Why π^π^π^π could be an integer (for all we know!).
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- čas přidán 26. 02. 2021
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3^3^3^3 on wolfram alpha:
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www.wolframalpha.com/input/?i...
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“We set pi equal to 3”
Engineers: *applause*
What an original joke
As an enginer I feel insulted. I use 355/113
pi is exactly 3, because the bible says so: 1 Kings 7,23
Pi is 3.14. I don't need more accuracy than that.
Pi is 3+1 for a bit of room.
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.
Wait another 1000 years of course
And this sir is why you are not invited to such a party!
@@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
I doubt that π^^5 ϵ *Z*
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)
6:48 I love how Matt just casually referred to the two people as Emma and Timothy like if they were close friends
Well Emma did feature in his "calculating pi by hand" video so they do know each other
2:00 - calling them "irrationals" is indirect, since π or e are irrationals as well. Numbers like √2 are algebraic, an antonym to transcendental.
For what it's worth, integers are also rational
Technically, the algebraic numbers include some imaginary numbers too, since the criterion is simply being a root of a polynomial with rational coefficients
Look up why pi is transcendental ,because it is
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.
“Everyone remembers where they were when they noticed that”
Ah, yes. This takes me back to two seconds ago.
It was my only takeaway from this video
So it is klickbate?
And, similarly, "irrationals" are called that because they're not ratios
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'.
"We set pi equal to 3”
I felt a great disturbance in the force.
Well, the Bible says that pi equals 3; and the Bible also says that the Bible is never wrong. QED.
How about when Indiana almost legally declared pi is equal to 3.2?
@@efulmer8675 'Cause godless heathens they are down there?
@@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.
I'm pretty sure this is only allowed under martial law.
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.
Tbh im kinda disappointed because the entire point of the video was just "yeah we just cant know"
@@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.
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.
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"!
"several corners" is one way to translate it, but it's understood to mean "several angles" by greek people
methylgon, ethylgon, propylgon, butylgon, amilgon, isopropylgon, isobutylgon, isoamilgon, sek-butylgon, tert-butylgon, sek-amilgon, tert-amilgon, etc, list goes on
Didn't they teach you what it means in school when you started it
@@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
@@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.
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.
SAME LOL 😂
Lmao same
Same too
You can remember more digits than that with "I need a drink, alcoholic of course, after the heavy lectures."
What I did (although a decade later) was using all 12 digits of the calc by 999.... and then multiplying by itself lol
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”
That totally sounds like a joke a math teacher would tell.
Kinda genius ngl
@@troodon1096 Damnit, apparently, I'm destined to become a math teacher
heh
Haha!
Thank you for making complicated math concepts fun and entertaining. Peace and Love Matt
This was such a fun video to watch. Definitely one of my favorites from Matt.
Never before have I seen someone have so much fun with a stock studio audience, and I love it so much
you look sus ngl
@@longpham-sj5sv Now that was the comment I was looking for
When the pretender is mistrustful
Unliked this comment due to the likes beong 456
Icarly? Sam Puckett?
I'm surprised that you didn't save this for March 14.
Hopefully that means there's something even cooler for then
I mean, traditionally he's calculating pi in March 14
He was too hyped. Or there will be a super amazing video
March 14 is reserved for calculating pi using non-standard ways.
March 14 is 14/03/21 in Britain.
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 ...
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.)
what an absolutely stunning comment and quote, I hadnt heard or seen that finished thought before, thank you for sharing@@hyperbaroque
Infinities are infinitely harder to deal with.
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! 😅
"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.
I'll gladly take his word for it, but I have never seen a proof
to be fair, we have harvard grad students who will argue against this
LMAO 🤣
tetris person poggers
It's more than we know about 0.1+0.2
As soon as I saw the title, I went to WolframAlpha, haha!
Blackpenredpen: “do not trust wolfram alpha, trust algebra”
Also blackpenredpen:
you are our favorite pokemon math youtuber
You had to fight evil Not join it!
Michael Wu
COMRADE !
@@captainsnake8515 I trust wolfram alpha with my life
Well really my school work, but that's pretty much my life right now. Yay college!
