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Hello, calculus is the mathematics of change.
The word comes from Latin meaning small stones
because it deals with problems
where one big thing needs to be broken down
into lots of little things.
Without it, humans would never have left our own planet.
Engineers use it to determine
how much material they need for a job
or how much power they should give a motor.
Biologists use it to determine the rate of disease spread
and chemists use it to measure reactions.
But where did this magical tool come from?
The earliest questions that calculus was invented to answer
came in the form of paradoxes.
In this video, we're going to look at three paradoxes
that led to three key concepts in calculus.
Zeno of Elea.
A fifth century BCE thinker is known exclusively
for a number of ingenious paradoxes.
He had a strange but interesting view of the world.
He thought that we couldn't trust our senses
but only logic and invented a paradox to demonstrate this.
If you want to travel one stade
the distance covered in ancient Greek footraces,
do you agree that you first must travel halfway there? Yes.
Do you agree that you must travel
to the halfway point between your current position
and the one stade mark before you reach the end?
Yes. You need to go a further quarter of a stade
on your way to finishing your journey.
But once you're there,
there's another halfway point for you to reach.
Now an eighth of a stade in front of you,
where are you going with this center?
But then halfway again is a 16th of a stade and so on.
Once you reached that halfway point
there'll be another one in front of you
no matter how far you go.
Before you can reach your final end point,
you must first reach the infinitely many half points
that come before it.
How do you manage that?
It seems that what the logic is telling us
is that the journey takes infinitely many steps to complete
therefore it, and in fact, all motion is simply impossible
but you're moving right now.
It's an illusion.
Well, Zeno's conclusion
isn't super convincing.
It's difficult to pinpoint the flow in his argument.
Something about Zeno's logic has to be wrong, but what?
Let's take a closer look .
At the first halfway point
your total distance is equal to half a stade.
Then once you reach the second halfway point
your total distance is that half plus a quarter of a stade.
By the next halfway point, the pattern is clear
and we can ride the total distance
as a bunch of fractions added together.
The numbers we're adding on keep getting smaller
but the sum never ends.
There's an infinite amount of fractions to add on.
And if a sum gets bigger every time you add on a fraction
and you have an infinite amount of numbers to add on,
shouldn't the total distance get infinitely large?
And if the distance is infinitely large
surely you can never traverse it.
This was one of the early problems
that led to the invention of a mathematical tool
that is the backbone of calculus.
The limit.
Let's take a closer look at how this sum breaks down.
The first two terms, add to get 0.75
adding one more term gives us 0.875
and adding another gives 0.9375.
The sum is just getting closer and closer to one
and there doesn't seem to be
any risk of it running off to finish.
Even though we have an infinite number of fractions to add
they seem to be approaching the number one.
We might make it to our final destination after all
but how do we know for sure.
No matter how many terms we add onto the sum by hand
there will always be another one we can add on after that.
And then another one and another one
without actually ever reaching one.
It's exactly this type of problem
that the limit was created to deal with.
To help us understand
what the mathematical limit really means,
let's rewrite the problem a little.
We've been writing out total distance as a sum like this
but it can be written much more conveniently like this.
Here's the derivation if you're interested.
Written like this, this term is much easier to work with.
The N stands for the number of fractions
or terms in our sum.
The more terms there are, the larger N gets
which means that this term here, two to the power N
gets really, really large.
The bigger two to the power N gets,
the smaller the fraction gets
and the smaller the fraction gets
the close the sum gets to equaling one.
And that logic is exactly what it means
to take the mathematical limit.
It never actually solves what happens at infinity,
it just takes you arbitrarily close to infinity.
Or another way to think of it
is asking what happens on the threshold of infinity
just before we actually reach it.
This idea is extremely useful
in a lot of mathematics where we need to resolve
what happens as a sum of purchase a number
but never actually reaches it.
This idea of approaching something
but never actually reaching it
is the big idea at the heart of calculus.
Which brings us to our next paradox.
Our next paradox begins with
trying to find the area of shape.
For simple and regular shapes like rectangles and circles,
formulae for the areas
have been around for thousands of years
but what about this shape?
How would you go about finding its area?
There's no neat formula,
the shape doesn't even have a name.
A mathematician named Bonaventura Cavalieri
wondered if we could use
the formula for shapes we already knew,
to calculate shapes we didn't know.
For example, take this semicircle.
It has a known area of half Pi R squared.
He imagined the semicircle as being made up
of lots and lots of width-less lines
and then created a new shape
by only moving those lines up or down.
The new shape is clearly different
but if you draw any vertical line
it will overlap with the old shape
by the same amount that overlaps with the new one
as if both shapes are made up of the same vertical segments.
This theory seems okay on the surface
but it has a few problems.
For instance, let's try drawing a vertical line
in this triangle.
Because this triangle has the same height,
it doesn't matter where we drew the line.
We could slide that line horizontally across
and find a spot where it fits in this triangle.
