- This episode was made
possible by Brilliant. Hey there, welcome to up "Up and Atom" I'm Jade and I want you to imagine that you're an ancient mathematician, trying to figure out the circumference, the outer edge of this wheel. Now you can't exactly use a ruler and measuring tapes don't exist yet. So, what can you do? Well, one way would be to
roll it one full rotation and measure the distance it traveled. You're feeling extra curious today and you want to try the same thing with the differently sized wheel but then you notice something interesting. There is actually a smaller
circle inside the larger wheel which seems to have
traveled the same amount. So, does that mean it has
the same circumference? Well, that doesn't seem right. Intuition tells you
that the smallest circle should have a smaller conference and therefore should have
traveled a shorter distance. So, what's going on? This is a paradox known
as Aristotle's Wheel and it originated in an ancient Greek
text called "Mechanica". It tortured mathematicians and
philosophers for centuries. And in this video,
we're going to solve it. One attempt to figure
out what was happening was made by a French mathematician
named Gilles de Roberval. Now I'm way too Aussie to say that without probably offending
some French people. So, I'm gonna get my
French husband to say it. - Gilles de Roberval. - And he focused on the path traced out by the point
on the circumference of a rolling circle. (light-hearted music) This distinctive shape
is called a cycloid. He then wanted to see what happened when we traced out the path
made by the inner circle too. The line traced out by the smallest circle is kind of stretched out. In fact, the smaller the inner circle is the more stretched out
the path seems to get. So, what does this mean? While you think about that, let's take a look at another
attempt to solve the problem. This puzzle also captivated the mind of the great astronomer Galileo and to figure out what was happening, he decided to look at a
wheel made of hexagons. Just like with the circles, the wheel has a large outer hexagon and a small inner hexagon. When you roll the large hexagon, it travels a distance equal to the sum of the length of its sides. Just like a circle rolls out the length of its circumference. Now let's take a closer look at what the inner hexagon is doing. We can clearly see it
lifting off the path, even though it travels the same distance as the bigger wheel, it's sides aren't
touching the entire time. I wanted to make this
extra clear for you guys, so we're going to mark out
the track made by the wheels. (light-hearted music) There, a nice clean skip. Galileo then imagined what would happen if you repeated the
experiment with a shape with even more sides, like an octagon. Here we see that the skips
are shorter and more frequent. He then imagined a circle as a polygon with an infinite number of sides. He concluded that the inner circle must be making an
infinite number of skips. While Galileo wasn't exactly right, it does get us closer to
the modern day solution. When we put Galileo and
Roberval's conclusions together we get only one possible answer. The inner circle must be slipping. Unlike the hexagon and the octagon the inner circle is not skipping as it never leaves the surface. So, the only possibility is
that it's getting dragged along with the outer circle. This seems like a pretty simple answer. So, why did it baffle some of
the greatest minds of history? Well, first, this isn't how we
generally think of slipping. Usually we think of it
as something like this, where one point on the circle stays in contact with the path, but anytime the outer circle rotates, the inner circle rotates
by the same amount. The outer circle never pauses to let the inner circle
slide along to catch up. It's all happening in the same
continuous rolling motion. Another way to say this is
that each point of contact between the circles
and the path is unique. But how do two circles of different sizes both make unique points of contact with the surface through
the entire rotation? Do the two circles have
an equal number of points? As one scholar put it, "This is where the problem
begins, not where it ends." Let's talk about what it really means for one thing to be equal to another. If I have this many objects, how do I know if I have
an equal number of each? Well, one way is to pair
the objects together and see if there are any leftover. If there aren't any leftover, I have an equal number of each. This is called a
one-to-one correspondence. And it was thought that
if two sets of objects have a one-to-one
correspondence, they're equal. Seems legit. Now let's try the thing with circles. To simplify things, let's roll them out into their circumferences. A circle is just a rolled
up line, after all. Now we have what would appear to be two differently sized lines. Let's see what happens when we try to make a one-to-one correspondence between them to make sure that we really are choosing unique points on the line. We have to make a triangle. Draw a line from the tip of the triangle, through the shorter and longer line. No matter what angle the line is drawn it will always intersect
each line at a unique point. If you don't believe me try finding a unique point on either line and drawing back up to
the tip of the triangle. A unique point on one line
will always correspond to a unique point on the other. So, then there does exist
