@babilon6097]

But we can already define it using integers. Matt does it every year for 14th of March (which people using skewed date notation call a pi day).

[@Errenium]

​@@babilon6097get back to me on april 31st /lh

[@perplexedon9834]

​@@babilon6097you're right, I meant a finite algebraic expression, like how the golden ratio can be.

[@typicwhisper6569]

@perplexedon9834 Tetration and its inverse are transcendental functions, so it could never be a finite algebraic expression.

[@samueljehanno]

But why couldn't it be finite algebraic expression ?@@typicwhisper6569

[@hyperbaroque]

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.

[@WaterShowsProd]

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

[@Eric_The_Cleric]

Virtual audiences are vicious.

[@johnchessant3012]

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.

[@kddanstars9288]

What's your source

[@CreeperDeLux]

@@kddanstars9288 if you know the quadratic formula, you can see that he ia right

[@daicon2k6]

Yes, but the question at 12:20 wasn't whether pi+e and pi*e are irrational, but rather whether they are transcendental.

[@ratlinggull2223]

Pi+e = pie, but because pi is already pronounced pie, we prove that e=0.

[@thassalantekreskel5742]

@@ratlinggull2223 And in a cylinder with a radius Z and a height A, the volume equals Pi*Z*Z*A

[@kano4ka]

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!

[@KpxUrz5745]

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

[@Thror251]

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

[@samuelthecamel]

That's great until you try to see if the extremely large result in base pi is an integer

[@EebstertheGreat]

In base π, π^π (i.e. 10^10) is equal to 1012.031000012..., because π^π = π³ + π + 2 +3 π⁻² + π⁻³ + π⁻⁸ + 2 π⁻⁹ + · · · . So that's not really helpful.

[@vigilantcosmicpenguin8721]

cursed

[@Syuvinya]

@@EebstertheGreat This is a big brain moment

[@briant7265]

@@samuelthecamel It WOULD be an integer, of course. The problem would be that all of the numbers that are currently nice, simple integers would become transcendental. Counting would become impossible.

[@RC32Smiths01]

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

[@misiekeloo6114]

Error: Result is too big

[@du42bz]

I thought my phone crashed

[@RC32Smiths01]

@@misiekeloo6114 Indeed haha

[@RC32Smiths01]

@@du42bz same

[@catfort.dragon]

Google says it's undefined

[@regimeoftruth]

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.

[@samueljehanno]

This is interesting !

[@CafeMuyCaliente]

And if all pi^^n wont be integers, what about pi^^pi ?

[@samueljehanno]

@@CafeMuyCaliente interesting

[@brianolsen5435]

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.

[@stargazer7644]

So what’s stopping you from punching in pi^pi^pi^pi?

[@lerntuspel6256]

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

[@antoniocoulton5017]

Don't be silly, it is very obviously equal to "MATH error"

[@sadkritx6200]

@@antoniocoulton5017 math error on casio calculators. Don't know what others say though

[@tiem217]

@@sadkritx6200 TI says Error: Overflow

[@tobiasbrohl5958]

it's "overflow - huge result is out of SpeedCrunch's number range"

[@giovanicampos4120]

Yeah I did it in my calculator and get math error

[@aykborstelmann8623]

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

[@pankajbhambhani2268]

My reaction exactly. Surely we can approximate pi^pi^pi^pi to within a few orders of magnitude?

[@CarbonRollerCaco]

Why do they of all people use 10, anyway? Everyone knows base 10's just a cultural bias inspired by our hands. And 3×3 squares, but those aren't that much more relevant to physics.

[@pankajbhambhani2268]

@@CarbonRollerCaco I guess because base 10 is the standard in scientific notation. If a star has mass x * 10^y, astronomers usually can't precisely measure x, so they don't care about. They only care about y, the order of magnitude, which they can estimate properly.

[@user-mv2nn6rw2w]

@@CarbonRollerCaco Because other people use 10. No number is better than another in a vacuum. Same reason why you use lightyear instead of inches to talk about astronomy, despite the calculation to change basis is trivially easy.

