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