This video was amazing. So many fascinating thoughts. Absolutely loved it!❤
Enjoy this channel immensely. Most of us need tutoring when it comes to mathematics.
"How about we start by setting pi equal to 3..."
What is this, stand-up engineering?
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.
@@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.
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
that seems right, but you do need to include 30% safety factor and round up to the next standard size.
@@Aeronwor or use 4. Depends which side is conservative
"For simplicity's sake, why don't we start with setting pi equal to 3." Engineers everywhere rejoiced
.... and cried....
Trust me, I'm an engineer: π=3
@@billwhoever2830 But for some reason my wheels always fall off
@@persilious81 “I want a refund”
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)
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.
But we aren’t calculating pi here, we’re calculating pi to a power.
@@stargazer7644that is still an nth of pi
@@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.
@@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.
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
Hey look its me again watching the first 10 seconds
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!
Captions take a few hours to show up on videos
@@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..?
@@frankjosephjr3722 yeah, atuo-generated ones
@@EcceJack they may be referring to the CZcams auto generated captions
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!
Matt: "It is complex..."
Me: "Okay, explain it."
Matt: "...literally."
Me: "Oh."
fear not, the complexity is merely _imaginary_
*Applause from crowd*
@@BattousaiHBr Quarternions be like:
@@BattousaiHBr boo! boo!
@@BattousaiHBr only part of it is. The other is the real part.
I’ve watched this video 3 or 4 times since it came out. Great quality and fun video
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.
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.
@@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.
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.
Base-Pi that'd still be a ten-billion-digit number.
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
@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.
10 in base pi is NOT an integer
This is a galaxy brain meme lol
√2 is the only irrational number in existence, now confirmed
I was surprised as well
π^π^π^π is rational. Proof: It isn't √2.
@@Luca_5425 You know he was joking, right?
@@cpotisch of course
@@usernamenotfound80 QED 😎👌
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!
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.
Don't give that virtual audience CG tomatoes, whatever you do.
Virtual audiences are vicious.
"Pie to the pie to the pie to the pie"
My doctor didn't like this diet plan
Underrated
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.
We say n^2 is n squared, ^3 is cubed, and ^4 is hypercubed. I think this might be pi hyperpied.
Also pie to the pie to the pie to the pie sounds like a rapper saying pie pie pie pie
@@gurrrn1102 sick rhyme
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.
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.
“RATIOnal. Everyone remembers where they were when they first noticed that.”
I was here, watching this video.
"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
Same
*Me putting the expression in a calculator to see if it's an integer before watching the video*
Error: Result is too big
I thought my phone crashed
@@misiekeloo6114 Indeed haha
@@du42bz same
Google says it's undefined
Still the best opening to any CZcams video
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.
This is interesting !
And if all pi^^n wont be integers, what about pi^^pi ?
@@CafeMuyCaliente interesting
Worried that the pandemic is finally getting to Matt and he's building an army of imaginary audience friends
Don’t worry. He’s not building them.
He’s already built them.
There is no problem, as long as he multiplies the imaginary audience by itself, he will get a real audience
@@simonecatenacci726 Although it will be negative, so not much applause alas
I think you mean Lateral.
His audience is quite complex.
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.
What's your source
@@kddanstars9288 if you know the quadratic formula, you can see that he ia right
Yes, but the question at 12:20 wasn't whether pi+e and pi*e are irrational, but rather whether they are transcendental.
Pi+e = pie, but because pi is already pronounced pie, we prove that e=0.
@@ratlinggull2223 And in a cylinder with a radius Z and a height A, the volume equals Pi*Z*Z*A
4:44
Mathematics dictator
😅😅😅
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.
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).
@@babilon6097get back to me on april 31st /lh
@@babilon6097you're right, I meant a finite algebraic expression, like how the golden ratio can be.
@perplexedon9834 Tetration and its inverse are transcendental functions, so it could never be a finite algebraic expression.
But why couldn't it be finite algebraic expression ?@@typicwhisper6569
"We set pi to 3"
Astrophysicists: Wait what, thought the approximation was 10?
My reaction exactly. Surely we can approximate pi^pi^pi^pi to within a few orders of magnitude?
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.
@@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.
@@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.
@@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.
