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  • I wanna start in a nice, familiar, comfortable place, Brady.

  • Which is this series:

  • 1.. +2 .. +3 ..+4 Oh no! Oh no!

  • Alright, all the way up to infinity. You remember what the answer to that was?

  • Yeah. Yeah, what was it?

  • it's a, uh, divergent sum. Yeah, it's -1/12, isn't it?

  • Which is totally uncontroversial.

  • that's that's not weird at all. so I'm not actually going to talk about that series

  • but I want to talk about something thats related to that series so I want to

  • consider what happens when we take the reciprocal of all these numbers and take

  • the sum of those so we take 1 + a half + a third + a quarter + a fifth.

  • what would we get if we added all these numbers together you can look at this

  • and you might think even though someone goes alright there's no way that's -1/12th

  • but it'd surely be infinity. you look at this and you think what is this?

  • i mean the numbers are getting... it's clearly getting bigger but the numbers

  • that you're adding again smaller and smaller and smaller so you might think

  • this goes to a finite answer, but actually it doesn't. this is infinite.

  • this is called the harmonic series, and it and it is infinite. it was first shown to

  • be infinite. and back in.. by a medieval, french mathematician Nicole Oresme. he

  • showed it in quite an elegant way, which we can go through now, he came up with

  • another series in which each element of the other series was less than the ones

  • you've got here and showed that guy was divergent. the other series that he

  • looked at, was the one where you asked what is the largest power of a half

  • that's less than the number you're interested in. so let me show you that's

  • probably easiest way. so 1, well it's less than or equal to that we're interested in. 1 is is

  • already a power of a half so we'll leave that as one. a half is of course a

  • power of a half so we'll leave that there. a third, well the largest power of

  • a half which is less than or equal to a third, is of course a quarter.

  • quarter, well we can leave that as a quarter. a fifth what's the largest power of a half,

  • that's uh, that is less than a fifth, well it's an eighth, which is a half cubed right.

  • for a sixth that would also be an eighth. for a seventh, that will also be an eighth. for an eighth.. it's an eighth

  • The next one would actually be a sixteenth, and you would go on like that, right. so this is

  • his alternative series, and now you look at this series and you just see that,

  • It actually has some nice structure so you got one plus a half, but then this combination

  • here gives you another half. these four here give you another half. there's

  • going to be eight of these 16ths, they'll give you another half, and what you're

  • doing is just adding a half loads of times, so it's pretty clear that this this

  • indeed diverges. and so by comparison, because each element here is larger than

  • the element here,

  • this guy must also diverge as well. So that's the proof that that this diverges.

  • now.. but hang on, but each element of what you're doing up here.. *inaudible* I knew you were going to say that!

  • there's this is sort of nice illustrated way to think about this, this sum and

  • how it diverges. We've got a lousy band here right. That's what we call it in (Liverpool?).

  • in order to describe this, it can tell us something about this this sequence right. so what I

  • want you to imagine is, that, imagine this this lousy band is, is like a meter in

  • circumference. I know it's not, but imagine it is. And imagine we put an ant on him and

  • this ant, can travel one centimeter a second. okay now, after each

  • second we're going to stretch the lousy band, so that it's...

  • we add and another meter to it's circumference.

  • okay, so after the first second, the ant has moved one centimeter.

  • We stretch the elastic band by a meter, ok after the next second, the ants moved

  • another centimeter. we stretch the lousy band by another meter and so on

  • and so on and so on.

  • ok, question is, does the ant ever get all the way around the elastic band?

  • (Brady) I can't see how it could. it seems like you couldn't, it couldn't, right. but it can right and

  • it's because of this sequence, right. so let's just see why it can right. so think

  • about what happens in in the first second. think about, think about the

  • percentage of the elastic band that it manages to to traverse, okay. well it's

  • traveled.. the elastic band is 1 meter around it travels one centimeter that's 1%, right.

  • in the 1st second travels 1%. What happens in the 2nd second?

  • well in the 2nd second the last band is now 2 meters long

  • it travels another centimeter, so that's 0.5%. in the 3rd second

  • the lousy band is now 3 meters long

  • it travels 1 centimeter, that's 1/3th of a percent. their total length of the

  • lousy band, and so on.

  • ok, the same sequence, but this goes to infinity. It only has to get to a 100%, never mind infinity right?

  • clearly he's going to get all the way around the lousy band, and actually you

  • can work out how long it takes him, right. it takes him about e^100 seconds,

  • which is about 10^50 seconds which is quite a long time. it's so amusing

  • to think what would happen if you start the experiment now

  • and by the time that the ant finished it. well before we got to the end, you'd have..