[@CarbonRollerCaco]

@@pankajbhambhani2268 It's still ironic that scientific notation, which is supposed to be unbiased, uses a scientifically wonky base informed by culture only because of evolutionary happenstance. But it is understandable in a sense as they need to quickly relate things to what's already convenient, even if it's an anachronism. Even still, it sounds wrong as THE base for magnitude.

[@echoawoo7195]

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.

[@takemyhand1988]

⁴^π

[@samueljehanno]

Yeah sure

[@mischa7406]

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

[@mischa7406]

Hey look its me again watching the first 10 seconds

[@emileheskey2754]

"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

[@rups251195]

Same

[@PanduPoluan]

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

[@ratlinggull2223]

my mind became numb py

[@jacquesstoop2587]

*Sigh py*

[@PanduPoluan]

@@jacquesstoop2587 Daaaaamn I was racking my brain trying to pun SciPy 😆

[@kakyoindonut3213]

damn, is this Py Game or something?

[@Orincaby]

@@kakyoindonut3213 don't worry it's just a joke from the programmer's PyPline

[@bikramkalsi1]

you are doing gods work my friend

[@XeiDaMoKaFE]

7:50 wrong 3 is infinite

[@phitsf5475]

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

[@hello_iam_potato]

Underrated

[@michaelsmith4904]

I remember a song some years back that went something like "moe to the e to the.." or something like that, and was thinking that if we knew the value of "moe" we could calculate the value of the equation.

[@azlandpilotcar4450]

We say n^2 is n squared, ^3 is cubed, and ^4 is hypercubed. I think this might be pi hyperpied.

[@gurrrn1102]

Also pie to the pie to the pie to the pie sounds like a rapper saying pie pie pie pie

[@hello_iam_potato]

@@gurrrn1102 sick rhyme

[@jerwahjwcc]

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

[@danielhenderson9719]

Don’t worry. He’s not building them. He’s already built them.

[@simonecatenacci726]

There is no problem, as long as he multiplies the imaginary audience by itself, he will get a real audience

[@syllogism5843]

@@simonecatenacci726 Although it will be negative, so not much applause alas

[@AmaroqStarwind]

I think you mean Lateral.

[@ailaG]

His audience is quite complex.

[@AbouTaim-Lille]

To understand the power of transcendental numbers compared to the set of algebraic numbrrs (that include the roots) actually the set of algebraic numbers are countable. And while the set of transcendental numbers are have the cardinality of continuity (i.e the card(IR) ). That makes the borel measure of transcendental numbers in the interval (0,1) equal to 1 while the the algebraic numbers are negligible with measure equal to zero. As if for each algebraic number you can find an iinfinitely nfinite number of transcendentals.

[@Fun_maths]

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

[@ZevEisenberg]

-1/12 was also a callback to a Numberphile video

[@meltingkeith7046]

@@ZevEisenberg don't know if I'd call the -1/12 a callback or just generally a controversial result in mathematics in general and hence worth putting in

[@trogdorstrngbd]

@@meltingkeith7046 The result itself isn't controversial. The sloppy/misleading presentation of it to the general populace was.

[@Garbaz]

Well spotted, didn't notice that one!

[@onradioactivewaves]

@@ZevEisenberg that was actually - 1/( 4 π)

[@FourthDerivative]

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

[@LittleEinsteinAdi]

yea

[@NoriMori1992]

Same.

[@woollycomet]

Haha so trueee

[@1mol831]

If we can find a phenomenon in nature... ... That we can do with ... via the use of sciences?

[@umgeburstet8161]

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

[@user-fb2qr4ru6i]

Eu gosto disso! Boa explicação detalhada!