a one-to-one correspondence between two lines of different length. This complicates our idea of what it means for two things to be equal. In fact, if we take a segment of the line we can still draw a
one-to-one correspondence between any point on the line
and any segment of the line. So, any segment of a line has
a one-to-one correspondence with the line itself. So, does this mean that
all lines are equal? That even a segment of a line is the same size as the line itself? Well, actually it's just the opposite. All of this leads to the
very strange conclusion that a one-to-one correspondence does not always mean that
two things are equal. The implications of this run very deep when we consider that numbers
can be represented on a line. But wait, let's take a step back. What about these guys? We showed that a one-to-one correspondence between them meant they were equal, right? So, why does it work for
these guys and not these guys? Can you spot the difference? Well, these objects are discreet, you can split them up into
neat individual packages while these objects are continuous, uninterrupted, you can't split them up. Let's see how this fares when we talk about a very
specific type of object, numbers. Take all positive whole numbers. Whole numbers are discrete because there exists spaces between them. They're better represented
as dots rather than a line. There are an infinity of them because no matter how high you count, you can always add one more. Now, take all the negative
whole numbers, same deal. No matter how low you count,
you can always go lower. Every positive whole number can be paired with a negative whole number
in a one-to-one correspondence. So, the number of positive whole numbers is equal to the number of
negative whole numbers. The sizes of their infinity are equal. This might seem like a weird
thing to say, but just hold on. We can also count them even though we would never stop counting, we can at least start counting. Now, let's start adding numbers. Let's add the fractions, the decimals, the irrational numbers,
the transcendentals. In fact, let's add every number until there are no more gaps. This makes up what is
called the real number line. Unlike the whole numbers
which are discreet, the real numbers lie
on one continuous line from negative infinity
to positive infinity. If you point to any way
on the real number line, there will always be a number there. Now let's do something. Let's take a segment of the
real number line, zero to one and then let's take the
segment one to 1 million. Here's the thing, just like with our two differently
sized lines from before we can always find a
one-to-one correspondence between any point from zero to one and any point from one to 1 million. This leads to the conclusion that there are equal amount
of numbers between zero to one as there are one to 1 million. In fact, there are an
equal amount of numbers from zero to one as zero to infinity. To really hit this home,
let's say what happens if we try to count the
numbers between zero and one. What is the first number after zero? Well, let's just ball
park it and say it's 0.01 but then we can just add another zero to make 0.001 or 0.0000000000001. In fact, the question, "What
is the next number after zero?" is impossible to answer because
for any number we choose there will always be a number
that is closer to zero. So, we can't even count the
numbers between zero and one let alone zero and infinity. The amount of numbers that
lie between zero and one and zero and infinity are
both uncountably infinite. This is very different to the
infinity of the whole numbers, which we can count and list. So, here we have two
different types of infinities, countably infinite, and
uncountably infinite. The surprising result that
there are different kinds of infinities was the
work of Georg Cantor. I've made a video which
goes more into this idea and gives you the proof of
it if you're interested. The paradox of Aristotle's Wheel took over two millennia to solve. While the physical answer
turned out to be pretty simple, it led to some very strange ideas about what it means for
two things to be equal and the strange nature of infinity. It's a reminder of the
mysteries that surround us in everyday life if you
just take a closer look. If you're curious about
the world around you, but are not sure where to start, one resource I use is Brilliant. So, Brilliant is a problem
solving based website and app with a hands-on approach. It's math, science and
computer science content help guide you to mastery
by taking complex concepts and breaking them up into
bite-sized understandable chunks. I find that it's a
really good way to learn because it forces you to think hard by answering questions as you go. In fact, if you liked today's episode, you'll like their course on number theory. It goes into more detail
about the nature of infinity and best of all you can work
through it at your own pace. They have over 60 courses
and are always adding more, including an entire one dedicated
to infinity coming soon. I will definitely be checking that out as I'm sure there is
still so much to learn. If you'd like to go ahead and sign up, the first 200 people to
sign up with this link will get a 20% discount
off premium membership. Just go to brilliant.org/UpAndAtom. Great, thank you for watching
and I'll see you next time. Bye. (upbeat music)
i don't like the way she introduces the notion of uncountability.
rationals are countable, but similarly, you couldn't state the smallest positive rational #