But that suggests that the two shapes
are made up of the same vertical lines
and it's an area is just made up of many vertical lines
that would suggest that these triangles have the same areas,
which we can see isn't true.
This is a paradox.
By definition, the line segments have zero width,
the one dimensional objects determined only by their length.
No matter how many zeros you add together ,
you always get zero.
You can never get some actual size or area.
Modern calculus uses this width-less line idea.
The inspiration for a more mathematically rigorous way
of finding areas.
Integration.
We still break shapes up into vertical pieces
but those pieces no longer have zero width.
So for example, if you want to find the area under a curve
you might start with an approximation.
Maybe you draw a bunch of rectangles under the curve.
All we have to do to find out area estimate
is add together the areas of the rectangles we've drawn.
This gives us a pretty decent guess
but it's only a guess because
now we have to deal with something
that we didn't run into with the line segments.
There are gaps near the top of our rectangles
where some of the area isn't being filled in.
The simple solution to this
is to make the rectangles narrower.
We have to draw more of them
but once they're all drawn out
the open space near the curve has decreased.
It's still an estimate,
but it's closer to the actual area under the curve
than our first estimate.
In fact ,what we'd really like to do
is get as close to zero width as possible
without ever actually getting there.
Does this remind you of anything?
Yes limits.
Instead of letting the width of the rectangles equal to zero
we ask what happens when we let the width approach zero?
As the width of the rectangles approaches zero,
the number of rectangles needed to fill the space
approaches infinity.
This technique is called integration
because we're combining lots and lots of small things
to create a larger whole.
So here we've come full circle
from width-less lines to, width-full lines,
to infinitely close to being width-less lines
Okay, so this paradox actually called Zeno's arrow
but I couldn't find a bone arrow
so we've got a Nerf gun instead.
Just think of this as the modern day version.
So far we've dealt with paradoxes
that consider how space is divided.
But in this one, Zeno considers time.
If you fire enough bullet,
it seems to glide through the air moving the whole time.
Well, let's take a closer look.
What is it doing one second after it was fired?
Intuitively we want to say that it's moving
but if we phrase on that exact instance
can you actually show that it's moving.
By definition something that moves changes its position
but frozen in a single moment of time
it doesn't move at all.
Like a movie, it's just a sequence of still photos
one after the other played, very rapidly
nothing is actually moving.
It just gives the illusion of movement.
Likewise, since you could have frozen the bullet
at any point along its path,
then when exactly does it move?
How can a collection of motionless moments make up motion?
This question seems pretty similar
to our previous two paradoxes.
So can we use what we've learned
and apply the same logic here?
Let's stop by trying to figure it out the bullet speed
at any given time instance,
let's start with an easy case where its speed is constant.
This looks like a straight diagonal line
on our distance versus time graph.
You don't need to check each individual point,
you just need to know the total distance traveled
in a given time and divide the two numbers.
Now let's say the motion of the bullet looks like this.
It's not so simple as just taking the average
because it's traveling different space at different points.
So how do we find out it's speed at a specific point?
Let's say we want to know the speed at five seconds.
We can pick two points,
one a tiny bit before the five second mark,
and one a tiny bit after
and draw a straight line between them
to get a very good idea of the speed at five seconds.
As the two points get closer and closer together
the more accurately you'll be calculating
the speed at five seconds.
So what is the smallest gap?
Well, it can't be zero because there isn't a time interval
but it needs to be small, really small.
Can you guess what we're about to do?
Take the limit .
As the distance between our two points approaches zero,
this slope value that we're calculating
approaches the tangent to the curve at a point.
This is the derivative
and it's what we use to calculate
what we're calling instantaneous speed.
So there we have it, three paradoxes
which led us to three foundational pillars of calculus.
Paradoxes are a great way
for us to explore new ideas and challenge our intuition.
Sometimes they turn out to be
just tricks of logic with simple solutions.
And in other cases
they have a huge impact changing the way we think
and carrying on the centuries.
Calculus confuses me in a philosophical sense.
I was a big believer that math is a discovery
that we discover mathematical patterns in nature
and use them to describe our wells.
But calculus seems purely man made,
an invention that doesn't exist anywhere but our minds.
So it's not an invention or a discovery.
This question fascinated me so much
that I made a feature video about it on Nebula.
Nebula is a streaming platform
built by a bunch of us educational YouTube creators,
where we can explore making different
and experimental content.
CuriosityStream, a streaming service
with award-winning documentaries
is so supportive of our idea
that they offer offering Nebula completely free
when you sign up with them.
In fact, there's a really cool documentary
by Hannah Fry about this exact question
of whether math is an invention or a discovery.
It's visually beautiful,
explores the history of math in a fun way.
And Hannah Fry is just charming.
I aspire to be like her one day.
I watched the entire three hour series in one go.
So if you do get the two for one deal,
make sure to watch Hannah Fry's magic numbers
and also my video on Nebula
because I think it's pretty good.
They offering a 26% of holiday promotion at the moment.
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That's it for me and I'll see you in the next video, bye.
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