[@neilruston8796]

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

[@urnoob5528]

Well try to plot a complex power without formula U cant Complex power is defined by infinite series Complex number and cos, sin is easy to plot But u cant plot a complex power without converting to cos and sin It is unintuitive, someone just wanted to give it a definition and so they did by infinite series If u want to prove me wrong otherwise, try to plot 7^(3+i8) without converting it or anything

[@2eanimation]

@@urnoob5528 "It is unintuitive" So are PDEs, that doesn't make them wrong. "someone just wanted to give it a definition and so they did by infinite series" Everything related to e^x(or better say, the exponential function, without knowing that exp(1) = e) can be derived from its power series alone. Or (I) exp(a + b) = exp(a)exp(b) and (II) 1 + x inf] (1 + x/n)^n Or continuous growth Or y = y' Or... If the power series is a perfectly fine way to define exp(x), exp(i) is perfectly fine as well. Let's not forget about the useful math thanks to exp(i). Laplace/Fourier transform comes to mind ;) "But u cant plot a complex power without converting to cos and sin" 7^(3 + i8) = 7^3 * e^(i8ln(7)). Vector with length 7^3, x-axis and vector enclose 8ln(7) rads, that is (360 * 8ln(7) / 2π)° ~= 891.94° ^= 171.94° (mod 360). Look mom, without trigs! "without converting it or anything" try to plot x^2 * y'' + x * y' + 4 * y = 0, y(-1) = 3, y'(0) = 0 wItHoUt CoNvErTiNg It Or AnYtHiNg

[@aguyontheinternet8436]

Well _I_ think he should have mentioned e^(i*τ)=1 instead, which is the far superior formula

[@mikeoxmall69420]

God is the greatest troll ever😂

[@fahrenheit2101]

@@aguyontheinternet8436 ew no

[@outsidestuff5283]

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

[@semiclassical7620]

Pfft, pi=3 is crude. Now pi^2=10, that’s where the money is! (More like 9.9 but that’s not as catchy)

[@floop_the_pigs2840]

3=e=π=√g (on earth)

[@georgelionon9050]

Pi is 22/7... that's probably good enough for anything a normal person does.

[@floop_the_pigs2840]

@@georgelionon9050 honestly yeah

[@Runoratsu]

Astrophysicists: Pi = 1 is close enough.

[@gordonmichaels1042]

Base 10 may be bad for persistence, because 10 is a composite number and thus has two of the lower digits that multiply to '0 mod10,' namely 2*5. I think that prime number bases, such as 7, 11 or 13 might be better for finding high persistence numbers. Hexadecimal would not be promising because you would have to avoid having 4 even digits in any of the numbers in the persistence calculation, otherwise the product of the digits of that number will end in 0.

[@gordonmichaels1042]

Notice that for the number 277777788888899, if you take any (or all) of the '8' digits and replace them with 222 you will have a number with more digits but the same 11 products in the persistence calculation. Similarly, you can replace any of the '9' digits by 33, and it will yield a larger number but equivalent to 277777788888899 in persistence. This trick, combined with reordering the digits, will give many more 'starting points" for finding a number with digits that multiply into a 'persistence-11' number.

[@christopherquigley5468]

I feel like everyone is thinking in circles here.

[@calebclunie4001]

On a pie chart, it might be assumed, but on a donut chart, one might come across phi. I tried bringing this up, but I was told to "shut my blooming phi HOLE!"

[@vblaas246]

@@calebclunie4001 Thanks, now I'm imagining a fractal donut of a donut... define the emerging donut. And the ratio of the radii. Someone calculate?

[@zbnmth]

semicircles...

[@Dowlphin]

Don't be a square.

[@Dowlphin]

@@vblaas246 That sounds totally radiical!

[@Treviisolion]

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.

[@RGC_animation]

Thrust me, in 20-30 years, most big cloud server would have that amount of storage.

[@triciaf61]

@@RGC_animation thrust you? oh my.

[@joshuathomasmacalintalsoli6307]

@@RGC_animation Moore's law just proves so

[@danieljensen2626]

My impression from what Matt was saying was we probably could calculate it if we dedicated all of Earth's computing resources to it for long enough (but like less than a human lifetime). But that's not exactly a reasonable thing to do.