  • the Sun would have become a red giant and it's probably swallowed up the earth

  • (Brady) you're adding one meter every second

  • yeah (Brady) and it's only moving one centimeter every second

  • (Brady) so every second the finish line is getting further away, every second.

  • but what it's already.. what was behind it also gets further away

  • so it's this is not just the distance in front but also the

  • distance behind and you can see what's really important is the percentage that

  • that it accumulates. and you can see that it just grows and grows and

  • grows and grows and grows. eventually it hits a 100. but you know after e^100 seconds

  • roughly, more or less. it gets it gets there. but yeah I mean you know, to

  • be honest, you know, we will even... by the time you get to the end of this uhm...

  • this game, the universe has entered a black hole area where all the earth,

  • matter and stars and including our experiment has ended up inside black

  • holes. so this is a gargantuan time, and what that illustrates is is really how

  • slowly this series grows. it grows very very slowly.. it grows, it grows indefinitely but

  • it does so very slowly and you can see that by how long it takes to go all the

  • way around. it actually grows logarithmically slowly. and what you can

  • show is, you can take, you know, this sequence all the way up to maybe the n'th

  • form. and you take off log(n) I'm taking the natural logarithm here,right.

  • what you get if you look at this guy, so each stage you take off that... the log.

  • well you know 1 minus.. you take the first step, 1-log(1) .. well that's just 1

  • 1+1/2-log(2) is about... 0.8. 1+1/2+1/3-log(3)

  • that's about 0.7. if you carried on ad infinitum, you actually

  • approach, a finite number. it's called gamma. it's the Euler-Mascheroni

  • constant. this is about 0.577, which is a.. if you take its square it's about 1/3rd

  • ok but not very much is known about this number. we know that it exists.

  • we know it exists, we can prove it exists.

  • basically the proof, it relies on the fact that this sequence is decreasing.

  • you can show also, that it's bounded from below.

  • ok, so it's decreasing and bounded from below. it's gotta stop somewhere. so you can

  • prove it exists but but we know hardly anything about it.

  • we don't know if it's a transcendental we don't even know if it's a rational

  • number or not. kinda mad right. we can describe it in certain number

  • of digits, and Euler came up with about 15 digits. Mascheroni came up with 90.

  • He then got the 20th 21st and 22nd one wrong

  • nowadays we know it's about a hundred billion digits, but we don't even know if its

  • rational... if it if it is rational

  • we know that the denominator is greater than 10^242080. the denominator

  • gotta... that if it's rational, but it might not even be rational, we don't know.

  • ok, so we know very little about it and yet this number, it appears all over

  • everywhere. all over physics, for example. this is, I know it from physics right.

  • when you calculate quantum corrections, to say, the mass of the electron or

  • the mass of the Higgs. or you calculate contractions to the charge of the

  • electron. these sorts of things, this guy pops out. it always pops out. so it's

  • there. it's in nature, it's in physics. and yet we don't know very much

  • about it

  • it also... if you take e^γ (e^gamma), take e^γ

  • this knows about products of prime numbers, which is, what what's that got to do anything?

  • so it seems to know about stuff. there's a connection there, and actually some some

  • physicists have actually sort of said that, there might be something deep

  • there. that they're ultimately cosmology is is gonna be very closely

  • related to number theory. it's very speculative this but, you know, in

  • the future maybe this is going to be a direction we're going.

  • if you're interested in numbers that govern the universe you might just like this

  • book called "Just six numbers" by Martin Rees. famous astronomer shows how six

  • numbers can explain the origins and behaviours of the entire

  • universe. as an added bonus the audiobook is narrated by Lord Rees himself. now

  • audible.com, which has supported this video, is a leading source of spoken

  • material. they've got just a huge range of titles. anything you can imagine can

  • be downloaded onto the device of your choice

  • it doesn't matter what you're into and if you sign on for a free 30-day trial

  • membership, your first audio book is included. so why not try "just six numbers"

  • it's quite a short one, about three-and-a-half hours. if you enjoyed

  • the video we just showed you, then I think this audiobook might be up your alley. go

  • to audible.com/numberphile when you visit them so they'll know you came from

  • here. that's audible.com/numberphile thanks to them for supporting this video

I wanna start in a nice, familiar, comfortable place, Brady.

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0.577的奧祕 - 數字發燒友 (The mystery of 0.577 - Numberphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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