I know what pi to the pi to the pi to the pi is. Its "Error: Overflow"
Don't be silly, it is very obviously equal to "MATH error"
@@antoniocoulton5017 math error on casio calculators. Don't know what others say though
@@sadkritx6200 TI says Error: Overflow
it's "overflow - huge result is out of SpeedCrunch's number range"
Yeah I did it in my calculator and get math error
What an elightening video...
I am, now, more convinced than ever that e^e^e^e^e^e = Elephant (for all we know!).
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.
To calculate pi^pi^pi^pi more easily, mathematicians should just work in base pi....
That's great until you try to see if the extremely large result in base pi is an integer
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.
cursed
@@EebstertheGreat This is a big brain moment
@@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.
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
-1/12 was also a callback to a Numberphile video
@@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
@@meltingkeith7046 The result itself isn't controversial. The sloppy/misleading presentation of it to the general populace was.
Well spotted, didn't notice that one!
@@ZevEisenberg that was actually - 1/( 4 π)
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.
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
Let me tell you: PyPy to the PyPI results in a lot of incompatible libraries. (Thankfully, the most important ones are compatible.)
my mind became numb py
*Sigh py*
@@jacquesstoop2587 Daaaaamn I was racking my brain trying to pun SciPy 😆
damn, is this Py Game or something?
@@kakyoindonut3213 don't worry it's just a joke from the programmer's PyPline
Matt: lets set Pi equal to 3
Everyone: boooooo
Engineers: this is my time to shine...
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)
3=e=π=√g (on earth)
Pi is 22/7... that's probably good enough for anything a normal person does.
@@georgelionon9050 honestly yeah
Astrophysicists: Pi = 1 is close enough.
Still in the adverts but I can tell I’m gonna love this channel.
the insertion of studio audience for example is simply revolutionary.
"What kind of clickbait is this???"
A seriously nerdy kind
I can never hear "three to the three to the three" without having bad flashbacks to Graham's Number
yea
Same.
Haha so trueee
If we can find a phenomenon in nature...
... That we can do with ...
via the use of sciences?
1:42 I was todays years old, when I finally learned about why it is called rational numbers
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.
So what’s stopping you from punching in pi^pi^pi^pi?
"We know 3, beacuse it is equal to 3"
Yes the floor here is made of floor
But 3 + 3 equals 7, for large values of 3.
But we're talking about 3, not 4.
@@RWZiggy However, it is also worth noting that the limit of 3 as 3 approaches 0 is 4.
Hi, Vsauce here...
floor(3) = 3
I feel like everyone is thinking in circles here.
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!"
@@calebclunie4001 Thanks, now I'm imagining a fractal donut of a donut... define the emerging donut. And the ratio of the radii. Someone calculate?
semicircles...
Don't be a square.
@@vblaas246 That sounds totally radiical!
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.
⁴^π
Yeah sure
This is the mathematical content I’ve been waiting for
I'm surprised there was no mention of the fact that e^(i.pi) = -1
Transcendental AND imaginary numbers combined to produce an integer.
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
@@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
Well _I_ think he should have mentioned e^(i*τ)=1 instead, which is the far superior formula
God is the greatest troll ever😂
@@aguyontheinternet8436 ew no
The fact that he pauses just before saying each number makes me think he's actually calculating them all in his head
You've fallen into his trap -- that's what Matt *wants* you to think ;)
you are doing gods work my friend
7:50 wrong 3 is infinite
e^(i*pi) is an integer. I'm surprised you didn't mention it. Great video!
i is imaginary though. I think he purposefully kept the categories in real numbers.
i is not a transcendental number tho
@@JackiTheOne i*π is transcendental
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.
Nice, don't touch the like, please
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
Same! What are the odds?!
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) ?