[@techrev9999]

Today I learned that cloud storage is some sort of magical entity that is not a computer. Interesting. Do you have more hocus pocus to share.

[@simplyme5324]

Upper and lower bounds : exponential functions are monotonous and either increasing or decreasing. So calculate 3^3^3^3 and 4^4^4^4 for the interval in which the solution lies. Try the first decimal 3.1^3.1^3.1^3.1 to check whether it is in the interval. Maybe reformulate the equation to basis e^x. So something like (pi^pi^pi)^x = (whatever it is in natural base)^y. Iterate for (pi^pi)^(pi) etc. Something like that. Check whether it is in the interval. These are the first approaches that come to my mind.

[@zachhoy]

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

[@DannyGottawa]

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

[@IBlewUpYourHouse]

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

[@RWZiggy]

But 3 + 3 equals 7, for large values of 3.

[@timothymclean]

But we're talking about 3, not 4.

[@shreebatsachaturvedi5376]

@@RWZiggy However, it is also worth noting that the limit of 3 as 3 approaches 0 is 4.

[@LA-MJ]

Hi, Vsauce here...

[@underrated1524]

floor(3) = 3

[@aidenbagshaw5573]

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

[@Thitadhammo]

This made me think of the contruction of Graham's Number, which sorta kinda starts with 3^3^3 and that is already fairly large. 3^thatlargenumber would not be suitable for a calculator or any decent computer.

[@seanmurphy8435]

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

[@bman5257]

i is imaginary though. I think he purposefully kept the categories in real numbers.

[@JackiTheOne]

i is not a transcendental number tho

[@AuroraNora3]

@@JackiTheOne i*π is transcendental

[@YTEdy]

But it's a different pi. It's not the number pi it's an arc of pi radians, or 180 degrees or 1/2 circle, both of those definitions are no longer transcendental.

[@mr.cheese5697]

Nice, don't touch the like, please

[@a_cats]

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

[@peNdantry]

You've fallen into his trap -- that's what Matt *wants* you to think ;)

[@sidkemp4672]

The whole discussion around minute 11 of calculating pi to the pi using an approximation of pi, I think, runs into another problem at a theoretical level, not about computing power. Any approximation of pi we can put into a computer will be a rational number. It will either end in terminating digits or in a repeating digit. So we are not testing pi (or pi to the pi), we are testing a rational number raised to a rational number.

[@ptorq]

True, but we don't necessarily need to know the exact answer to know if it's not an integer; knowing it to one decimal place might be enough (or it might not). The hilarious thing would be if we calculate it accurate to 8 decimal places and it ends with .00000000something. It's PROBABLY an integer, but we wouldn't know for certain that the next digit isn't a 4.

[@sidkemp4672]

@@ptorq My point is that, even if we did that, we would simply be converting a rational number to a (probable) integer, and that would still say nothing about the issue of transforming transcendentals into integers or rational numbers.

[@graemetang4173]

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

[@vin_fm2354]

When COVID happens people get bored

[@METALSCAVENGER78]

It is broken again.Now, it's 62.8 trillion digits.Exciting times

[@greatorionbelt]

@@METALSCAVENGER78 thats twice pi

[@kijete]

@@greatorionbelt tau

[@David-gu8hv]

:)

[@klikkolee]

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

[@aidanhennessey5586]

Your impression is corrext

[@harrisonbaguley5691]

transcendental numbers are by definition irrational, since they can't be expressed in a ratio. i'm assuming this video separated transcendentals from other irrational numbers to simplify the difference between numbers like root 2 and pi

[@willgaj]

I'm glad it's not just me 😅

[@trogdorstrngbd]

Everyone needs to upvote this comment! Matt has repeatedly made this mistake (I'm pretty sure he does it out of convenience) and needs to stop. It's mostly OK when he says it verbally in the presence of an accurate graphic depicting the number set relationships, but otherwise it's just wrong.

