Vsauce!
Kevin here.
This ant needs to get to the end of this 20cm rubber rope.
It’s really more of an elastic latex string but… work with me here.
If he moves 5cm per second, he’ll get there in…1, 2, 3, 4 seconds.
Okay.
Uhh.
No.
No paradox there.
But what if, as the ant walks 5cm per second, I stretch the rope 10cm per second?
Will he ever get to the end of the rope and fulfill his ultimate ant ambitions?
Ant-bitions?
It doesn’t seem like it.
It actually seems completely impossible.
How could he ever reach the end of the rope if I’m increasing the distance he needs
to travel by more than his progress?
Oh, also this ant’s name is…
Billy.
Speaking of ants, I started reading about them and found out that some ant queens can
live for 28 or 30 years.
Which means there are ants alive today that were 10 years old when the first Harry Potter
came out.
My voice cracked.
Okay, back to stretching.
At 0 seconds, Billy starts walking.
And at 1 second, he’s 5cm toward his goal.
That’s when we stretch the rope another 10cm.
And then Billy moves.
Billy moves Uhh…
I can't stretch the rope and move Billy at the same time I really need like another arm
or something Ahhhh!
Woah!
Hey!
No! Come back, come back, come back.
Come back, mystery arm.
I need your help.
Look, just hold this end of the rope.
Like that.
And then.
Yeah!
This is frightening but helpful!
Now after this stretch, the rope is 30cm long.
And look!
Billy’s no longer at 5cm because the key to all of this is that the ant is on the rope
and he stretches with the rope.
So to make sure he moves exactly 5cm every second I’ll just grab another ruler.
There we go.
Billy moves another 5cm.
And now we stretch to 40cm…
Billy moves another 5.
50cm.
Another 5 for Billy.
60cm.
5 more for Billy.
70cm.
And you can see that despite stretching the rope much more than the ground he covers each
second, Billy is making progress toward the end.
So a 5cm speed and 10cm stretch really isn't much of a paradox here either.
But.
What if the rope is 1km long and Billy moves at only a 1cm per second pace?
And we stretch it 1km every second?
Then can Billy ever reach the end of the rope?
Obviously not!
Or definitely yes?
And there’s our paradox.
Ant On A Rubber Rope is a veridical paradox, which we learned about in my What Is A Paradox?
video.
It’s the type of paradox that packs a surprise, because obviously the ant can’t reach the
end of the rope if we stretch it that much every second… but that certainty dissipates
as we ponder the proof.
Actually, if I stretch it 1 kilometer per second and the ant only moves 1cm per second,
he will still reach the end.
As long as old Billy here lives forever.
Okay.
Let’s ponder that proof.
It’s important to think about this as the fraction of rope that Billy has left to travel
instead of the raw distance.
I’ll show you.
Let's put Billy at the halfway mark of the rope.
Like that.
And I'm just gonna mark where he's standing with a Sharpie on the rope.
Okay.
So that mark is our Billy.
And this time Billy doesn’t move at all.
He just stands there being Billy.
Every time we stretch the rope, distance is added behind and in front of the rope, but
he’s still at the halfway point because his relative position doesn’t change.
Which means that in spite of the stretching, every time Billy steps forward from this point,
he’s making progress toward reaching the end of the rubber rope.
By continuing his journey forward, he can only get closer to the end, and eventually
he will.
Because he’s always shrinking the fraction of the rope he has left.
If you still don’t believe then this we can totally get algebra-y and calculicious.
First I want to briefly mention the harmonic series.
It’s a divergent series, meaning it’s infinite and the partial sums of the series
don’t have a finite limit.
It was first proven over 600 years ago and there’s been a whole list of different proofs
since that I'll link you to
down in the description below.
But the important thing you need to know is that it’s like a neverending addition problem
where the sum of these fractions eventually surpasses 1.
Okay let’s talk about Billy and his stretching rope.