Very nice video. A small consideration about the number of necessary digits to compute a power. Take an approximation q of pi with n digits after 3. Every number with the same n digit approx can be written as q+10^(-n)s for s in (0,1). Then (q+10^(-n)s)^L =q^L(1+10^(-n)s/q)^L is between
q^L(1+10^(-n)s/q) = q^L + 10^(-n)s q^(L-1)
and
q^L(1+10^(-n)s L/q) = q^L + 10^(-n)s q^(L-1)L
That is, the percentual error is quite well understood (proportional to the original one and multiplied by a factor between 1/q and L/q), but the absolute error is proportional to q^(L-1). I found that counterintuitive at the beginning because I expected the error in the n digit to produce an error in the n*L digit, but it's actually the opposite: in the process of elevating to a power the error gets bigger and bigger, up to the point it can eat all the precision you had (almost)!! I guess this gives an idea of why computing powers with high precision is hard.
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.
Thrust me, in 20-30 years, most big cloud server would have that amount of storage.
@@RGC_animation thrust you? oh my.
@@RGC_animation Moore's law just proves so
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.
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.
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
When COVID happens people get bored
It is broken again.Now, it's 62.8 trillion digits.Exciting times
@@METALSCAVENGER78 thats twice pi
@@greatorionbelt tau
:)
I love how Douglas Adams' 42 always gets a reference 🙂
Yes, I love the final "digts" of powers of 3, too. ;-)
I was under the impression that "irrational" included "transcendental", and that things like root-2 were more specifically "algebraic".
Your impression is corrext
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
I'm glad it's not just me 😅
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.
Yup, I reacted to this as well when he essentially claimed pi is not an irrational number.
0:09 : "An integer?"
*Someone puts a hat on it*
"Perry the integer?!"
this is not for normies ,... only few people would understand
@@tomcat1184 one of the most normie memes around
@@brahadkokad5424 you’re 10
Nice one
I like the entertainment style - in an intellectual way. No cheap clickbait.
11^6/13 as a rational is a nice touch at 1:43
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
Yeah i was like how the heck 8 showed up
And also "digits" is spelled "digts"!
Matt: "What type of click bait is this?"
Me (who clicked the video): "The good type, obviously."
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?
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.
Wouldn't "π + e" just be "pie"?
But is pie an integer or not?
That sounds rational to me.
Since when "ab" can mean "a+b"?
So "pie" is actually π×e
Give this man a nobel price!
@@JayOhm No, that's pixie minus i.
pi^pi^pi = Dream's luck when speedrunning.
Mmm
Ee
about that
That's only like over 30 my guy
Nvm it's xE+18
@dang bro it aged very well
Would mechanical or quantum computing be able to solve this problem in the future? Especially mechanical, as the real world doesn't have digits.
Mechanical won't work for transcendentals, because of plank length. At some point nothing is a perfect circle, our universe is pixelated :
”Let’s set π equal to 3.”
I sense some Graham’s Numbery stuff approaching 😨.
“I’m gonna do what’s called an engineer move”
*Sets Pi equal to 3*
*Sets 3 equal to 3*
Nope we engineers don't do that, ever.
"we set pi equal to 3"
Pappa flammy has entered the chat
"We're doing calculations tho"
Flammy has left the chat
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
Square root four times won’t work as that inverts (((π^2)^2)^2)^2 or π^16, not π^π^π^π
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…
"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...
i/2 is a ratio but not a rational number.
@@happygimp0 rational numbers are ratios of integers
This is like trying to rebuild after a hurricane by sending three more hurricanes through
With an infinite number of hurricanes eventually everything will be blown back into place.
Yes...chimps given enough time and a keyboard will mash out the collected works of Shakespeare.
Yea it's just like evolution, you can get something complex and structured from pure chaos
yeah....send more hurricanes hoping that they ALL could eventually fix those buildings and revive those killed people .....absolutely stunning :))
@@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
"negative 1/12"
I see what you're getting at. It's good to see a fellow believer.
what happens if you do circumference to the power of the circumference to the power of the circumference to the power of the circumference and then divide that diamter ^ diameter ^ diameter ^ diameter. Would that be an easier way to work it out? Or is it just as tricky?
Correction: Irrationals include transcendental numbers. "Things that are a solution to a nice polynomial equation" are called algebraic numbers (2:05)
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?
@@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 😀
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.
@@chaosredefined3834 sqrt(2) is an algebraic irrational
@@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.
"This is even more straight foward!"
*E*
E
e^e^e^e
Eu gosto disso! Boa explicação detalhada!
I love the fact that this video ha 2,7 million views & 83k likes. Great job !