[@theWebWizrd]

Yup, I reacted to this as well when he essentially claimed pi is not an irrational number.

[@pinklady7184]

Interesting. After seeing this, I am imagining a graph with x^x^x^x = y. I oftentimes use graphic art software (Inkscape) for creating graphs with intricate equations.

[@iHATEbigots666]

2:50 it IS weird! To under stand why, you can always go back to this: Plotting F(x + iy) = e^(t * i *2pi) on the complex plane is a circle. Going from t=0 to t=2pi represents traveling around the circumference of the circle. Why on earth? Well, multiplying by i takes you from the real axis to the imaginary axis (or vice verse). Multiplying by i is literally a 90 degree = pi/2 rotation. 4 of those rotations is 2pi a.k.a. a circle. Why does e show up in this parameterization? I'm still trying to get an intuitive feel for this.... e^x is its own derivative of course, so maybe it somehow describes the perfect roundness of a circle or something??? For a non-intuitive feel, you can write out the Taylor series for e^(ix), cosx and i * sinx and see that the terms are equal combined through addition.

[@moparacker]

Wouldn't "π + e" just be "pie"?

[@catfort.dragon]

But is pie an integer or not?

[@TheNameOfJesus]

That sounds rational to me.

[@JayOhm]

Since when "ab" can mean "a+b"? So "pie" is actually π×e

[@psy0rz]

Give this man a nobel price!

[@mtgradwell]

@@JayOhm No, that's pixie minus i.

[@ffggddss]

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

[@devd_rx]

Yeah i was like how the heck 8 showed up

[@M0jibake]

And also "digits" is spelled "digts"!

[@JayKubiakGuitarstep]

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…

[@Ironman-ko5vw]

I think I might have worked it out. My answer led me to this equation: Pi = y^(1/y) But knowing this is impossible as the highest value or y^(1/y) is e, I plugged it into maple calculator which game me a complex number with tons of decimal places.

[@BlackDeath2812]

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

[@Youaveragecountryhumansfan]

Same! What are the odds?!

[@anthonyisom7468]

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

[@tomcat1184]

this is not for normies ,... only few people would understand

[@kroolini3678]

@@tomcat1184 one of the most normie memes around

[@kroolini3678]

@@brahadkokad5424 you’re 10

[@yuvi6034]

Nice one

[@bleikeze]

The trick with sqrt(2) raised to itself 3 times giving an integer (2), works with the cube root of 3 raised to itself 4 times (like the pi thing… gives 3); and any Nth root of N raised to itself N+1 times (gives you back N)!

[@gnomehead2073]

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

[@GerinoMorn]

Mechanical won't work for transcendentals, because of plank length. At some point nothing is a perfect circle, our universe is pixelated :

[@judedavis92]

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

[@psychopompous489]

*Sets 3 equal to 3*

[@e.s.6275]

Nope we engineers don't do that, ever.

[@d0themath284]

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

[@GaussianEntity]

"We're doing calculations tho" Flammy has left the chat

[@Chris.Davies]

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.

[@QueenAfyn]

It feels like we should be able to prove this with contradiction setting Pi tetrated four times (Personally I write this as pi|4) to a number n and just declaring it an integer and then messing with it till we reach a contradiction. However I'm sure thats been tried and the fact that this is still an open problem probably means it doesn't work

[@balsoft01]

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

[@infinemyself5604]

Algebraic numbers also include rational numbers. And rational numbers include integers.. so really right things to say would have been "integers", "non-integer rationals", "algebraic irrationals" and "transcendentals" But this is just too crowded, don't ya think?

[@ElvisTranscriber2]

@@infinemyself5604 no two-word terms, if they are more specific and avoids wrongly excluding a number from a group it actually belongs to is justified 😀

[@chaosredefined3834]

But this now leads to an interesting question. The proof that he gave that irrational ^ irrational = rational worked because sqrt(2)^sqrt(2) is either rational or irrational, and either way, we got an irrational ^ irrational = rational. However, are there two algebraic irrational numbers, a and b, such that a^b = rational.