Let’s say the rope is initially c units long, and the ant moves a units towards the
other end of the rope every second but the rope itself stretches v units longer every
single second.
During the first second, the ant will have moved a units forward and the rope will have
stretched to c plus v units long.
Cool?
Cool.
In the second second, the ant will again move a units forward and the rope stretch another
v units longer, making its new length its original length c plus v plus v again.
Which we can just write as c+2v.
In the third second, the ant will have moved another a units forward and the rope will
be c+v+v+v units long or c+3v.
During any second, the fraction of the rope the ant covers is just the ratio between the
two lengths in that second’s row.
After the first second, the ant covers a units of the total c+v units the rope is long.
During the second second, the ant covers a units of the rope’s now total c+2v units
of length.
And so on.
If you add these two fractions, you get the fraction of the rope covered after the first
and second second.
The number of fractions we add corresponds to how many seconds have elapsed and their
sum tells us the total fraction of the rope the ant has covered after that many seconds.
One way to think about adding fractions to represent a sum is eating pizza.
Okay?
Take one big bite of pizza and then a smaller bite of pizza, add those two bites together
and their sum equals the total amount of pizza you’ve just eaten.
So if we represent seconds as k, during the kth second, the ant covers 1 of the total
c+kv units the rope is long.
Okay cool story, Kevin.
But the question is: if we wait long enough, if we add up enough of these diminishing fractions
of the rope’s length the ant covers during each next second, will the sum ever equal
1?
One whole of the rope’s length?
YES.
And we can prove that by using a COMPARISON TEST.
Let’s compare this series with one whose behavior we know: the harmonic series.
We can do that by creatively tweaking the general formula for the fraction of the rope
the ant covers during any given second k.
If we multiply not only v but ALSO c by k, then for any natural number k, like 1,
2, 3, 4, etc., this new formula will give either the same result as the original formula
. Well, it'll give us the same result when k equals 1.
Because if k equals 1 then this is just c and if k equals 1 here that's just v so that
will be the same so that will be equal.
Or it'll give us a number that is smaller.
And this new formula -- a/(kc+kv) -- is equal to a over c plus v times 1 over k.
Because 1 times a is just a and we still have the k times c and k times v. Okay?
Got it?
And if we generate a new series using this formula, when we start plugging in number
for k we get a over c plus v time 1 over 1 plus 1 over 2 plus 1 over 3 plus 1 over 4
and so on...
Which is the harmonic series!
Exciting!
So this new series we’ve created diverges.
As long as you keep adding in new values for bigger and bigger k’s, the sum can reach
any number you want -- including 1.
That's a big one.
Because this is a big deal.
And since every single element of this new series is always equal to OR LESS than an
element in the series that describes the ant’s progress, our ant’s progress MUST ALSO DIVERGE.
So no matter how tiny the fractions get, no matter how long it takes the ant to cover
any proportion of the rope, he will eventually cover 1/1th of whatever the rope’s length
has become.
H will reach the end.
But it will take A LONG, LONG TIME since a smaller and smaller portion is covered every
step of the way.
In our example of an ant traveling 1cm every second and a rope stretching 1 km longer every
second, the ant will reach the end after about 8.9 × 10 raised to the 43,421 years.
To put that number in perspective.
The known universe is about 13.8 billion years old, which is 1.38 x 10^10.
All the known atoms in the observable universe number is about 10^80.
A Googol is still only 10^100.
So what does a number with 43,421 zeroes actually look like?
This.
The real world analog to the ant on a rubber rope would be light from distant galaxies
traveling through space.
If photons are traveling through a universe that is constantly expanding, will their light
ever reach Earth?
The ant on a rubber rope teaches us that yes, yes the light will eventually reach us, or
it would, if the universe were expanding at a constant rate.
But the metric expansion of the universe is actually accelerating, which means there are
ant photons traveling through the universe’s rubber rope that will never crawl into your
eyes.
So be sure to enjoy the starlight that does make that journey.
And as always -- thanks for watching.