[@tantarudragos]

@@chaosredefined3834 sqrt(2) is an algebraic irrational

[@chaosredefined3834]

@@tantarudragos This is true. But sqrt(2)^sqrt(2) is not. So, he ends up with a is transcendental, b is algebraic, and got a^b is rational.

[@rmschad5234]

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

[@ajc389]

You say that it is not possible to work out at the moment, but I wonder if it possible to work out if it will be possible to work out in the future?

[@EquaTechnologies]

Ideas to calculate it: Reverse Engeneering - for example: you have the number 1,000,000. You square root it four times and if the number is closer to pi, you add +1 and do the calculation again. As you get closer, you add less and less to match pi. Physically - Have a physical circle, that is spinned the requered amout and see the answer as π=circumference/diameter

[@jhgvvetyjj6589]

Square root four times won’t work as that inverts (((π^2)^2)^2)^2 or π^16, not π^π^π^π

[@masvindu]

"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...

[@happygimp0]

i/2 is a ratio but not a rational number.

[@blauesserpiroyal2887]

@@happygimp0 rational numbers are ratios of integers

[@mattsnyder4754]

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

[@Kishmond]

With an infinite number of hurricanes eventually everything will be blown back into place.

[@xavariusquest4603]

Yes...chimps given enough time and a keyboard will mash out the collected works of Shakespeare.

[@bennyblubman9476]

Yea it's just like evolution, you can get something complex and structured from pure chaos

[@rcsibiu]

yeah....send more hurricanes hoping that they ALL could eventually fix those buildings and revive those killed people .....absolutely stunning :))

[@rstriker21]

@@rcsibiu if the difference between life and death is just having your atoms in the right places, the chances of a hurricane reviving someone is technically more than 0 lol

[@TheHighborn]

at those numbers, you should be happy if you get the interger part correctly. Multiplying VERY big numbers with small numbers (or adding or dividing, or anything) your precision will drop, even in the integer part.

[@Paraselene_Tao]

In a similar manner to how there's knuth up-arrow notation for higher operators, is there a formal notation to describe a "pi-step" in the operations between 3 (exponentiation) and 4 (tetration)? Is there a "pi-ation", and what do we get when we raise pi to the pi with "pi-ation"? I might be asking a nonsical question, but it's fun to imagine. First, can there be fractional operators? I recall that there's fractional derivatives and of course there's fractional exponents & logarithms. Second, can there be irrational operators? I've never see irrational derivatives, but I've seen irrational exponents & logarithms. Third can "pi-ation" work? What do I get if I operate pi through pi-ation?

[@Wrenosaur_]

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

[@God-gi9iu]

Mmm

[@God-gi9iu]

Ee

[@flameseeker4058]

about that

[@insertnamehere4775]

That's only like over 30 my guy Nvm it's xE+18

[@incription]

@dang bro it aged very well

[@rektor1farta739]

One of your best introes! Love your tone and delivery of "to the Pi" and "an integer?!". Haha, fantastic!

[@Kris_not_Chris]

based naively on those pi computation times per digits, back of the envelop estimate suggests it would take O(10 million) years to calculate pi arrow 4

[@gg2324]

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?

[@MathAndComputers]

Wait, aren't what he called the "irrational numbers" specifically the "algebraic numbers"? I thought that all transcendental numbers are also irrational numbers. 🤔

[@MoiMagnus1er]

Yes, you are correct. Irrational numbers includes transcendental numbers by definition.

[@rhysdenno5993]

All transcendentals are irrational, but all integers are rational for that matter.

[@sebastianmestre8971]

Yeah, and integers are also rational

[@leadnitrate2194]

@@MoiMagnus1er you got that the wrong way round if I'm not wrong?

[@wellshit9489]

Integers and rationals are also algebraic arent they?

[@ottolehikoinen6193]

My calculator says it's a variable called 'out of range', but let's hear this out.